Free vibration analysis of saturated porous circular micro-plates integrated with piezoelectric layers; differential transform method

Abstract

The current work investigates axisymmetric vibration of saturated porous circular micro-plates subjected to uniform in-plane force and coupled with piezoelectric sensor and actuator layers. Including the material length scale parameter, the modified coupled stress theory is chosen to capture the size-dependent behavior of the model. The micro-structure is embedded on an elastic substrate medium modeled by Winkler–Pasternak foundation. In addition, the fluid pressure in the pores is partitioned in the formulation via the linear poroelasticity theory of Biot. Based on first-order shear deformation theory, governing equations of motion and corresponding boundary conditions are obtained by employing Hamilton’s principle and solved by a semi-analytical approach called the differential transform method. To ensure the accuracy and reliability of the developed model, natural frequencies of the presented micro-plate are compared with available data in the literature. Finally, numerical examples are presented for studying the effect of some parameters including material length scale parameter, in-plane force, porosity, fluid pressure, electro-mechanical interaction, and the ratio of the elastic core thickness to piezoelectric layers thickness on vibrational responses of the proposed model for two cases of clamped and simply supported edges. The obtained results show that the proposed method is a reliable way of studying the free vibration response of the micro-structures. On the other hand, it is observed that porosity, fluid pressure, tensile in-plane force, size-effect, and open-circuit electrical condition increase the natural frequency of the system.

Appendix

$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {\sigma_{rr}^{P} } \\ {\sigma_{\theta \theta }^{P} } \\ {\sigma_{rz}^{P} } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} {C_{11} } & {C_{12} } & 0 \\ {C_{12} } & {C_{11} } & 0 \\ 0 & 0 & {C_{55} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{rr} } \\ {\varepsilon_{\theta \theta } } \\ {\varepsilon_{rz} } \\ \end{array} } \right\} - \left[ {\begin{array}{*{20}c} 0 & 0 & {e_{31} } \\ 0 & 0 & {e_{31} } \\ {e_{15} } & 0 & 0 \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {E_{r} } \\ {E_{\theta } } \\ {E_{z} } \\, \end{array} } \right\}, \\ \left\{ {\begin{array}{*{20}c} {D_{r} } \\ {D_{\theta } } \\ {D_{z} } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} 0 & 0 & {e_{15} } \\ 0 & 0 & 0 \\ {e_{31} } & {e_{31} } & 0 \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{rr} } \\ {\varepsilon_{\theta \theta } } \\ {\varepsilon_{rz} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} { \in_{11} } & 0 & 0 \\ 0 & { \in_{11} } & 0 \\ 0 & 0 & { \in_{33} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {E_{r} } \\ {E_{\theta } } \\ {E_{z} } \\ \end{array} } \right\}. \\ \end{aligned}$$