Green’s functions for a trigonal piezoelectric half-plane belonging to 3m crystal class

Abstract

Piezoelectric materials have a wide range of industrial applications in different branches of engineering due to their electromechanical coupling. So, investigating their responses to either mechanical or electric loadings helps engineers for efficient design of smart systems. However, most of the studies have assessed the well-known 6 mm piezoelectric materials or piezoceramics and few papers have studied other piezoelectric crystals despite of their application in industry. In this paper, fundamental solutions of a trigonal piezoelectric half-plane belonging to 3m crystal class is obtained. The governing differential equations are derived and solved analytically using potential method. It is shown that the solution for the 3m material can be degenerated to 6 mm solution as a special case. The contour lines were depicted for two practical piezoelectric materials belonging to 3m and 6 mm crystal classes including lithium niobate and PZT-4 and they were compared to each other. The numerical results showed that the response of the trigonal material is asymmetric due to anisotropy and the effect of anisotropy on some responses is considerable causing totally different behavior from 6 mm piezoelectric material.

Appendices

Appendix A

The coefficients $a_{\mathrm {1}}$ to $a_{\mathrm {6}}$ are determined in terms of material constants as follows

$$\begin{aligned} a_{1}&=c_{44} (c_{33} \kappa _{33} +e_{33}^{2} ), \nonumber \\ a_{2}&=2c_{14} (c_{33} \kappa _{33} +e_{33}^{2} ), \nonumber \\ a_{3}&=c_{33} [c_{44} \kappa _{11} +(e_{15} +e_{31} )^{2}]+\kappa _{33} [c_{11} c_{33} +c_{44}^{2} -(c_{13} +c_{44} )^{2}] \nonumber \\&\quad \times e_{33} [2c_{44} e_{15} +c_{11} e_{33} -2(c_{13} +c_{44} )(e_{15} +e_{31} )], \nonumber \\ a_{4}&=2(c_{13} +c_{44} )(e_{22} e_{33} -c_{14} \kappa _{33} )-2(e_{15} +e_{31} )(c_{33} e_{22} +c_{14} e_{33} ) \nonumber \\&\quad \times 2c_{14} (c_{33} \kappa _{11} +c_{44} \kappa _{33} +2e_{15} e_{33} ), \nonumber \\ a_{5}&=c_{44} [c_{11} \kappa _{33} +(e_{15} +e_{31} )^{2}]+\kappa _{11} [c_{11} c_{33} +c_{44}^{2} -(c_{13} +c_{44} )^{2}] \nonumber \\&\quad +e_{15} [2c_{11} e_{33} +c_{44} e_{15} -2(c_{13} +c_{44} )(e_{15} +e_{31} )]+c_{14} [2e_{22} e_{33} -c_{14} \kappa _{33} ]+c_{33} e_{22}^{2}, \nonumber \\ a_{6}&=2c_{14} (c_{44} \kappa _{11} +e_{15}^{2} )+2(c_{13} +c_{44} )(e_{22} e_{15} -c_{14} \kappa _{11} ), -2(e_{15} +e_{31} )(c_{44} e_{22} +c_{14} e_{15} ), \nonumber \\ a_{7}&=c_{11} (e_{15}^{2} +c_{44} \kappa _{11} )+c_{14} (2e_{15} e_{22} -c_{14} \kappa _{11} )+c_{44} e_{22}^{2}. \end{aligned}$$

(A.1)

The parameters $m_{i}$ ($i=1,2,3$) and $n_{jl}$ ($j=1,2,\ldots ,5; l=1,2$) in Eq. (2.12) can be determined as follows:

$$\begin{aligned} m_{1}&=e_{15} e_{22} -\kappa _{11} c_{14}, \nonumber \\ m_{2}&=\kappa _{11} (c_{13} +c_{44} )+e_{15} (e_{31} +e_{15} ), \nonumber \\ m_{3}&=e_{33} e_{22} -\kappa _{33} c_{14}, \nonumber \\ m_{4}&=\kappa _{33} (c_{13} +c_{44} )+e_{33} (e_{31} +e_{15} ), \nonumber \\ n_{11}&=-(c_{11} \kappa _{11} +e_{22}^{2} ), \nonumber \\ n_{21}&=2(c_{14} \kappa _{11} -e_{22} (e_{31} +e_{15} )), \nonumber \\ n_{31}&=-(c_{44} \kappa _{11} +c_{11} \kappa _{33} +(e_{31} +e_{15} )^{2}),\nonumber \\ n_{41}&=2c_{14} \kappa _{33}, \nonumber \\ n_{51}&=-c_{44} \kappa _{33}, \nonumber \\ n_{12}&=-(c_{11} e_{15} +c_{14} e_{22} ), \nonumber \\ n_{22}&=(c_{14} (e_{15} -e_{31} )+e_{22} (c_{13} +c_{44} )), \nonumber \\ n_{32}&=-(c_{11} e_{33} +c_{44} e_{15} -(c_{13} +c_{44} )(e_{31} +e_{15} )), \nonumber \\ n_{42}&=2c_{14} e_{33}, \nonumber \\ n_{52}&=-c_{44} e_{33}, \end{aligned}$$

