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A second-order theory for lithium niobate piezoelectric plates with a ferroelectric inversion layer in coupled extensional, thickness-stretch and symmetric thickness-shear motions

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Abstract

A second-order plate theory for a lithium niobate piezoelectric plate with a ferroelectric inversion layer is established. The theory describes coupled extensional, thickness-stretch and symmetric thickness-shear motions of the plate. The two-dimensional theory obtained is validated by comparing the dispersion relations of the relevant waves with the three-dimensional exact theory. For long waves with small wave numbers, the dispersion curves obtained from the plate theory and the three-dimensional theory have the same cutoff frequencies and curvatures. Therefore, the plate theory is useful in the design of devices operating with these waves. A piezoelectric gyroscope based on symmetric thickness-shear modes is analyzed as an example.

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Acknowledgements

This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY18A020004) and the K. C. Wong Magana Fund through Ningbo University.

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Authors and Affiliations

  1. Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, 315211, Zhejiang, China

    Dejin Huang

  2. Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, 68588-0526, USA

    Jiashi Yang

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  1. Dejin Huang
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  2. Jiashi Yang
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Correspondence to Dejin Huang or Jiashi Yang.

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Appendix: Material matrices and plate constitutive relations for z-cut \(\hbox {LiNbO}_{{{3}}}\)

Appendix: Material matrices and plate constitutive relations for z-cut \(\hbox {LiNbO}_{{{3}}}\)

The material constants of \(\hbox {LiNbO}_{{3}}\) and \(\hbox {LiTaO}_{{3}}\) can be described by the following matrices [21]:

$$\begin{aligned}&\left( {{\begin{array}{lllllll} {c_{11} } &{}\quad {c_{12} } &{}\quad {c_{13} } &{}\quad {c_{14} } &{}\quad 0 &{}\quad 0 \\ {c_{21} } &{} \quad {c_{11} } &{}\quad {c_{13} } &{}\quad {-c_{14} } &{}\quad 0 &{}\quad 0 \\ {c_{13} } &{}\quad {c_{13} } &{}\quad {c_{33} } &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ {c_{14} } &{}\quad {-c_{14} } &{}\quad \quad 0 &{}\quad {c_{44} } &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad {c_{44} } &{}\quad {c_{14} } \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad {c_{14} } &{}\quad {c_{66} } \\ \end{array} }} \right) ,\\&\left( {{\begin{array}{llllll} 0 &{}\quad 0 &{}\quad 0 &{} 0 &{}\quad {e_{15} } &{}\quad {-e_{22} } \\ {-e_{22} } &{}\quad {e_{22} } &{}\quad 0 &{}\quad {e_{15} } &{}\quad 0 &{}\quad 0 \\ {e_{31} } &{}\quad {e_{31} } &{}\quad {e_{33} } &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \end{array} }} \right) , \left( {{\begin{array}{lll} {\varepsilon _{11} } &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad {\varepsilon _{11} } &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad {\varepsilon _{33} } \\ \end{array} }} \right) . \end{aligned}$$

For both materials, \(c_{{14}}\) is much smaller than the other elastic constants. Therefore, in the following we make the approximation that \(c_{14} \cong 0\). Then, the plate’s effective material constants in (17) and (22) take the following form:

