An investigation over the effect of piezoelectricity and porosity distribution on natural frequencies of porous smart plates

Abstract

The eigenvibration characteristics of a smart plate with piezoelectric layers and porous-cellular core are investigated in the present article. The core plate is assumed to be composed of materials that contain pores and the porosities may be distributed according to different mathematical rules. Variational principle is applied in order to derive the continuous system equations on the basis of Mindlin plate theory. A highly efficient analytical modeling for eigenfrequency analysis of the smart plate is presented under the assumption that both Skempton’s pore pressure coefficient and normal elongation through the thickness are negligible. Unlike numerical methods that require huge computational cost, this approach enables us to find the system’s response for rectangular plates with arbitrary dimensions. To examine the validity of the present framework, multiple comparison studies are made between the extracted results and those available in the literature. It is shown that the type of porosity distribution influences strongly on the way that frequency changes. Furthermore, it is found out that it is necessary to consider electrical effects for plates with open circuit condition unlike the other electrical condition.

Keywords

Free vibration smart plates porous-cellular materials piezoelectric materials porosity distribution

Introduction

Porous materials are a type of lightweight materials in which pores may be distributed along one or multiple directions; consequently, the material properties may follow a certain rule to achieve desired functionalities. Several methods for fabrication of cellular materials such as steel foams and cellular aluminum, are proposed in literature [1 –3]. In this context, Wei et al. [4] evaluated the thermal conductivity and heat transfer properties of C/SiC pyramidal core lattice sandwich panel experimentally and numerically. They [5] also designed different planar lattices of multi-fold rotational symmetry so as to reach various tailorable thermal expansions. Besides, piezoelectric materials are usually employed to control the vibration behavior of the host system.

Hundreds of numerical, semi-analytical, and analytical approaches have been proposed to analyze the dynamic response of various structures such as beams, plates, and shells made of different materials. Considering the plate as the structure under studying, thickness–length ratio is definitely one of the most important parameters for identifying the system’s response accurately. Therefore, several plate theories have been put forward [6 –11]. Investigations related to non-porous plates have been presented over the past 50 years. In this regard, one of the most prominent and fundamental studies has been carried out by Leissa [12] who presented a solution for free vibration of rectangular plates with all 21 possible combination of classical boundary conditions.

Regarding the dynamic analysis of plates composed of piezoelectric layers, electromechanical vibration behavior of simply supported laminated piezoelectric plates has been investigated by Heyliger and Saravanos [13] who solved the three-dimensional equations of motion as well as the charge equation. In view of the difficulties in dealing with determining the dynamic response of piezoelectric functionally graded sandwich plates resting on elastic foundation, an accurate approach has been proposed by Duc and Cong [14] which makes it possible to investigate the vibration and nonlinear dynamic response of the plate by considering stress function, the Galerkin method, and the Runge–Kutta method. In order to study the shape and vibration control of FG plates integrated with piezoelectric layers, He et al. [15] used finite element method as well as a constant velocity feedback control algorithm. Based on Kirchhoff and Mindlin plate theories, Pietrzakowski [16] dealt with the vibration analysis of composites containing piezoelectric sensor/actuator layers. An electromechanical model for free vibration analysis of open circuit piezoelectric coupled circular plate has carried out by Wu et al. [17], with the help of Kirchhoff plate theory as well as Maxwell’s equation.

In recent years, mechanical analysis of porous structures has received attention due to their excellent strength to weight ratio. The bending motion of fluid-saturated simply supported rectangular poroelastic plates was analyzed by Theodorakopoulos and Beskos [18] who used Kirchhoff plate model. Etchessahar et al. [19] carried out the flexural vibration of porous plates under Kirchhoff’s hypotheses by deriving a mixed displacement–pressure relations. In order to study the transverse vibration of rectangular isotropic porous plates, the coupled dynamic equations relating the solid skeleton deflection and the fluid–solid relative displacement have been given by Leclaire et al. [20] Considering a non-linear hypothesis for displacement field, Magnucka-Blandzi [21] has found that the critical buckling load decreases as the porosity is increasing by analyzing the problem of buckling and deflection of simply supported circular porous plates under uniform compression and uniformly distributed pressure. Chen et al. [22 –24] developed exact formulations for beams according to Timoshenko beam theory, and studied the free and forced vibration as well as elastic buckling of beams made of functionally graded porous materials. Barati et al. [25] carried out the electro-mechanical vibration of smart plates using a refined four-variable plate theory and some admissible functions. Farzaneh Joubaneh et al. [26] have obtained the critical thermal buckling loads of a porous circular plate and investigated the effect of different parameters such as porous thermal expansion coefficient and feedback gain on the thermal stability of the plate. Very recently, Askari et al. [27] presented a very precise analysis regarding the free vibration behavior of sandwich plates with porous-cellular core surrounded by piezoelectric layers by considering a nonlinear porosity distribution for the core plate as well as a higher-order shear deformation plate theory.

According to the above literature review, up to now, no study has been investigated the effect of porosity distribution and piezoelectricity on the free vibration behavior of hybrid piezoelectric porous plates. Therefore, in this article, a well-known plate model, i.e. Mindlin plate theory, is used for studying such effects. Based on the Hamilton’s principle and Maxwell equation, six highly coupled linear partial differential equations of motion are extracted and solved analytically following a decoupling procedure. Due to some comparison studies, the accuracy of the presented formulation is validated. It is shown that the variation trends of natural frequency against different parameters are highly dependent on the porosity distribution along the core plate’s thickness.

Functionally graded porous plate

The Cartesian coordinate system (x, y, z) is chosen such that x and y are in-plane coordinates and z defines the direction along the plate’s thickness. A rectangular plate with lengths a and b is assumed (see Figure 1). The thickness of core plate and each piezoelectric layer is 2h and h_p, respectively. Different porosity distributions are taken into account as [22] (see Figure 1)

{\begin{matrix} E (z) = E_{\max} [1 - e_{0} cos (π (z + h) / (4 h))] \\ ρ (z) = ρ_{\max} [1 - e_{m} cos (π (z + h) / (4 h))] \end{matrix}

(1a)

{\begin{matrix} E (z) = E_{\max} [1 - e_{0} cos (π z / 2 h)] \\ ρ (z) = ρ_{\max} [1 - e_{m} cos (π z / 2 h)] \end{matrix}

(1b)

{\begin{matrix} E (z) = E_{\max} [1 - e_{0} α] \\ ρ (z) = ρ_{\max} \sqrt{1 - e_{0} α} \end{matrix}

(1c)

where E and

ρ

are elastic modulus and mass density of the porous plate, respectively.

e_{0}

and

e_{m}

are the porosity coefficient for Young’s modulus and mass density which can be expressed as [22]

\begin{array}{l} e_{0} = 1 - \frac{E_{\min}}{E_{\max}}, 0 \leq e_{0} < 1 \\ e_{m} = 1 - \sqrt{1 - e_{0}} \end{array}

(2)

Here $E_{max}$ ( $ρ_{max}$ ) is the corresponding Young’s modulus (mass density) of the top face of the plate and $E_{min}$ ( $ρ_{min}$ ) denoted this parameter at either bottom or mid-plane of the plate for the above-mentioned distributions, respectively. With uniform porosity distribution, the material properties of the core are constant along the plate thickness and they are only dependent on porosity coefficient. The coefficient $α$ is obtained on the basis of equivalent mass of sandwich porous plates as follow [22]

α = \frac{1}{e_{0}} [1 - {(\frac{2}{π} \sqrt{1 - e_{0}} - \frac{2}{π} + 1)}^{2}]

(3)

Theoretical formulation

Kinematics and constitutive relations

According to the Mindlin plate Model, the displacement field of the plate, denoted as $U_{x} (x, y, z, t)$ , $U_{y} (x, y, z, t)$ and $U_{z} (x, y, z, t)$ can be expressed as

\begin{array}{l} U_{x} (x, y, z, t) = {\bar{U}}_{x} (x, y, t) + z {\bar{Γ}}_{x} (x, y, t) \\ U_{y} (x, y, z, t) = {\bar{U}}_{y} (x, y, t) + z {\bar{Γ}}_{y} (x, y, t) \\ U_{z} (x, y, z, t) = {\bar{U}}_{z} (x, y, t) \end{array}