(A.2)

$\alpha _{j}$ and $\beta _{lj}$ ($l=$ 1,2) in Eq. (2.15) are calculated as follows:

$$\begin{aligned} \alpha _{j}&=m_{1} -m_{2} s_{j} +m_{3} s_{j}^{2} -m_{4} s_{j}^{3}, \nonumber \\ \beta _{lj}&=n_{1l} -n_{2l} s_{j} +n_{3l} s_{j}^{2} -n_{4l} s_{j}^{3} +n_{5l} s_{j}^{4} ,\,\,\, l=1,2. \end{aligned}$$

(A.3)

The coefficients $\omega _{mj}$ (m = 1,2,...,5) are obtained in terms of material constants as

$$\begin{aligned} \omega _{1j}&=-(c_{11} +c_{14} s_{j} )+k_{1j} (c_{14} +c_{13} s_{j} )+k_{2j} (e_{31} s_{j} -e_{22} ), \nonumber \\ \omega _{2j}&=-c_{13} +c_{33} s_{j} k_{1j} +e_{33} s_{j} k_{2j}, \nonumber \\ \omega _{3j}&=-e_{31} +e_{33} s_{j} k_{1j} -\kappa _{33} s_{j} k_{2j}, \nonumber \\ \omega _{4j}&=c_{14} +c_{44} s_{j} -c_{44} k_{1j} -e_{15} k_{2j}, \nonumber \\ \omega _{5j}&=-e_{22} +e_{15} s_{j} -e_{15} k_{1j} +\kappa _{11} k_{2j}, \end{aligned}$$

(A.4)

Appendix B

In the case of 6 mm piezoelectric material, $k_{lj}$, $\omega _{mj}$, $\lambda _{j}$, $\eta _{j}$, and $\gamma _{j}$ coefficients have the following relations

$$\begin{aligned} k_{l1}&=k_{l1} \qquad k_{l2} =k_{l2}\qquad k_{l3} =-\bar{{k}}_{l1} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( {\mathrm{Re}\left( {k_{l2} } \right) =0} \right) \left( {l=1,2} \right) , \nonumber \\ k_{l4}&=\bar{{k}}_{l1} \qquad k_{l5} =-k_{l2} \qquad k_{l6} =-k_{l1} \nonumber \\ \omega _{m1}&=\omega _{m1} \qquad \omega _{m2} =\omega _{m2} \qquad \omega _{m3} =\bar{{\omega }}_{m1} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( {\mathrm{Im}\left( {\omega _{m2} } \right) =0} \right) \left( {m=1,2,3} \right) , \nonumber \\ \omega _{m4}&=\bar{{\omega }}_{m1} \qquad \omega _{m5} =\omega _{m2} \omega _{m6} =\omega _{m1} \nonumber \\ \omega _{m1}&=\omega _{m1} \qquad \omega _{m2} =\omega _{m2} \qquad \omega _{m3} =-\bar{{\omega }}_{m1} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left( {\mathrm{Re}\left( {\omega _{m2} } \right) =0} \right) \left( {m=4,5} \right) , \nonumber \\ \omega _{m4}&=\bar{{\omega }}_{m1} \qquad \omega _{m5} =-\omega _{m2} \qquad \omega _{m6} =-\omega _{m1} \nonumber \\ \lambda _{1}&=\lambda _{1} \qquad \lambda _{2} =\lambda _{2} \qquad \lambda _{3} =\bar{{\lambda }}_{1} \qquad \qquad \,\,\, \left( {\mathrm{Im}\left( {\lambda _{2} } \right) =0} \right) ,\nonumber \\ \eta _{1}&=\eta _{1} \qquad \eta _{2} =\eta _{2} \qquad \eta _{3} =\bar{{\eta }}_{1} \qquad \qquad \,\,\, \left( {\mathrm{Im}\left( {\eta _{2} } \right) =0} \right) , \nonumber \\ \gamma _{1}&=\gamma _{1} \qquad \gamma _{2} =\gamma _{2} \qquad \gamma _{3} =-\bar{{\gamma }}_{1} \qquad \quad \,\,\, \left( {\mathrm{Re}\left( {\gamma _{2} } \right) =0} \right) . \end{aligned}$$

(B.1)