Wave dispersion relations in laminated fiber-reinforced composite plates with surface-mounted piezoelectric materials

Abstract

This study presents the dispersion relations for wave propagation in laminated fiber-reinforced composite plates coupled with the piezoelectric actuators at the top and bottom surfaces based on the Kirchhoff plate theory. Piezoelectric layers are poled in the transverse (thickness) direction to apply flexural action and a cosine distribution for the electric potential in the transverse direction is assumed to satisfy the closed (short)-circuit electrical boundary condition. The Maxwell static electricity equation is applied for the piezoelectric layers bonded on the surfaces of the host laminated composite plate. Based on the Kirchhoff plate theory, a linear relation between the non-dimensional wave phase velocity and the non-dimensional wavenumber is obtained which can present the effects of piezoelectricity, wavenumbers, laminate stacking sequence, and material properties of the host laminated plate on the wave dispersion relations. The main contribution of this study is proposing a new analytical approach to determine wave propagation characteristics in smart composite plates considering transverse polarization of piezoelectric materials. The results of this study indicate that the presence of piezoelectric actuators has a significant effect on the wave phase velocity variations within various wavenumbers as well as the effects of the laminate stacking sequence and the host plate material properties.

Appendices

Appendix 1

The components of lamina transformed reduced stiffness matrix ${\left[{Q}_{ij}\right]}_{K} (i,j=x,y,s)$ as a function of the components of the lamina principal stiffness matrix ${\left[{Q}_{ij}\right]}_{K} (i,j=\mathrm{1,2},6)$ are given by

$$\begin{aligned} Q_{xx} & = m^{4} Q_{11} + n^{4} Q_{22} + 2m^{2} n^{2} Q_{12} + 4m^{2} n^{2} Q_{66} , \\ Q_{yy} & = n^{4} Q_{11} + m^{4} Q_{22} + 2m^{2} n^{2} Q_{12} + 4m^{2} n^{2} Q_{66} , \\ Q_{xy} & = m^{2} n^{2} Q_{11} + m^{2} n^{2} Q_{22} + \left( {m^{4} + n^{4} } \right)Q_{12} - 4m^{2} n^{2} Q_{66} , \\ Q_{xs} & = m^{3} nQ_{11} - mn^{3} Q_{22} - mn\left( {m^{2} - n^{2} } \right)Q_{12} - 2mn\left( {m^{2} - n^{2} } \right)Q_{66} , \\ Q_{ys} & = mn^{3} Q_{11} - m^{3} nQ_{22} + mn\left( {m^{2} - n^{2} } \right)Q_{12} + 2mn\left( {m^{2} - n^{2} } \right)Q_{66} , \\ Q_{ss} & = m^{2} n^{2} Q_{11} + m^{2} n^{2} Q_{22} - 2m^{2} n^{2} Q_{12} + \left( {m^{2} - n^{2} } \right)^{2} Q_{66} . \\ \end{aligned}$$

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where

$$Q_{11} = \frac{{E_{11} }}{{1 - v_{12} v_{21} }}, Q_{12} = \frac{{v_{21} E_{11} }}{{1 - v_{12} v_{21} }}, Q_{21} = \frac{{v_{12} E_{22} }}{{1 - v_{12} v_{21} }}, Q_{22} = \frac{{E_{22} }}{{1 - v_{12} v_{21} }} , Q_{66} = G_{12} ,$$

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and

$$m = \cos \theta , n = \sin \theta .$$

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Appendix 2

$$\begin{aligned} \overline{c}_{11} & = c_{11} - \frac{{c_{13}^{2} }}{{c_{33} }} , \overline{c}_{12} = c_{12} - \frac{{c_{13} c_{23} }}{{c_{33} }} , { }\overline{c}_{22} = c_{22} - \frac{{c_{23}^{2} }}{{c_{33} }} , \overline{c}_{66} = c_{66} , \\ \overline{e}_{31} & = e_{31} - \frac{{c_{13} e_{33} }}{{c_{33} }}, \;\overline{e}_{32} = e_{32} - \frac{{c_{23} e_{33} }}{{c_{33} }}, \\ \overline{ \in }_{11} & = \in_{11} + \frac{{e_{15}^{2} }}{{c_{55} }}, \overline{ \in }_{22} = \in_{22} + \frac{{e_{24}^{2} }}{{c_{44} }} , \overline{ \in }_{33} = \in_{33} + \frac{{e_{33}^{2} }}{{c_{33} }} \\ \end{aligned}$$

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Appendix 3

$$\begin{aligned} A_{1} & = - D_{xx} - \overline{c}_{11} \left( {\frac{{h_{p} h^{2} }}{2} + hh_{{\text{p}}}^{2} + \frac{{2h_{{\text{p}}}^{3} }}{3}} \right) \\ A_{2} & = - D_{xy} - \overline{c}_{12} \left( {\frac{{h_{p} h^{2} }}{2} + hh_{{\text{p}}}^{2} + \frac{{2h_{{\text{p}}}^{3} }}{3}} \right) \\ A_{3} & = - 2D_{xs} \\ A_{4} & = - \frac{{4h_{{\text{p}}} }}{\pi }\overline{e}_{31} \\ B_{1} & = - D_{yx} - \overline{c}_{12} \left( {\frac{{h_{{\text{p}}} h^{2} }}{2} + hh_{{\text{p}}}^{2} + \frac{{2h_{{\text{p}}}^{3} }}{3}} \right) \\ B_{2} & = - D_{yy} - \overline{c}_{22} \left( {\frac{{h_{{\text{p}}} h^{2} }}{2} + hh_{{\text{p}}}^{2} + \frac{{2h_{{\text{p}}}^{3} }}{3}} \right) \\ B_{3} & = - 2D_{ys} \\ B_{4} & = - \frac{{4h_{p} }}{\pi }\overline{e}_{32} \\ C_{1} & = - D_{sx} \\ C_{2} & = - D_{sy} \\ C_{3} & = - 2D_{ss} - \overline{c}_{66} \left( {h_{p} h^{2} + 2hh_{{\text{p}}}^{2} + \frac{{4h_{p}^{3} }}{3}} \right). \\ \end{aligned}$$

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where

$$\left[ D \right]_{ij} = \frac{1}{3}\mathop \sum \limits_{K = 1}^{N} \left[ Q \right]_{ij}^{K} \left( {z_{K}^{3} - z_{K - 1}^{3} } \right) i,j = x,y,s.$$

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Appendix 4

$$\begin{aligned} L_{11} & = I_{0} \left( {c\gamma } \right)^{2} + \gamma^{4} \left( {A_{1} + A_{2} + A_{3} + B_{1} + B_{2} + B_{3} + 2C_{1} + 2C_{2} + 2C_{3} } \right) \\ L_{12} & = - \gamma^{2} \left( {A_{4} + B_{4} } \right) \\ L_{21} & = 2h_{{\text{p}}} \gamma^{2} \left( {\overline{e}_{31} + \overline{e}_{32} } \right) \\ L_{22} & = \frac{{4h_{{\text{p}}} }}{\pi }\gamma^{2} \left( {\overline{ \in }_{11} + \overline{ \in }_{22} } \right) + \frac{4\pi }{{h_{p} }}\overline{ \in }_{33} . \\ \end{aligned}$$

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