(4)

where

{\bar{U}}_{x} (x, y, t)

and

{\bar{U}}_{y} (x, y, t)

are the in-plane displacements of the mid-plane,

{\bar{U}}_{z} (x, y, t)

is the transverse displacement and

{\bar{Γ}}_{x} (x, y, t)

and

{\bar{Γ}}_{y} (x, y, t)

denote the rotations about in-plane coordinates. The strain–displacement relations are given by t

\begin{matrix} {\begin{matrix} {\hat{ε}}_{x x} \\ {\hat{ε}}_{y y} \\ {\hat{γ}}_{x y} \end{matrix}} = {\begin{matrix} {\bar{U}}_{x, x} \\ {\bar{U}}_{y, y} \\ {\bar{U}}_{x, y} + {\bar{U}}_{y, x} \end{matrix}} + z {\begin{matrix} {\bar{Γ}}_{x, x} \\ {\bar{Γ}}_{y, y} \\ {\bar{Γ}}_{x, y} + {\bar{Γ}}_{y, x} \end{matrix}} \\ {\begin{matrix} {\hat{γ}}_{x z} \\ {\hat{γ}}_{y z} \end{matrix}} = {\begin{matrix} {\bar{Γ}}_{x} + {\bar{U}}_{z, x} \\ {\bar{Γ}}_{y} + {\bar{U}}_{z, y} \end{matrix}} \end{matrix}

(5)

The linear constitutive equations for porous materials are

\begin{array}{l} [\begin{matrix} σ_{x x}^{H} \\ σ_{y y}^{H} \end{matrix}] = [\begin{matrix} \frac{E (z)}{1 - υ^{2}} & \frac{υ E (z)}{1 - υ^{2}} \\ \frac{υ E (z)}{1 - υ^{2}} & \frac{E (z)}{1 - υ^{2}} \end{matrix}] [\begin{matrix} {\hat{ε}}_{x x} \\ {\hat{ε}}_{y y} \end{matrix}] \\ [\begin{matrix} σ_{y z}^{H} \\ σ_{x z}^{H} \\ σ_{x y}^{H} \end{matrix}] = [\begin{matrix} \frac{K^{2} E (z)}{2 (1 + υ)} & 0 & 0 \\ 0 & \frac{K^{2} E (z)}{2 (1 + υ)} & 0 \\ 0 & 0 & \frac{E (z)}{2 (1 + υ)} \end{matrix}] [\begin{matrix} {\hat{γ}}_{y z} \\ {\hat{γ}}_{x z} \\ {\hat{γ}}_{x y} \end{matrix}] \end{array}

(6)

The superscript “H” denotes the variables for the host structure and $K^{2}$ is the shear correction factor. Researchers usually assume the piezoelectric materials to be homogeneous and transversely isotropic. Considering a piezoelectric layer which is polarized in the thickness direction, the constitutive relations describing the electromechanical interactions can be represented by disregarding the through-the-thickness stress as [28]

\begin{array}{l} [\begin{matrix} σ_{x x}^{P} \\ σ_{y y}^{P} \end{matrix}] = [\begin{matrix} {\tilde{c}}_{11} & {\tilde{c}}_{12} \\ {\tilde{c}}_{12} & {\tilde{c}}_{11} \end{matrix}] [\begin{matrix} {\hat{ε}}_{x x} \\ {\hat{ε}}_{y y} \end{matrix}] - [\begin{matrix} 0 & {\tilde{e}}_{31} \\ 0 & {\tilde{e}}_{31} \end{matrix}] [\begin{matrix} 0 \\ E_{z} \end{matrix}] \\ [\begin{matrix} σ_{y z}^{P} \\ σ_{x z}^{P} \\ σ_{x y}^{P} \end{matrix}] = [\begin{matrix} K^{2} c_{55} & 0 & 0 \\ 0 & K^{2} c_{55} & 0 \\ 0 & 0 & \frac{({\tilde{c}}_{11} - {\tilde{c}}_{12})}{2} \end{matrix}] [\begin{matrix} {\hat{γ}}_{y z} \\ {\hat{γ}}_{x z} \\ {\hat{γ}}_{x y} \end{matrix}] - [\begin{matrix} 0 & e_{15} & 0 \\ e_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] \\ [\begin{matrix} D_{x} \\ D_{y} \\ D_{z} \end{matrix}] = [\begin{matrix} \begin{matrix} \begin{matrix} 0 \\ 0 \\ {\tilde{e}}_{31} \end{matrix} & \begin{matrix} 0 \\ 0 \\ {\tilde{e}}_{31} \end{matrix} & \begin{matrix} 0 \\ e_{15} \\ 0 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} e_{15} \\ 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}] [\begin{matrix} \begin{matrix} {\hat{ε}}_{x x} \\ {\hat{ε}}_{y y} \end{matrix} \\ \begin{matrix} {\hat{γ}}_{y z} \\ {\hat{γ}}_{x z} \end{matrix} \\ {\hat{γ}}_{x y} \end{matrix}] + [\begin{matrix} \begin{matrix} Ξ_{11} \\ 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ Ξ_{11} \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \\ {\tilde{Ξ}}_{33} \end{matrix} \end{matrix}] [\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] \end{array}

(7)

The superscript “P” denotes the variables for the piezoelectric layers. Here, $[D]$ and $[E]$ are the electrical displacement and electrical field vectors, respectively. In addition, $c_{i j}$ , $Ξ_{i j}$ and $e_{i j}$ are the elasticity tensor, the dielectric permittivity coefficient tensor and the piezoelectric constant tensor components, respectively. The coefficients ${\tilde{c}}_{11}$ , ${\tilde{c}}_{12}$ , ${\tilde{e}}_{31}$ and ${\tilde{Ξ}}_{33}$ are given in the relation (36) of Appendix 1.

Electric potential distribution and Maxwell Equation

In this study, the electric potential $\tilde{Φ}$ is assumed to be a quadratic function of the thickness coordinate. If both major surfaces of each piezoelectric layer are kept at zero voltage (short or closed circuit electrical condition), the following function for $\tilde{Φ}$ may be assumed

\tilde{Φ} (x, y, z, t) = φ (x, y, t) \times {\begin{array}{l} [1 - {(\frac{2 z - 2 h - h_{p}}{h_{p}})}^{2}] (h \leq z \leq h + h_{p}) \\ [1 - {(\frac{- 2 z - 2 h - h_{p}}{h_{p}})}^{2}] (- h - h_{p} \leq z \leq - h) \end{array}

(8)

Here, $φ (x, y, t)$ denotes the electric potential in the mid-surface of piezoelectric layers. In open circuit electrical boundary condition, the outer surface is electrically insulated and the inner one is held at zero voltage, and the variation of electric potential can be considered as

\tilde{Φ} (x, y, z, t) = φ (x, y, t) \times {\begin{array}{l} [1 - {(\frac{2 z - 2 h - h_{p}}{h_{p}})}^{2}] + Λ_{1} (h \leq z \leq h + h_{p}) \\ [1 - {(\frac{- 2 z - 2 h - h_{p}}{h_{p}})}^{2}] + Λ_{2} (- h - h_{p} \leq z \leq - h) \end{array}

(9)

where

Λ_{1} (= A z + B)

and

Λ_{2} (= A^{'} z + B^{'})

are two unknown linear functions that can be obtained by applying electrical boundary conditions, i.e.

