High temperature electromechanical response of multilayer piezoelectric laminates under AC electric fields for fuel injector applications

Abstract

This paper examines theoretically and experimentally the dynamic electromechanical response of multilayer piezoelectric laminates for fuel injectors at high temperatures. A phenomenological model of depolarization at high temperatures was used, and the temperature dependent piezoelectric coefficients were obtained. The high temperature electromechanical fields of the multilayer piezoelectric actuators under AC electric fields were then calculated by the finite element method. In addition, test data on the electric field induced strain at high temperatures, which verify the model, were presented.

Appendix

For piezoelectric ceramics which exhibit symmetry of a hexagonal crystal of class 6 mm with respect to principal x₁, x₂ and x₃ (poling) axes, the constitutive relations can be written in the following form:

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{11} } \\ {\varepsilon_{22} } \\ {\varepsilon_{33} } \\ {2\varepsilon_{23} } \\ {2\varepsilon_{31} } \\ {2\varepsilon_{12} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {s_{11} } & {s_{12} } & {s_{13} } & 0 & 0 & 0 \\ {s_{12} } & {s_{11} } & {s_{13} } & 0 & 0 & 0 \\ {s_{13} } & {s_{13} } & {s_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {s_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {s_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {s_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{33} } \\ {\sigma_{23} } \\ {\sigma_{31} } \\ {\sigma_{12} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} 0 & 0 & {\bar{d}_{31} } \\ 0 & 0 & {\bar{d}_{31} } \\ 0 & 0 & {\bar{d}_{33} } \\ 0 & {\bar{d}_{15} } & 0 \\ {\bar{d}_{15} } & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {E_{1} } \\ {E_{2} } \\ {E_{3} } \\ \end{array} } \right\} $$

(12)

$$ \left\{ \begin{array}{c} {D_{1} } \\ {D_{2} } \\ D_{3} \\ \end{array} \right\} = \left[ {\begin{array}{cccccc} 0 & 0 & 0 & 0 & {\bar{d}_{15} } & 0 \\ 0 & 0 & 0 & {\bar{d}_{15} } & 0 & 0 \\ {\bar{d}_{31} } & {\bar{d}_{31} } & {\bar{d}_{33} } & 0 & 0 & 0 \\ \end{array} } \right]\left\{ {\begin{array}{c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{33} } \\ {\sigma_{23} } \\ {\sigma_{31} } \\ {\sigma_{12} } \\ \end{array} } \right\} + \left[ \begin{array}{ccc} \epsilon_{11}^{T} & 0 & 0 \\ 0 & \epsilon_{11}^{T} & 0 \\ 0 & 0 & \epsilon\epsilon_{33}^{T} \\ \end{array} \right]\left\{ {\begin{array}{c} {E_{1} } \\ {E_{2} } \\ {E_{3} } \\ \end{array} } \right\} $$

(13)

where

$$ \sigma_{23} = \sigma_{32} ,\quad \sigma_{31} = \sigma_{13} ,\quad \sigma_{12} = \sigma_{21} $$

(14)

$$ \varepsilon_{23} = \varepsilon_{32} ,\quad \varepsilon_{31} = \varepsilon_{13} ,\quad \varepsilon_{12} = \varepsilon_{21} $$

(15)

$$ \begin{aligned} s_{11} & = s_{1111} = s_{2222} ,\quad s_{12} = s_{1122} ,\quad s_{13} = s_{1133} = s_{2233} ,\quad s_{33} = s_{3333} , \\ s_{44} & = 4s_{2323} = 4s_{3131} ,\quad s_{66} = 4s_{1212} = 2\left( {s_{11} - s_{12} } \right) \\ \end{aligned} $$

(16)

$$ \bar{d}_{15} = 2\bar{d}_{131} = 2\bar{d}_{223} ,\quad \bar{d}_{31} = \bar{d}_{311} = \bar{d}_{322} ,\quad \bar{d}_{33} = \bar{d}_{333} $$

(17)