$$\begin{aligned} {[\bar{{c}}_{pq} ]}= & {} \left[ {{\begin{array}{llllll} {c_{11} -{c_{13}^{2} } / {c_{33} }} &{} {c_{12} -{c_{13}^{2} } / {c_{33} }} &{} 0 &{} 0 &{} 0 &{} 0 \\ {c_{12} -{c_{13}^{2} } / {c_{33} }} &{} {c_{11} -{c_{13}^{2} } / {c_{33} }} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {c_{44} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {c_{44} } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {c_{66} } \\ \end{array} }} \right] ,\\ {[\hat{{c}}_{pq} ]}= & {} \left[ {{\begin{array}{llllll} {c_{11} -\frac{5}{9}\frac{c_{13}^{2} }{c_{33} }} &{} {c_{12} -\frac{5}{9}\frac{c_{13}^{2} }{c_{33} }} &{} {\frac{4}{9}c_{13} } &{} 0 &{} 0 &{} 0 \\ {c_{12} -\frac{5}{9}\frac{c_{13}^{2} }{c_{33} }} &{} {c_{11} -\frac{5}{9}\frac{c_{13}^{2} }{c_{33} }} &{} {\frac{4}{9}c_{13} } &{} 0 &{} 0 &{} 0 \\ {\frac{4}{9}c_{13} } &{} {\frac{4}{9}c_{13} } &{} {\frac{4}{9}c_{33} } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} {\frac{4}{9}c_{44} } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} {\frac{4}{9}c_{44} } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {c_{66} } \\ \end{array} }} \right] ,\\ {[\tilde{{c}}_{pq} ]}= & {} \left[ {{\begin{array}{llllll} {c_{11} -{c_{13}^{2} } / {c_{33} }} &{} {c_{12} -{c_{13}^{2} } / {c_{33} }} &{} 0 &{} 0 &{} 0 &{} 0 \\ {c_{12} -{c_{13}^{2} } / {c_{33} }} &{} {c_{11} -{c_{13}^{2} } / {c_{33} }} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {c_{66} } \\ \end{array} }} \right] ,\\ {[\bar{{e}}_{iq} ]}= & {} \left[ {{\begin{array}{llllll} 0 &{} 0 &{} 0 &{} 0 &{} {e_{15} } &{} {-e_{22} } \\ {-e_{22} } &{} {e_{22} } &{} 0 &{} {e_{15} } &{} 0 &{} 0 \\ {e_{31} -\frac{c_{13} }{c_{33} }e_{33} } &{} {e_{31} -\frac{c_{13} }{c_{33} }e_{33} } &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] ,\\ {[\tilde{{e}}_{iq} ]}= & {} \left[ {{\begin{array}{llllll} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {-e_{22} } \\ {-e_{22} } &{} {e_{22} } &{} 0 &{} 0 &{} 0 &{} 0 \\ {e_{31} -\frac{e_{33} c_{13} }{c_{33} }} &{} {e_{31} -\frac{e_{33} c_{13} }{c_{33} }} &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] ,\\ {[\hat{{e}}_{iq} ]}= & {} \left[ {{\begin{array}{lllllll} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{e_{15} }{6}} &{} {-e_{22} } \\ {-e_{22} } &{} {e_{22} } &{} 0 &{} {\frac{e_{15} }{6}} &{} 0 &{} 0 \\ {e_{31} -\frac{5}{6}\frac{e_{33} c_{13} }{c_{33} }} &{} {e_{31} -\frac{5}{6}\frac{e_{33} c_{13} }{c_{33} }} &{} {\frac{e_{33} }{6}} &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] , \end{aligned}$$
$$\begin{aligned} {[\bar{{\varepsilon }}_{ij} ]}= & {} \left[ {{\begin{array}{lllllll} {\varepsilon _{11} } &{} 0 &{} 0 \\ 0 &{} {\varepsilon _{11} } &{} 0 \\ 0 &{} 0 &{} {\varepsilon _{33} +\frac{3}{4}\frac{e_{33}^{2} }{c_{33} }} \\ \end{array} }} \right] , \ [\tilde{{\varepsilon }}_{ij} ]=\left[ {{\begin{array}{lllllll} {\varepsilon _{11} } &{} 0 &{} 0 \\ 0 &{} {\varepsilon _{11} } &{} 0 \\ 0 &{} 0 &{} {\varepsilon _{33} +\frac{9}{8}\frac{e_{33}^{2} }{c_{33} }} \\ \end{array} }} \right] ,\\ {[\hat{{\varepsilon }}_{ij} ]}= & {} \left[ {{\begin{array}{lllllll} {\varepsilon _{11} } &{} 0 &{} 0 \\ 0 &{} {\varepsilon _{11} } &{} 0 \\ 0 &{} 0 &{} {\varepsilon _{33} +\frac{15}{16}\frac{e_{33}^{2} }{c_{33} }} \\ \end{array} }} \right] , \ [\breve{{\varepsilon }}_{ij} ]=\left[ {{\begin{array}{lllllll} {\varepsilon _{11} +\frac{15}{16}\frac{e_{15} e_{33} }{c_{44} }} &{} 0 &{} 0 \\ 0 &{} {\varepsilon _{11} +\frac{15}{16}\frac{e_{15}^{2} }{c_{44} }} &{} 0 \\ 0 &{} 0 &{} {\varepsilon _{33} +\frac{15}{16}\frac{e_{15} e_{33} }{c_{33} }} \\ \end{array} }} \right] . \end{aligned}$$