\tilde{Φ} (at z = \pm (h)) = 0

and

D_{3} (at z = \pm (h + h_{p})) = 0

Also, the relations between electric potential and electric field components are given as

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] = - [\begin{matrix} {\tilde{Φ}}_{, x} \\ {\tilde{Φ}}_{, y} \\ {\tilde{Φ}}_{, z} \end{matrix}]

(10)

Integral form of the Maxwell equation is considered in order to obtain the last necessary equation

\int_{- h - h_{p}}^{- h} (D_{x, x} + D_{y, y} + D_{z, z}) d z + \int_{h}^{h + h_{p}} (D_{x, x} + D_{y, y} + D_{z, z}) d z = 0

(11)

Governing equations of motion

The variation of stored energy for the hybrid plate may be determined as

δ U = \iint_{A} \int_{- h - h_{p}}^{h + h_{p}} {σ_{i j} δ {\hat{ε}}_{i j}} d A d z, i, j = x, y, z

(12)

In view of equations (5) and (7), the above integral becomes

\begin{array}{l} \int_{t_{1}}^{t_{2}} δ U d t = \int_{t_{1}}^{t_{2}} [\iint_{A} - {δ {\bar{U}}_{x} (\vec{\nabla} . {\vec{V}}_{1}) + δ {\bar{U}}_{y} (\vec{\nabla} . {\vec{V}}_{2}) + δ {\bar{U}}_{z} (\vec{\nabla} . {\vec{V}}_{3}) + δ {\bar{Γ}}_{x} (\vec{\nabla} . {\vec{V}}_{4} - {\bar{\bar{H}}}_{x}) \\ + δ {\bar{Γ}}_{y} (\vec{\nabla} . {\vec{V}}_{5} - {\bar{\bar{H}}}_{y})} d A \\ + \int_{Γ} {δ {\bar{U}}_{x} (\vec{n} . {\vec{V}}_{1}) + δ {\bar{U}}_{y} (\vec{n} . {\vec{V}}_{2}) + δ {\bar{U}}_{z} (\vec{n} . {\vec{V}}_{3}) + δ {\bar{Γ}}_{x} (\vec{n} . {\vec{V}}_{4}) + δ {\bar{Γ}}_{y} (\vec{n} . {\vec{V}}_{5})} dΓ] d t \end{array}

(13)

where

\begin{array}{l} {\vec{V}}_{1} = {\bar{\bar{N}}}_{x x} {\vec{e}}_{x} + {\bar{\bar{N}}}_{x y} {\vec{e}}_{y} \\ {\vec{V}}_{2} = {\bar{\bar{N}}}_{x y} {\vec{e}}_{x} + {\bar{\bar{N}}}_{y y} {\vec{e}}_{y} \\ {\vec{V}}_{3} = {\bar{\bar{H}}}_{x} {\vec{e}}_{x} + {\bar{\bar{H}}}_{y} {\vec{e}}_{y} \\ {\vec{V}}_{4} = {\bar{\bar{M}}}_{x x} {\vec{e}}_{x} + {\bar{\bar{M}}}_{x y} {\vec{e}}_{y} \\ {\vec{V}}_{5} = {\bar{\bar{M}}}_{x y} {\vec{e}}_{x} + {\bar{\bar{M}}}_{y y} {\vec{e}}_{y} \end{array}

(14)

Here $({\vec{e}}_{x}, {\vec{e}}_{y})$ are the unit normal to the boundary $Γ$ of the domain of the plate and ${\bar{\bar{N}}}_{i j}$ , ${\bar{\bar{M}}}_{i j}$ , ${\bar{\bar{H}}}_{i}$ are the stress resultants

\begin{matrix} {{\bar{\bar{N}}}_{x x}, {\bar{\bar{N}}}_{x y}, {\bar{\bar{N}}}_{y y}} = \int_{- h - h_{p}}^{- h} {σ_{x x}^{P}, σ_{x y}^{P}, σ_{y y}^{P}} d z + \int_{- h}^{+ h} {σ_{x x}^{H}, σ_{x y}^{H}, σ_{y y}^{H}} d z + \int_{h}^{h + h_{p}} {σ_{x x}^{P}, σ_{x y}^{P}, σ_{y y}^{P}} d z \end{matrix}

\begin{matrix} {{\bar{\bar{M}}}_{x x}, {\bar{\bar{M}}}_{x y}, {\bar{\bar{M}}}_{y y}} = \int_{- h - h_{p}}^{- h} {σ_{x x}^{P}, σ_{x y}^{P}, σ_{y y}^{P}} z d z + \int_{- h}^{+ h} {σ_{x x}^{H}, σ_{x y}^{H}, σ_{y y}^{H}} z d z \\ + \int_{h}^{h + h_{p}} {σ_{x x}^{P}, σ_{x y}^{P}, σ_{y y}^{P}} z d z \\ {{\bar{\bar{H}}}_{x}, {\bar{\bar{H}}}_{y}} = \int_{- h - h_{p}}^{- h} {σ_{x z}^{P}, σ_{y z}^{P}} d z + \int_{- h}^{+ h} {σ_{x z}^{H}, σ_{y z}^{H}} d z + \int_{h}^{h + h_{p}} {σ_{x z}^{P}, σ_{y z}^{P}} d z \end{matrix}

(15)

The variation of kinematic energy of the plate is

δ T = \iint_{A} \int_{- h - h_{p}}^{h + h_{p}} {ρ (z) {\ddot{U}}_{i} δ U_{i}} d A d z, i = x, y, z

(16)

Using equation (4), the above integral can be rewritten as

\begin{matrix} \int_{t_{1}}^{t_{2}} δ T d t = - \int_{t_{1}}^{t_{2}} \iint_{A} {[I_{00} {\ddot{\bar{U}}}_{x} + I_{11} {\ddot{\bar{Γ}}}_{x}] δ {\bar{U}}_{x} + [I_{00} {\ddot{\bar{U}}}_{y} + I_{11} {\ddot{\bar{Γ}}}_{y}] δ {\bar{U}}_{y} \\ + [I_{00} {\ddot{\bar{U}}}_{z}] δ {\bar{U}}_{z} + [I_{11} {\ddot{\bar{U}}}_{x} + I_{22} {\ddot{\bar{Γ}}}_{x}] δ {\bar{Γ}}_{x} \\ + [I_{11} {\ddot{\bar{U}}}_{y} + I_{22} {\ddot{\bar{Γ}}}_{y}] δ {\bar{Γ}}_{y}} d A d t \end{matrix}

(17)

And inertia terms are defined as

{I_{00}, I_{11}, I_{22}} = \int_{- h - h_{p}}^{h + h_{p}} ρ (z) {1, z, z^{2}} d z

(18)

By using equations (13), (14) and (17) and applying the Hamilton’s principle, the equations of motion could be obtained as

δ {\bar{U}}_{x} : {\bar{\bar{N}}}_{x x, x} + {\bar{\bar{N}}}_{x y, y} = I_{00} {\ddot{\bar{U}}}_{x} + I_{11} {\ddot{\bar{Γ}}}_{x}

(19a)

δ {\bar{U}}_{y} : {\bar{\bar{N}}}_{x y, x} + {\bar{\bar{N}}}_{y y, y} = I_{00} {\ddot{\bar{U}}}_{y} + I_{11} {\ddot{\bar{Γ}}}_{y}

(19b)

δ {\bar{Γ}}_{x} : {\bar{\bar{M}}}_{x x, x} + {\bar{\bar{M}}}_{x y, y} - {\bar{\bar{H}}}_{x} = I_{11} {\ddot{\bar{U}}}_{x} + I_{22} {\ddot{\bar{Γ}}}_{x}

(19c)

δ {\bar{Γ}}_{y} : {\bar{\bar{M}}}_{x y, x} + {\bar{\bar{M}}}_{y y, y} - {\bar{\bar{H}}}_{y} = I_{11} {\ddot{\bar{U}}}_{y} + I_{22} {\ddot{\bar{Γ}}}_{y}

(19d)

δ {\bar{U}}_{z} : {\bar{\bar{H}}}_{x, x} + {\bar{\bar{H}}}_{y, y} = I_{00} {\ddot{\bar{U}}}_{z}

(19e)

And mechanical boundary conditions are

Essential Natural

δ {\bar{U}}_{x} = 0 or n_{x} {\bar{\bar{N}}}_{x x} + n_{y} {\bar{\bar{N}}}_{x y} = 0 on Γ

(20a)

δ {\bar{U}}_{y} = 0 or n_{x} {\bar{\bar{N}}}_{x y} + n_{y} {\bar{\bar{N}}}_{y y} = 0 on Γ

(20b)

δ {\bar{Γ}}_{x} = 0 or n_{x} {\bar{\bar{M}}}_{x x} + n_{y} {\bar{\bar{M}}}_{x y} = 0 on Γ

(20c)

δ {\bar{Γ}}_{y} = 0 or n_{x} {\bar{\bar{M}}}_{x y} + n_{y} {\bar{\bar{M}}}_{y y} = 0 on Γ

(20d)

δ {\bar{U}}_{z} = 0 or n_{x} {\bar{\bar{H}}}_{x} + n_{y} {\bar{\bar{H}}}_{y} = 0 on Γ

(20e)

By expressing the stress resultants in term of displacement components, the governing equations of motion are rewritten as

\begin{array}{l} Q_{11} ({\bar{U}}_{x, x x} + {\bar{U}}_{y, x y}) + Q_{66} ({\bar{U}}_{x, y y} - {\bar{U}}_{y, x y}) \\ + {\hat{Q}}_{11} ({\bar{Γ}}_{x, x x} + {\bar{Γ}}_{y, x y}) + {\hat{Q}}_{66} ({\bar{Γ}}_{x, y y} - {\bar{Γ}}_{y, x y}) = I_{00} {\ddot{\bar{U}}}_{x} + I_{11} {\ddot{\bar{Γ}}}_{x} \end{array}

(21a)

\begin{array}{l} Q_{11} ({\bar{U}}_{x, x y} + {\bar{U}}_{y, y y}) - Q_{66} ({\bar{U}}_{x, x y} - {\bar{U}}_{y, x x}) \\ + {\hat{Q}}_{11} ({\bar{Γ}}_{x, x y} + {\bar{Γ}}_{y, y y}) - {\hat{Q}}_{66} ({\bar{Γ}}_{x, x y} - {\bar{Γ}}_{y, x x}) = I_{00} {\ddot{\bar{U}}}_{y} + I_{11} {\ddot{\bar{Γ}}}_{y} \end{array}

(21b)

\begin{array}{l} {\hat{Q}}_{11} ({\bar{U}}_{x, x x} + {\bar{U}}_{y, x y}) + {\hat{Q}}_{66} ({\bar{U}}_{x, y y} - {\bar{U}}_{y, x y}) + ({\tilde{Q}}_{11} + {\bar{μ}}_{1}) ({\bar{Γ}}_{x, x x} + {\bar{Γ}}_{y, x y}) \\ + {\tilde{Q}}_{66} ({\bar{Γ}}_{x, y y} - {\bar{Γ}}_{y, x y}) - K^{2} Q_{55} ({\bar{Γ}}_{x} + {\bar{U}}_{z, x}) + ({\tilde{μ}}_{2} - {\tilde{μ}}_{1}) φ_{, x} = I_{11} {\ddot{\bar{U}}}_{x} + I_{22} {\ddot{\bar{Γ}}}_{x} \end{array}

(21c)

\begin{matrix} {\hat{Q}}_{11} ({\bar{U}}_{x, x y} + {\bar{U}}_{y, y y}) - {\hat{Q}}_{66} ({\bar{U}}_{x, x y} - {\bar{U}}_{y, x x}) + ({\tilde{Q}}_{11} + {\bar{μ}}_{1}) ({\bar{Γ}}_{x, x y} + {\bar{Γ}}_{y, y y}) \\ - {\tilde{Q}}_{66} ({\bar{Γ}}_{x, x y} - {\bar{Γ}}_{y, x x}) - K^{2} Q_{55} ({\bar{Γ}}_{y} + {\bar{U}}_{z, y}) + ({\tilde{μ}}_{2} - {\tilde{μ}}_{1}) φ_{, y} = I_{11} {\ddot{\bar{U}}}_{y} + I_{22} {\ddot{\bar{Γ}}}_{y} \end{matrix}

(21d)

\begin{array}{l} K^{2} Q_{55} ({\bar{U}}_{z, x x} + {\bar{U}}_{z, y y} + {\bar{Γ}}_{x, x} + {\bar{Γ}}_{y, y}) - {\bar{μ}}_{1} ({\bar{Γ}}_{x, x x x} + {\bar{Γ}}_{y, y x x} + {\bar{Γ}}_{x, x y y} + {\bar{Γ}}_{y, y y y}) \\ - {\tilde{μ}}_{2} (φ_{, x x} + φ_{, y y}) = I_{00} {\ddot{\bar{U}}}_{z} \end{array}

(21e)

where the coefficients in equation (21) are given in relations (37) to (39) of Appendix 1. Using equation (11), the last partial differential equation may be obtained as follow

\begin{array}{l} μ_{1} ({\bar{Γ}}_{x, x} + {\bar{Γ}}_{y, y}) + μ_{2} ({\bar{U}}_{z, x x} + {\bar{U}}_{z, y y}) + μ_{3} φ + {\bar{μ}}_{2} ({\bar{Γ}}_{x, x x x} + {\bar{Γ}}_{y, y x x} + {\bar{Γ}}_{x, x y y} + {\bar{Γ}}_{y, y y y}) \\ + {\tilde{μ}}_{3} (φ_{, x x} + φ_{, y y}) = 0 \end{array}

(22)

where the coefficients

μ_{1}

μ_{2}

μ_{3}

{\bar{μ}}_{2}

and

{\tilde{μ}}_{3}

are defined in relations (40) and (41) of Appendix 1.

Solution procedure

The task of solving the equations which govern the system has become easier by introduction of new functions. Applying the functions and some algebraic operation on differentiation of equations (21) and (22), the following equations can be obtained

Q_{11} \nabla^{2} ϱ_{1} + {\hat{Q}}_{11} \nabla^{2} ϱ_{3} = I_{00} {\ddot{ϱ}}_{1} + I_{11} {\ddot{ϱ}}_{3}

(23a)

{\hat{Q}}_{11} \nabla^{2} ϱ_{1} + ({\tilde{Q}}_{11} + {\bar{μ}}_{1}) \nabla^{2} ϱ_{3} - K^{2} Q_{55} (\nabla^{2} {\bar{U}}_{z} + ϱ_{3}) + ({\tilde{μ}}_{2} - {\tilde{μ}}_{1}) \nabla^{2} φ = I_{11} {\ddot{ϱ}}_{1} + I_{22} {\ddot{ϱ}}_{3}

(23b)

K^{2} Q_{55} (\nabla^{2} {\bar{U}}_{z} + ϱ_{3}) - {\bar{μ}}_{1} \nabla^{2} ϱ_{3} - {\tilde{μ}}_{2} \nabla^{2} φ = I_{00} {\ddot{\bar{U}}}_{z}

(23c)

μ_{1} ϱ_{3} + μ_{2} \nabla^{2} {\bar{U}}_{z} + μ_{3} φ + {\bar{μ}}_{2} \nabla^{2} ϱ_{3} + {\tilde{μ}}_{3} \nabla^{2} φ = 0

(23d)

Q_{66} \nabla^{2} ϱ_{2} + {\hat{Q}}_{66} \nabla^{2} ϱ_{4} = I_{00} {\ddot{ϱ}}_{2} + I_{11} {\ddot{ϱ}}_{4}

(23e)

{\hat{Q}}_{66} \nabla^{2} ϱ_{2} + {\tilde{Q}}_{66} \nabla^{2} ϱ_{4} - K^{2} Q_{55} ϱ_{4} = I_{11} {\ddot{ϱ}}_{2} + I_{22} {\ddot{ϱ}}_{4}

(23f)

where

[\begin{matrix} \begin{matrix} ϱ_{1} \\ ϱ_{2} \end{matrix} \\ \begin{matrix} ϱ_{3} \\ ϱ_{4} \end{matrix} \end{matrix}] = [\begin{matrix} \begin{matrix} {\bar{U}}_{x, x} \\ {\bar{U}}_{x, y} \end{matrix} \\ \begin{matrix} {\bar{Γ}}_{x, x} \\ {\bar{Γ}}_{x, y} \end{matrix} \end{matrix}] + [\begin{matrix} \begin{matrix} {\bar{U}}_{y, y} \\ - {\bar{U}}_{y, x} \end{matrix} \\ \begin{matrix} {\bar{Γ}}_{y, y} \\ - {\bar{Γ}}_{y, x} \end{matrix} \end{matrix}]

(24)

To solve the free vibration problem, the unknown functions are considered in the following form

ϱ_{k} (x, y, t) = \sum_{m = 1}^{\infty} ϱ_{k m} (x, y) e^{i ω_{m} t}, k = 1, 2, 3, 4

(25a)

{\begin{matrix} φ (x, y, t) \\ {\bar{U}}_{z} (x, y, t) \end{matrix}} = \sum_{m = 1}^{\infty} {\begin{matrix} φ_{m} (x, y) \\ {\bar{U}}_{z m} (x, y) \end{matrix}} e^{i ω_{m} t}

(25b)

Here $ω_{m}$ is the eigenfrequency of the hybrid plate and $i = \sqrt{- 1}$ . The time dependency of equation (23) can be eliminated with the help of equation (25). Since the first four equations of equation (23) contain only four unknowns and the last two contain the remaining ones, it is possible to perform some mathematical manipulation to decouple the equations. Doing some hierarchical operation to find each parameter in term of another one, is the key point to the decoupling procedure. This way we are able to achieve to two decoupled equations, one for the first category (the first four equations) and the other one for the second category (the last two equations), as