The corresponding plate constitutive relations are

$$\begin{aligned} T_{11}^{(0)}= & {} B_{00} \left( {\hat{{c}}_{11} S_{1}^{(0)} +\hat{{c}}_{12} S_{2}^{(0)} +\hat{{c}}_{13} S_{3}^{(0)} } \right) +B_{20} \left( {\tilde{{c}}_{11} S_{1}^{(2)} +\tilde{{c}}_{12} S_{2}^{(2)} } \right) +\bar{{B}}_{10} \left( {-\hat{{e}}_{21} E_{2}^{(1)} -\hat{{e}}_{31} E_{3}^{(1)} } \right) \\= & {} B_{00} \left( {\hat{{c}}_{11} u_{1,1}^{(0)} +\hat{{c}}_{12} u_{2,2}^{(0)} +\hat{{c}}_{13} u_{3}^{(1)} } \right) +B_{20} \left( {\tilde{{c}}_{11} u_{1,1}^{(2)} +\tilde{{c}}_{12} u_{2,2}^{(2)} } \right) +\bar{{B}}_{10} \left( {\hat{{e}}_{21} \phi _{,2}^{(1)} +\hat{{e}}_{31} 2\phi ^{(2)}} \right) , \\ T_{22}^{(0)}= & {} B_{00} \left( {\hat{{c}}_{21} S_{1}^{(0)} +\hat{{c}}_{22} S_{2}^{(0)} +\hat{{c}}_{23} S_{3}^{(0)} } \right) +B_{20} \left( {\tilde{{c}}_{21} S_{1}^{(2)} +\tilde{{c}}_{22} S_{2}^{(2)} } \right) -\bar{{B}}_{10} \left( {\hat{{e}}_{22} E_{2}^{(1)} +\hat{{e}}_{32} E_{3}^{(1)} } \right) \\= & {} B_{00} \left( {\hat{{c}}_{21} u_{1,1}^{(0)} +\hat{{c}}_{22} u_{2,2}^{(0)} +\hat{{c}}_{23} u_{3}^{(1)} } \right) +B_{20} \left( {\tilde{{c}}_{21} u_{1,1}^{(2)} +\tilde{{c}}_{22} u_{2,2}^{(2)} } \right) +\bar{{B}}_{10} \left( {\hat{{e}}_{22} \phi _{,2}^{(2)} +\hat{{e}}_{32} 2\phi ^{(2)}} \right) , \\ T_{33}^{(0)}= & {} B_{00} \left( {\hat{{c}}_{31} S_{1}^{(0)} +\hat{{c}}_{32} S_{2}^{(0)} +\hat{{c}}_{33} S_{3}^{(0)} } \right) +B_{20} \tilde{{c}}_{36} S_{6}^{(2)} -\bar{{B}}_{10} \hat{{e}}_{33} E_{3}^{(1)} \\= & {} B_{00} \left( {\hat{{c}}_{31} u_{1,1}^{(0)} +\hat{{c}}_{32} u_{2,2}^{(0)} +\hat{{c}}_{33} u_{3}^{(1)} } \right) +B_{20} \tilde{{c}}_{36} \left( {u_{1,2}^{(2)} +u_{2,1}^{(2)} } \right) +\bar{{B}}_{10} \hat{{e}}_{33} 2\phi ^{(2)}, \\ T_{12}^{(0)}= & {} B_{00} \hat{{c}}_{66} S_{6}^{(0)} +B_{20} \tilde{{c}}_{66} S_{6}^{(2)} -\bar{{B}}_{10} \hat{{e}}_{16} E_{1}^{(1)} \\= & {} B_{00} \hat{{c}}_{66} \left( {u_{1,2}^{(0)} +u_{2,1}^{(0)} } \right) +B_{20} \tilde{{c}}_{63} \left( {u_{1,2}^{(2)} +u_{2,1}^{(2)} } \right) +\bar{{B}}_{10} \hat{{e}}_{16} \phi _{,1}^{(1)} , \\ T_{13}^{(1)}= & {} B_{11} \bar{{c}}_{55} S_{5}^{(1)} -\bar{{B}}_{01} \bar{{e}}_{15} E_{1}^{(0)} -\bar{{B}}_{21} \bar{{e}}_{15} E_{1}^{(2)} \\= & {} B_{11} \bar{{c}}_{55} \left( {u_{3,1}^{(1)} +2u_{1}^{(2)} } \right) +\bar{{B}}_{01} \bar{{e}}_{15} \phi _{,1}^{(0)} +\bar{{B}}_{21} \bar{{e}}_{15} \phi _{,1}^{(2)} , \\ T_{23}^{(1)}= & {} B_{11} \bar{{c}}_{44} S_{4}^{(1)} -\bar{{B}}_{01} \bar{{e}}_{24} E_{2}^{(0)} -\bar{{B}}_{21} \bar{{e}}_{24} E_{2}^{(2)} \\= & {} B_{11} \bar{{c}}_{44} \left( {u_{3,2}^{(1)} +2u_{2}^{(2)} } \right) +\bar{{B}}_{01} \bar{{e}}_{24} \phi _{,2}^{(0)} +\bar{{B}}_{21} \bar{{e}}_{24} \phi _{,2}^{(2)} , \\ T_{11}^{(2)}= & {} B_{02} \left( {\tilde{{c}}_{11} S_{1}^{(0)} +\tilde{{c}}_{12} S_{2}^{(0)} } \right) +B_{22} \left( {\tilde{{c}}_{11} S_{1}^{(2)} +\tilde{{c}}_{12} S_{2}^{(2)} } \right) -\bar{{B}}_{12} \left( {\tilde{{e}}_{21} E_{2}^{(1)} +\tilde{{e}}_{31} E_{3}^{(1)} } \right) \\= & {} B_{02} \left( {\tilde{{c}}_{11} u_{1,1}^{(0)} +\tilde{{c}}_{12} u_{2,2}^{(0)} } \right) +B_{22} \left( {\tilde{{c}}_{11} u_{1,1}^{(2)} +\tilde{{c}}_{12} u_{2,2}^{(2)} } \right) +\bar{{B}}_{12} \left( {\tilde{{e}}_{21} \phi _{,2}^{(1)} +\tilde{{e}}_{31} 2\phi ^{(2)}} \right) , \\ T_{22}^{(2)}= & {} B_{02} \left( {\tilde{{c}}_{21} S_{1}^{(0)} +\tilde{{c}}_{22} S_{2}^{(0)} } \right) +B_{22} \left( {\tilde{{c}}_{21} S_{1}^{(2)} +\tilde{{c}}_{22} S_{2}^{(2)} } \right) -\bar{{B}}_{12} \left( {\tilde{{e}}_{22} E_{2}^{(1)} +\tilde{{e}}_{32} E_{3}^{(1)} } \right) \\= & {} B_{02} \left( {\tilde{{c}}_{21} u_{1,1}^{(0)} +\tilde{{c}}_{22} u_{2,2}^{(0)} } \right) +B_{22} \left( {\tilde{{c}}_{21} u_{1,1}^{(2)} +\tilde{{c}}_{22} u_{2,2}^{(2)} } \right) +\bar{{B}}_{12} \left( {\tilde{{e}}_{22} \phi _{,2}^{(1)} +\tilde{{e}}_{32} 2\phi ^{(2)}} \right) , \\ T_{12}^{(2)}= & {} B_{02} \tilde{{c}}_{66} S_{6}^{(0)} +B_{22} \tilde{{c}}_{66} S_{6}^{(2)} -\bar{{B}}_{12} \tilde{{e}}_{16} E_{1}^{(1)} \\= & {} B_{02} \tilde{{c}}_{66} \left( {u_{1,2}^{(0)} +u_{2,1}^{(0)} } \right) +B_{22} \tilde{{c}}_{66} \left( {u_{1,2}^{(2)} +u_{2,1}^{(2)} } \right) +\bar{{B}}_{12} \tilde{{e}}_{16} \phi _{,1}^{(1)} , \\ D_{1}^{(0)}= & {} \bar{{B}}_{10} \left( {\bar{{e}}_{15} S_{5}^{(1)} +\bar{{e}}_{16} S_{6}^{(1)} } \right) +B_{00} \bar{{\varepsilon }}_{11} E_{1}^{(0)} +B_{20} \tilde{{\varepsilon }}_{11} E_{1}^{(2)} \\= & {} \bar{{B}}_{10} \left[ {\bar{{e}}_{15} \left( {u_{3,1}^{(1)} +2u_{1}^{(2)} } \right) +\bar{{e}}_{16} \left( {u_{1,2}^{(1)} +u_{2,1}^{(1)} } \right) } \right] -B_{00} \bar{{\varepsilon }}_{11} \phi _{,1}^{(0)} -B_{20} \tilde{{\varepsilon }}_{11} \phi _{,1}^{(2)} , \\ D_{2}^{(0)}= & {} \bar{{B}}_{10} \left( {\bar{{e}}_{21} S_{1}^{(1)} +\bar{{e}}_{22} S_{2}^{(1)} +\bar{{e}}_{24} S_{4}^{(1)} } \right) +B_{00} \bar{{\varepsilon }}_{22} E_{2}^{(0)} +B_{20} \tilde{{\varepsilon }}_{22} E_{2}^{(2)} \\= & {} \bar{{B}}_{10} \left[ {\bar{{e}}_{21} u_{1,1}^{(1)} +\bar{{e}}_{22} u_{1,1}^{(2)} +\bar{{e}}_{24} \left( {u_{3,2}^{(1)} +2u_{2}^{(2)} } \right) } \right] -B_{00} \bar{{\varepsilon }}_{22} \phi _{,2}^{(0)} -B_{20} \tilde{{\varepsilon }}_{22} \phi _{,2}^{(2)} , \\ D_{3}^{(0)}= & {} \bar{{B}}_{10} \left( {\bar{{e}}_{31} S_{1}^{(1)} +\bar{{e}}_{32} S_{2}^{(1)} } \right) +B_{00} \bar{{\varepsilon }}_{33} E_{3}^{(0)} +B_{20} \tilde{{\varepsilon }}_{33} E_{3}^{(2)} \\= & {} \bar{{B}}_{10} \left( {\bar{{e}}_{31} u_{1,1}^{(1)} +\bar{{e}}_{32} u_{2,2}^{(1)} } \right) -B_{00} \bar{{\varepsilon }}_{33} \phi ^{(1)}, \\ D_{1}^{(1)}= & {} \bar{{B}}_{01} \left( {\hat{{e}}_{15} S_{5}^{(0)} +\hat{{e}}_{16} S_{6}^{(0)} } \right) +\bar{{B}}_{21} \tilde{{e}}_{16} S_{6}^{(2)} +B_{11} \breve{{\varepsilon }}_{11} E_{1}^{(1)} \\= & {} \bar{{B}}_{01} \left[ {\hat{{e}}_{15} \left( {u_{3,1}^{(0)} +u_{1}^{(1)} } \right) +\hat{{e}}_{16} \left( {u_{1,2}^{(0)} +u_{2,1}^{(0)} } \right) } \right] +\bar{{B}}_{21} \tilde{{e}}_{16} \left( {u_{1,2}^{(2)} +u_{2,1}^{(2)} } \right) -B_{11} \breve{{\varepsilon }}_{11} \phi _{,1}^{(1)} , \\ D_{2}^{(1)}= & {} \bar{{B}}_{01} \left( {\hat{{e}}_{21} S_{1}^{(0)} +\hat{{e}}_{22} S_{2}^{(0)} +\hat{{e}}_{24} S_{4}^{(0)} } \right) +\bar{{B}}_{21} \left( {\tilde{{e}}_{21} S_{1}^{(2)} +\tilde{{e}}_{22} S_{2}^{(2)} } \right) +B_{11} \breve{{\varepsilon }}_{22} E_{2}^{(1)} \\= & {} \bar{{B}}_{01} \left[ {\hat{{e}}_{21} u_{1,1}^{(0)} +\hat{{e}}_{22} u_{2,2}^{(0)} +\hat{{e}}_{24} \left( {u_{3,2}^{(0)} +u_{2}^{(1)} } \right) } \right] +\bar{{B}}_{21} \left( {\tilde{{e}}_{21} u_{1,1}^{(2)} +\tilde{{e}}_{22} u_{2,2}^{(2)} } \right) -B_{11} \breve{{\varepsilon }}_{22} \phi _{,2}^{(1)} , \end{aligned}$$
$$\begin{aligned} D_{3}^{(1)}= & {} \bar{{B}}_{01} \left( {\hat{{e}}_{31} S_{1}^{(0)} +\hat{{e}}_{32} S_{2}^{(0)} +\hat{{e}}_{33} S_{3}^{(0)} } \right) +\bar{{B}}_{21} \left( {\tilde{{e}}_{31} S_{1}^{(2)} +\tilde{{e}}_{32} S_{2}^{(2)} } \right) +B_{11} \breve{{\varepsilon }}_{33} E_{3}^{(1)} \\= & {} \bar{{B}}_{01} \left( {\hat{{e}}_{31} u_{1,1}^{(0)} +\hat{{e}}_{32} u_{2,2}^{(0)} +\hat{{e}}_{33} u_{3}^{(1)} } \right) +\bar{{B}}_{21} \left( {\tilde{{e}}_{31} u_{1,1}^{(2)} +\tilde{{e}}_{32} u_{2,2}^{(2)} } \right) -B_{11} \breve{{\varepsilon }}_{33} 2\phi ^{(2)}, \\ D_{1}^{(2)}= & {} \bar{{B}}_{12} \left( {\bar{{e}}_{15} S_{5}^{(1)} +\bar{{e}}_{16} S_{6}^{(1)} } \right) +B_{02} \tilde{{\varepsilon }}_{11} E_{1}^{(0)} +B_{22} \hat{{\varepsilon }}_{11} E_{1}^{(2)} \\= & {} \bar{{B}}_{12} \left[ {\bar{{e}}_{15} \left( {u_{3,1}^{(1)} +2u_{1}^{(2)} } \right) +\bar{{e}}_{16} \left( {u_{1,2}^{(1)} +u_{2,1}^{(1)} } \right) } \right] -B_{02} \tilde{{\varepsilon }}_{11} \phi _{,1}^{(0)} -B_{22} \hat{{\varepsilon }}_{11} \phi _{,1}^{(2)} , \\ D_{2}^{(2)}= & {} \bar{{B}}_{12} \left( {\bar{{e}}_{21} S_{1}^{(1)} +\bar{{e}}_{22} S_{2}^{(1)} +\bar{{e}}_{24} S_{4}^{(1)} } \right) +B_{02} \tilde{{\varepsilon }}_{22} E_{2}^{(0)} +B_{22} \hat{{\varepsilon }}_{22} E_{2}^{(2)} \\= & {} \bar{{B}}_{12} \left[ {\bar{{e}}_{21} u_{1,1}^{(1)} +\bar{{e}}_{22} u_{2,2}^{(1)} +\bar{{e}}_{24} \left( {u_{3,2}^{(1)} +2u_{2}^{(2)} } \right) } \right] -B_{02} \tilde{{\varepsilon }}_{22} \phi _{,2}^{(0)} -B_{22} \hat{{\varepsilon }}_{22} \phi _{,2}^{(2)} . \end{aligned}$$

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Huang, D., Yang, J. A second-order theory for lithium niobate piezoelectric plates with a ferroelectric inversion layer in coupled extensional, thickness-stretch and symmetric thickness-shear motions. Acta Mech 231, 5239–5250 (2020). https://doi.org/10.1007/s00707-020-02794-5

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