ξ_{1} \nabla^{8} {\bar{U}}_{z m} + ξ_{2} \nabla^{6} {\bar{U}}_{z m} + ξ_{3} \nabla^{4} {\bar{U}}_{z m} + ξ_{4} \nabla^{2} {\bar{U}}_{z m} + ξ_{5} {\bar{U}}_{z m} = 0

(26a)

{\bar{ξ}}_{1} \nabla^{4} ϱ_{4 m} + {\bar{ξ}}_{2} \nabla^{2} ϱ_{4 m} + {\bar{ξ}}_{3} ϱ_{4 m} = 0

(26b)

The unknown parameters may be easily found with the help of above equations as

ϱ_{1 m} = Z_{5} \nabla^{2} ϱ_{3 m} + Z_{6} ϱ_{3 m} + Z_{7} \nabla^{2} {\bar{U}}_{z m} + Z_{8} {\bar{U}}_{z m}

(27a)

ϱ_{2 m} = {\bar{ξ}}_{4} \nabla^{2} ϱ_{4 m} + {\bar{ξ}}_{5} ϱ_{4 m}

(27b)

ϱ_{3 m} = ξ_{6} \nabla^{6} {\bar{U}}_{z m} + ξ_{7} \nabla^{4} {\bar{U}}_{z m} + ξ_{8} \nabla^{2} {\bar{U}}_{z m} + ξ_{9} {\bar{U}}_{z m}

(27c)

φ_{m} = Z_{9} ϱ_{3 m} + Z_{10} \nabla^{2} ϱ_{3 m} + Z_{11} \nabla^{2} {\bar{U}}_{z m} + Z_{12} {\bar{U}}_{z m}

(27d)

There is a decoupled partial differential equation in each set of equations. As shown above, the other unknown functions are obtained by using these two decoupled equations. The coefficients in equations (26) and (27) are given in relation (42) of Appendix 1.

Levy-type solution

Levy-type boundary conditions are considered here. For a plate with simply supported edges at $x = 0$ and $x = a$ , the following expressions for transverse displacement ${\bar{U}}_{z m}$ and the function $ϱ_{4 m}$ should be considered

{\begin{matrix} {\bar{U}}_{z m} (x, y) \\ ϱ_{4 m} (x, y) \end{matrix}} = \sum_{j = 1}^{\infty} {\begin{matrix} {\bar{U}}_{z m j} (y) \sin (β_{j} x) \\ ϱ_{4 m j} (y) \cos (β_{j} x) \end{matrix}}

(28)

Here $β_{j} = j π / a$ . Substituting equation (28) into equations (26a) and (26b) yields

a_{1} \frac{d^{8} {\bar{U}}_{z m j}}{d y^{8}} + a_{2} \frac{d^{6} {\bar{U}}_{z m j}}{d y^{6}} + a_{3} \frac{d^{4} {\bar{U}}_{z m j}}{d y^{4}} + a_{4} \frac{d^{2} {\bar{U}}_{z m j}}{d y^{2}} + a_{5} {\bar{U}}_{z m j} = 0

(29a)

b_{1} \frac{d^{4} ϱ_{4 m j}}{d y^{4}} + b_{2} \frac{d^{2} ϱ_{4 m j}}{d y^{2}} + b_{3} ϱ_{4 m j} = 0

(29b)

The coefficients $a_{i} (i = 1, 2, \dots, 5)$ and $b_{i} (i = 1, 2, 3)$ are given in relation (43) of Appendix 1. Solution of equations (29a) and (29b) is

{\bar{U}}_{z m j} (y) = \sum_{p = 1}^{4} {C_{2 p - 1} \sinh (Θ_{p} y) + C_{2 p} \cosh (Θ_{p} y)}

(30a)

ϱ_{4 m j} (y) = \sum_{p = 5}^{6} {C_{2 p - 1} \sinh (Θ_{p} y) + C_{2 p} \cosh (Θ_{p} y)}

(30b)

where

C_{2 p}

and

C_{2 p - 1}

(

p = 1, 2, \dots, 6

) are 12 unknown parameters. Also, the parameters

Θ_{p} = \sqrt{X_{p}} (p = 1, 2, 3, 4)

and

Θ_{p} = \sqrt{Y_{q}} ((p, q) = (5, 1), (6, 2))

are the roots of the following polynomials

a_{1} X^{4} + a_{2} X^{3} + a_{3} X^{2} + a_{4} X + a_{5} = 0

(31a)

b_{1} Y^{2} + b_{2} Y + b_{3} = 0

(31b)

After finding ${\bar{U}}_{z m j}$ and $ϱ_{4 m j}$ , all other functions and displacement components could be obtained easily as

{\bar{U}}_{x m j} (y) = λ_{1} β_{j} ϱ_{1 m j} + λ_{2} \frac{d ϱ_{2 m j}}{d y} + λ_{3} β_{j} ϱ_{3 m j} + λ_{4} \frac{d ϱ_{4 m j}}{d y} + λ_{5} β_{j} {\bar{U}}_{z m j} + λ_{6} β_{j} φ_{m j}

(32a)

{\bar{U}}_{y m j} (y) = λ_{1} \frac{d ϱ_{1 m j}}{d y} - λ_{2} β_{j} ϱ_{2 m j} + λ_{3} \frac{d ϱ_{3 m j}}{d y} - λ_{4} β_{j} ϱ_{4 m j} + λ_{5} \frac{d {\bar{U}}_{z m j}}{d y} + λ_{6} \frac{d φ_{m j}}{d y}

(32b)

{\bar{Γ}}_{x m j} (y) = {\bar{λ}}_{1} β_{j} ϱ_{1 m j} + {\bar{λ}}_{2} \frac{d ϱ_{2 m j}}{d y} + {\bar{λ}}_{3} β_{j} ϱ_{3 m j} + {\bar{λ}}_{4} \frac{d ϱ_{4 m j}}{d y} + {\bar{λ}}_{5} β_{j} {\bar{U}}_{z m j} + {\bar{λ}}_{6} β_{j} φ_{m j}

(32c)

{\bar{Γ}}_{y m j} (y) = {\bar{λ}}_{1} \frac{d ϱ_{1 m j}}{d y} - {\bar{λ}}_{2} β_{j} ϱ_{2 m j} + {\bar{λ}}_{3} \frac{d ϱ_{3 m j}}{d y} - {\bar{λ}}_{4} β_{j} ϱ_{4 m j} + {\bar{λ}}_{5} \frac{d {\bar{U}}_{z m j}}{d y} + {\bar{λ}}_{6} \frac{d φ_{m j}}{d y}

(32d)

where the coefficients

λ_{i} and {\bar{λ}}_{i} (i = 1, 2, \dots, 6)

are defined in relation (44) of Appendix 1.

Boundary conditions

Provided that the plate is insulated at the edges along y direction, the electrical boundary conditions are

\int_{- h - h_{p}}^{- h} D_{y} (x, y, z, t) d z + \int_{h}^{h + h_{p}} D_{y} (x, y, z, t) d z = 0

(33)

In view of equations (7) to (10), the electrical boundary condition along $y = \pm b / 2$ can be expressed as

T_{1} ({\bar{Γ}}_{y} + {\bar{U}}_{z, y}) + {\tilde{T}}_{1} φ_{, y} + {\bar{T}}_{1} ({\bar{Γ}}_{x, x y} + {\bar{Γ}}_{y, y y}) = 0

(34)

and the coefficients

T_{1}

{\tilde{T}}_{1}

and

{\bar{T}}_{1}

are defined in relation (45) of Appendix 1.

On the other hand, mechanical boundary conditions along the remaining edges of a rectangular porous-cellular plate integrated with piezoelectric layers are given as

\begin{array}{l} Clamped: {\bar{U}}_{x} = 0, {\bar{U}}_{y} = 0, {\bar{Γ}}_{x} = 0, {\bar{Γ}}_{y} = 0, {\bar{U}}_{z} = 0 \\ Free : {\bar{\bar{N}}}_{x y} = 0, {\bar{\bar{N}}}_{y y} = 0, {\bar{\bar{M}}}_{x y} = 0, {\bar{\bar{M}}}_{y y} = 0, {\bar{\bar{H}}}_{y} = 0 \\ Simply supported : {\bar{U}}_{x} = 0, {\bar{\bar{N}}}_{y y} = 0, {\bar{Γ}}_{x} = 0, {\bar{\bar{M}}}_{y y} = 0, {\bar{U}}_{z} = 0 \end{array}

(35)

By imposing the desired electrical and mechanical boundary condition at $y = \pm b / 2$ , a system of 12 homogeneous linear algebraic equations is obtained. The eigenfrequency of the plate could be calculated by setting the determinant of the coefficient matrix equal to zero.

Numerical results

For the sake of brevity, a four-letter notation is considered in order to define the conditions at the edges of the plate. The notation SCSF, for example, indicates that the edges $x = 0$ and $x = a$ are simply supported (S), and edges $y = - b / 2$ and $y = + b / 2$ are clamped (C) and free (F), respectively. Also, the shear correction factor $K^{2}$ is set to $π^{2} / 12$ .

The results are computed for porous-cellular plates comprised of Steel Foam, Cellular Aluminum and PZT4 (for piezoelectric layers) mainly because results are usually presented for plates composed of such materials in the literature [21,22,27]. Material properties are

Steel Foam : E = 200 GPa, υ = 1 / 3, ρ = 7850 kg / m^{3}

Cellular Aluminum : E = 70 GPa, υ = 0.3, ρ = 2707 kg / m^{3}

\begin{array}{l} PZT4 : c_{11} = 132 GPa, c_{12} = 71 GPa, c_{13} = 73 GPa, c_{33} = 115 GPa, \\ c_{55} = 26 GPa, e_{33} = 14.1 \frac{C}{m^{2}}, e_{31} = - 4.1 \frac{C}{m^{2}}, e_{15} = 10.5 \frac{C}{m^{2}}, \\ Ξ_{11} = 7.124 \frac{n C}{V m}, Ξ_{22} = 6.460 \frac{n C}{V m}, Ξ_{33} = 5.841 \frac{n C}{V m}, ρ = 7500 kg / m^{3} \end{array}

Validation of the method

In order to verify the obtained frequencies, a comparison between the results of this method and those of literature [29 –31] is shown in Table 1. These articles presented exact results for free vibration of rectangular plates with different types of boundary conditions by considering three-dimensional as well as shear deformation plate theories. It can be found that the extracted natural frequencies are in excellent agreement with the available results for the homogeneous rectangular plates. Table 2 presents the calculated results of natural frequencies of a simply supported square smart plate as compared with the finite element results by He et al. [15] It can be seen that the present results are in good agreement with those of He et al. [15] Moreover, the frequencies presented in the table indicates that they are more consistent with the ones calculated by Rouzegar and Abad [32] due to the type of plate theory. The accuracy of this approach is further investigated by performing a comparison study between its results for Levy-type boundary conditions with those obtained by using Reddy’s third-order shear deformation plate theory [27] in Table 3. A good consistency has been observed between the results.

Table 1.

Comparison of the first three non-dimensional natural frequencies, ( $\bar{ω} = ω a^{2} \sqrt{2 h ρ / D}$ ) of the Levy-type homogenous isotropic plates (2h/a = 0.1, a/b = 1).

BC’s	Method	First Mode	Second Mode	Third Mode
SFSF	Liew et al. [29]	9.4462	15.3995	33.9129
	Hashemi and Arsanjani [30]	9.4458	15.4054	33.9160
	Hosseini-Hashemi et al. [31]	9.4417	15.3915	33.8675
	Present	9.4388	15.3838	33.8408
SSSF	Hashemi and Arsanjani [30]	11.3810	26.1910	38.3610
	Hosseini-Hashemi et al. [31]	11.3737	26.1576	38.2904
	Present	11.3704	26.1420	38.2605
SFSC	Hashemi and Arsanjani [30]	12.2606	30.4743	38.7128
	Hosseini-Hashemi et al. [31]	12.2519	30.4372	38.6425
	Present	12.2454	30.3862	38.6074
SSSS	Liew et al. [29]	19.0898	45.6193	64.3834
	Hashemi and Arsanjani [30]	19.0840	45.5845	–
	Hosseini-Hashemi et al. [31]	19.0653	45.4869	–
	Present	19.0584	45.4478	69.7167
SSSC	Hashemi and Arsanjani [30]	22.426	47.2245	52.3247
	Hosseini-Hashemi et al. [31]	22.4018	47.1306	52.2324
	Present	22.3759	47.0628	52.0899
SCSC	Liew et al. [29]	26.8335	49.3589	59.7328
	Hashemi and Arsanjani [30]	26.7369	49.2606	59.4801
	Hosseini-Hashemi et al. [31]	26.7084	49.1756	59.4159
	Present	26.6449	49.0625	59.1183

Table 2.

Comparison of the first five natural frequencies (Hz) for a simply supported smart plate (2h/a = 0.0125, h_p/2h = 0.02, a/b = 1).

Method	First Mode	Second Mode	Third Mode	Fourth Mode	Fifth Mode
He et al. [15]	144.25	359.00	359.00	576.92	720.72
Rouzegar and Abad [32]	145.35	363.06	363.06	580.37	725.03
Present	144.94	362.01	362.01	578.69	722.92

Table 3.

Comparison of the fundamental natural frequency (Hz) of porous smart plates with Levi-type boundary conditions (2h/a = 0.1, h_p/2h = 0.05, a/b = 1).

BC’s	EC’s	Method	e₀
BC’s	EC’s	Method	0	0.25	0.5
SFSF	Closed	TSDT [27]	238.329	232.453	225.363
	Closed	FSDT (Present)	238.163	232.337	225.322
	Open	TSDT [27]	243.905	238.523	232.160
	Open	FSDT (Present)	243.642	238.305	232.014
SSSS	Closed	TSDT [27]	478.750	466.490	451.728
	Closed	FSDT (Present)	478.412	466.337	451.855
	Open	TSDT [27]	494.566	483.746	471.124
	Open	FSDT (Present)	493.833	483.176	470.823
SCSC	Closed	TSDT [27]	664.835	646.294	624.228
	Closed	FSDT (Present)	662.365	644.410	623.239
	Open	TSDT [27]	685.155	668.293	648.710
	Open	FSDT (Present)	681.186	664.823	646.078
SSSF	Closed	TSDT [27]	286.863	279.724	271.119
	Closed	FSDT (Present)	286.546	279.481	270.989
	Open	TSDT [27]	292.582	285.942	278.072
	Open	FSDT (Present)	292.130	285.557	277.794
SFSC	Closed	TSDT [27]	308.659	300.874	291.510
	Closed	FSDT (Present)	308.147	300.465	291.256
	Open	TSDT [27]	314.121	306.805	298.129
	Open	FSDT (Present)	313.419	306.195	297.666
SSSC	Closed	TSDT [27]	560.379	545.473	527.617
	Closed	FSDT (Present)	559.304	544.734	527.390
	Open	TSDT [27]	578.293	564.953	549.419
	Open	FSDT (Present)	576.541	563.506	548.478

Numerical examples and discussion

Based on the presented formulation, comprehensive results of the first three natural frequencies of porous-cellular square plates with $h_{p} / 2 h = 0.1$ under asymmetric boundary conditions for different porosity distributions and both electrical boundary conditions are listed in Tables 4 to 6. In each table, the core thickness–length ratio $2 h / a$ is taken to be 0.05 and 0.1, while three different values of porosity coefficient $e_{0} = 0.2, 0.5, 0.8$ are considered. As expected, the results show that the sequence of modes is highly dependent upon the type of boundary condition. The effect of porosity distribution on the natural frequencies is very interesting. It is found that if the pores are symmetrically distributed, the increase of porosity coefficient results in increment of the natural frequencies. The situation is different if the pores are distributed otherwise. It is noticeable from these tables that plates with symmetrically distributed pores possess the largest frequencies followed by asymmetric and uniform porosity distributions, respectively.

Table 4.

The first three natural frequencies (Hz) of a SSSF porous-cellular smart plate with different porosity distributions (a/b = 1, h_p/2h = 0.1, Steel foam – PZT4).

2h/a	EC’s	e₀	First Mode (1,1)	Second Mode (1,2)	Third Mode (2,1)
Symmetric Porosity Distribution
0.05	Closed	0.2	146.242	343.452	508.428
		0.5	147.421	345.729	511.400
		0.8	152.869	357.676	528.368
	Open	0.2	148.779	350.948	519.481
		0.5	150.210	353.974	523.475
		0.8	156.044	367.059	541.968
0.1	Closed	0.2	285.337	651.485	948.6430
		0.5	286.875	652.378	947.365
		0.8	296.162	669.107	967.396
	Open	0.2	289.918	664.680	966.194
		0.5	291.854	666.600	966.055
		0.8	301.735	684.970	987.628
Asymmetric Porosity Distribution
0.05	Closed	0.2	143.383	336.838	498.737
		0.5	137.944	323.838	479.363
		0.8	130.188	305.384	451.973
	Open	0.2	145.975	344.505	510.054
		0.5	140.964	332.795	492.530
		0.8	134.109	317.098	469.094
0.1	Closed	0.2	279.960	639.833	932.389
		0.5	269.121	614.066	894.186
		0.8	253.902	578.455	842.318
	Open	0.2	284.654	653.373	950.486
		0.5	274.571	629.819	915.085
		0.8	260.972	599.053	869.457
Uniform Porosity Distribution
0.05	Closed	0.2	142.780	335.437	496.676
		0.5	136.661	320.817	474.877
		0.8	128.893	302.115	446.882
	Open	0.2	145.380	343.128	508.031
		0.5	139.675	329.758	488.013
		0.8	132.656	313.335	463.212
0.1	Closed	0.2	278.814	637.314	928.820
		0.5	266.603	608.304	885.670
		0.8	250.922	570.407	828.718
	Open	0.2	283.524	650.905	946.989
		0.5	272.039	624.026	906.479
		0.8	257.644	589.819	854.022

Table 5.

The first three natural frequencies (Hz) of a SFSC porous-cellular smart plate with different porosity distributions (a/b = 1, h_p/2h = 0.1, Steel foam – PZT4).

2h/a	EC’s	e₀	First Mode (1,1)	First Mode (1,2)	Third Mode (2,1)
Symmetric Porosity Distribution
0.05	Closed	0.2	158.077	405.560	513.752
		0.5	159.300	407.761	516.687
		0.8	165.097	421.023	533.716
	Open	0.2	160.526	414.601	524.580
		0.5	161.988	417.659	528.510
		0.8	168.149	432.203	547.020
0.1	Closed	0.2	306.373	752.462	956.333
		0.5	307.717	750.886	954.762
		0.8	317.168	765.920	974.509
	Open	0.2	310.697	767.320	973.432
		0.5	312.385	766.519	972.926
		0.8	322.377	783.171	994.182
Asymmetric Porosity Distribution
0.05	Closed	0.2	155.006	397.876	503.983
		0.5	149.127	382.379	484.404
		0.8	140.775	360.542	456.761
	Open	0.2	157.507	407.104	515.067
		0.5	152.038	393.147	497.297
		0.8	144.554	374.623	473.521
0.1	Closed	0.2	300.694	739.724	940.037
		0.5	288.999	709.240	901.471
		0.8	272.713	668.044	849.237
	Open	0.2	305.109	754.819	957.650
		0.5	294.118	726.738	921.805
		0.8	279.351	690.924	875.635
Uniform Porosity Distribution
0.05	Closed	0.2	154.357	396.239	501.904
		0.5	147.738	378.790	479.867
		0.8	139.328	356.343	451.557
	Open	0.2	156.868	405.525	513.030
		0.5	150.646	389.571	492.734
		0.8	142.954	369.821	467.542
0.1	Closed	0.2	299.477	736.916	936.453
		0.5	286.280	702.460	892.872
		0.8	269.271	656.863	835.302
	Open	0.2	303.926	752.257	954.154
		0.5	291.412	720.163	913.144
		0.8	275.581	678.324	859.913

Table 6.

The first three natural frequencies (Hz) of a SSSC porous-cellular smart plate with different porosity distributions (a/b = 1, h_p/2h = 0.1, Steel foam – PZT4).

2h/a	EC’s	e₀	First Mode (1,1)	Second Mode (2,1)	Third Mode (1,2)
Symmetric Porosity Distribution
0.05	Closed	0.2	295.221	635.522	714.355
		0.5	297.114	638.522	716.839
		0.8	307.285	658.525	737.795
	Open	0.2	303.662	653.218	733.834
		0.5	306.371	657.829	738.049
		0.8	317.808	680.286	761.304
0.1	Closed	0.2	559.710	1164.319	1280.408
		0.5	559.897	1159.447	1270.798
		0.8	573.359	1178.761	1285.289
	Open	0.2	574.164	1191.966	1309.533
		0.5	575.464	1188.802	1298.969
		0.8	590.575	1210.490	1316.581
Asymmetric Porosity Distribution
0.05	Closed	0.2	289.521	623.532	701.123
		0.5	278.250	598.937	673.253
		0.8	262.210	564.253	634.289
	Open	0.2	298.1730	641.772	721.230
		0.5	288.354	620.208	696.674
		0.8	275.420	592.055	664.894
0.1	Closed	0.2	549.826	1145.175	1260.591
		0.5	527.495	1097.048	1206.595
		0.8	496.879	1032.580	1135.925
	Open	0.2	564.826	1174.438	1290.650
		0.5	544.918	1130.929	1241.274
		0.8	519.638	1076.850	1181.231
Uniform Porosity Distribution
0.05	Closed	0.2	288.312	620.973	698.280
		0.5	275.637	593.298	666.869
		0.8	259.365	557.521	626.054
	Open	0.2	296.974	639.162	718.327
		0.5	285.703	614.389	690.077
		0.8	271.965	583.805	654.928
0.1	Closed	0.2	547.669	1140.906	1256.066
		0.5	522.430	1086.438	1194.675
		0.8	489.247	1013.747	1111.876
	Open	0.2	562.588	1169.571	1286.100
		0.5	539.638	1119.309	1229.462
		0.8	510.459	1053.782	1151.581

For illustrative purposes, the effect of porosity coefficient on the natural frequency of a square plate with $2 h / a = 0.1$ , $h_{p} / 2 h = 0.1$ and various porosity distributions is demonstrated in Figure 2. Again, it is seen that by increasing $e_{0}$ , the natural frequency of plates with uniformly and asymmetrically distributed pores decreases; however, the opposite trends for variation of $ω$ against $e_{0}$ is observed for plates with symmetric porosity distribution regardless of the absence/presence of piezoelectric layers. Such behavior is due to the fact that the effect of mass density of host plate overcomes that of flexural rigidity when the pores are symmetrically distributed about the mid-plane of the core plate. Besides, one can see that the difference between the magnitudes of $ω$ increases as higher restraining boundary condition is applied at the other two remaining edges.

Variation of natural frequency of porous-cellular square plates with $e_{0} = 0.6$ versus core thickness–length ratio for various porosity distributions is depicted in Figure 3. The figure gives the information for Levy-type boundary conditions. For all studied porosity distributions and boundary conditions, natural frequency increases while the core thickness–length ratio is increasing. The incremental effect of $2 h / a$ on the natural frequency is the most for plates with symmetric porosity distribution and the least for uniform distribution while other parameters are kept constant.

To quantify the influence of the piezoelectric layers, a new parameter called the relative difference $β$ is defined as

β = 100 * (ω_{p p} - ω_{p}) / ω_{p}

Here $ω_{p}$ denotes the natural frequency of the core plate and $ω_{p p}$ is the fundamental frequency of the hybrid plate. In order to study the mechanical and electrical effect of piezoelectric layers, the changes in $β$ of hybrid square plates under symmetric boundary conditions with $2 h / a = 0.05$ and $e_{0} = 0.4$ due to the variation of $h_{p} / 2 h$ are presented in Figure 4. From the figure, it is found that the electrical effect of piezoelectric layers is almost negligible in case of closed circuit condition, whereas this effect plays a key role in natural frequency increment for the similar plate with open circuit piezoelectric layers. This can be attributed to the considered function for electric potential distribution for both electrical boundary conditions. This is to be expected because unlike open circuit condition, the electric discharge occurs in case of closed circuit piezoelectric layers which causes the plate to deform more easily. The same effects have been investigated in Figure 5 for the first three vibrational modes of a smart plate. The figures show the similar trends with respect to the previous figure indicating that piezo-effect is pretty much the same for different modes in such plates. For the porous-cellular square plate with $2 h / a = 0.15$ , the variation of fundamental natural frequency versus porosity coefficient for two different materials is plotted in Figure 6. The frequencies corresponding to cellular aluminum are higher than those of the steel foam. This figure shows the trends for the both materials are similar to each other. It is to be noted that the presented results are in compliance with those shown in Tables 4 to 6 and Figure 2.

The effect of thickness ratio $h_{p} / 2 h$ on the relative difference of a moderately thick porous-cellular square plate with open circuit piezoelectric layers is shown in Figure 7(a) and (b) by considering different porosity distributions and vibrational modes for two different mechanical boundary conditions ( $2 h / a = 0.1$ and $e_{0} = 0.5$ ). It is observed that the presence of piezoelectric layers causes the natural frequency to increase irrespective of boundary conditions (electrical/mechanical) and the type of porosity distribution. However, it is to be noted that the growth in the fundamental mode is more prominent as compared with other modes. The fundamental natural frequency as well as the relative difference $β$ of hybrid plates of different aspect ratio $a / b$ , thickness–length ratio $2 h / a$ , and thickness ratio $h_{p} / 2 h$ , various porosity distributions, and electrical/mechanical boundary conditions are listed in Tables 7 to 9. The tables show that the relative difference $β$ is larger in case of plates with softer boundary conditions meaning that the effect of piezoelectric layers on natural frequency growth is more considerable when less constraints are applied at plates’ edges. Furthermore, the listed results imply that $β$ decreases as the aspect ratio increases for all studied boundary conditions except for SFSF plates. In fact, the plates get softer when two parallel edges are free. Moreover, it can be deduced that the value of $β$ is the largest for plates with uniformly distributed pores followed by asymmetric and symmetric porosity distributions for all considered geometrical parameters as well as electrical and mechanical boundary conditions. Once again, one can deduce that the relative difference $β$ corresponding to open circuit condition is larger than that of plates with closed-circuit piezoelectric layers due to the cited reason for Figure 4.

Table 7.

The effect of piezoelectric layers on the natural frequency (Hz) of plates under SFSF and SSSF boundary conditions for both closed and open circuit conditions (e₀=0.3, Steel foam – PZT4).

				Natural Frequency (Hz) ( $β$ )
Porosity Distribution				Symm.	Asym.	Unif.	Symm.	Asym.	Unif.
BC’s	a/b	2h/a	h_p/2h	Closed			Open
SFSF	1	0.05	0.1	121.288	117.398	116.671	123.795	119.997	119.279
			0.1	(4.05)	(5.24)	(5.41)	(6.20)	(7.57)	(7.77)
			0.2	129.083	126.037	125.432	133.698	130.772	130.179
			0.2	(10.74)	(12.99)	(13.33)	(14.69)	(17.23)	(17.62)
		0.1	0.1	237.792	230.393	228.990	242.421	235.209	233.823
			0.1	(3.67)	(4.85)	(5.01)	(5.69)	(7.04)	(7.23)
			0.2	251.903	246.174	245.012	260.225	254.748	253.604
			0.2	(9.82)	(12.03)	(12.36)	(13.45)	(15.93)	(16.30)
	2	0.05	0.1	119.557	115.729	115.014	121.583	117.829	117.121
			0.1	(4.15)	(5.35)	(5.53)	(5.91)	(7.26)	(7.46)
			0.2	127.332	124.337	123.742	131.045	128.145	127.559
			0.2	(10.92)	(13.19)	(13.53)	(14.16)	(16.65)	(17.04)
		0.1	0.1	234.504	227.212	225.829	238.244	231.102	229.733
			0.1	(3.78)	(4.96)	(5.13)	(5.43)	(6.76)	(6.95)
			0.2	248.623	242.978	241.834	255.319	249.876	248.744
			0.2	(10.02)	(12.24)	(12.58)	(12.99)	(15.43)	(15.80)
SSSF	1	0.05	0.1	146.411	141.741	140.866	149.023	144.450	143.585
			0.1	(4.07)	(5.27)	(5.44)	(5.93)	(7.28)	(7.48)
			0.2	155.815	152.165	151.438	160.606	157.084	156.369
			0.2	(10.76)	(13.01)	(13.36)	(14.16)	(16.67)	(17.05)
		0.1	0.1	285.449	276.676	275.003	290.152	281.577	279.922
			0.1	(3.60)	(4.78)	(4.95)	(5.31)	(6.63)	(6.82)
			0.2	302.057	295.293	293.910	310.426	303.934	302.563
			0.2	(9.63)	(11.83)	(12.16)	(12.67)	(15.10)	(15.47)
	2	0.05	0.1	199.048	192.799	191.620	200.867	194.688	193.516
			0.1	(4.17)	(5.38)	(5.55)	(5.12)	(6.41)	(6.60)
			0.2	211.843	206.984	206.007	215.130	210.366	209.395
			0.2	(10.87)	(13.13)	(13.48)	(12.59)	(14.98)	(15.34)
		0.1	0.1	382.570	371.161	368.957	385.595	374.333	372.137
			0.1	(3.47)	(4.640)	(4.81)	(4.29)	(5.54)	(5.71)
			0.2	403.823	395.110	393.297	409.010	400.526	398.682
			0.2	(9.22)	(11.39)	(11.73)	(10.62)	(12.92)	(13.26)

Table 8.

The effect of piezoelectric layers on the natural frequency (Hz) of plates under SFSC and SSSS boundary conditions for both closed and open circuit conditions (e₀=0.3, Steel foam – PZT4).

				Natural Frequency (Hz) ( $β$ )
Porosity Distribution				Symm.	Asym.	Unif.	Symm.	Asym.	Unif.
BC’s	a/b	2h/a	h_p/2h	Closed			Open
SFSC	1	0.05	0.1	158.245	153.230	152.287	160.766	155.843	154.912
			0.1	(4.07)	(5.27)	(5.44)	(5.73)	(7.07)	(7.26)
			0.2	168.358	164.446	163.664	172.968	169.168	168.409
			0.2	(10.72)	(12.98)	(13.32)	(13.75)	(16.22)	(16.60)
		0.1	0.1	306.405	297.145	295.361	310.845	301.752	300.006
			0.1	(3.50)	(4.67)	(4.83)	(5.00)	(6.29)	(6.48)
			0.2	323.745	316.640	315.168	331.563	324.660	323.262
			0.2	(9.35)	(11.53)	(11.86)	(12.00)	(14.36)	(14.74)
	2	0.05	0.1	278.778	270.177	268.537	282.353	273.861	272.265
			0.1	(3.86)	(5.05)	(5.22)	(5.20)	(6.49)	(6.69)
			0.2	295.749	289.101	287.744	302.250	295.689	294.458
			0.2	(10.19)	(12.41)	(12.75)	(12.61)	(14.97)	(15.38)
		0.1	0.1	521.981	507.321	504.383	527.630	513.121	510.352
			0.1	(2.78)	(3.90)	(4.06)	(3.89)	(5.09)	(5.30)
			0.2	546.785	535.775	533.366	556.369	545.352	543.346
			0.2	(7.67)	(9.73)	(10.04)	(9.55)	(11.69)	(12.10)
SSSS	1	0.05	0.1	248.512	240.573	239.085	255.939	248.277	246.817
			0.1	(3.69)	(4.86)	(5.03)	(6.79)	(8.22)	(8.43)
			0.2	263.654	257.449	256.212	277.425	271.607	270.401
			0.2	(10.01)	(12.22)	(12.55)	(15.75)	(18.39)	(18.79)
		0.1	0.1	479.418	464.945	462.154	492.823	478.951	476.197
			0.1	(2.99)	(4.12)	(4.29)	(5.87)	(7.26)	(7.45)
			0.2	504.354	493.266	490.964	528.281	518.147	515.749
			0.2	(8.34)	(10.46)	(10.79)	(13.48)	(16.04)	(16.38)
	2	0.05	0.1	609.916	590.990	587.391	627.523	609.336	605.783
			0.1	(3.32)	(4.48)	(4.64)	(6.31)	(7.72)	(7.92)
			0.2	644.214	629.578	626.599	676.178	662.717	659.622
			0.2	(9.13)	(11.30)	(11.63)	(14.55)	(17.16)	(17.51)
		0.1	0.1	1125.123	1094.291	1088.031	1152.913	1123.992	1117.434
			0.1	(1.91)	(2.97)	(3.12)	(4.42)	(5.76)	(5.91)
			0.2	1169.017	1146.042	1140.923	1213.795	1196.353	1187.498
			0.2	(5.88)	(7.84)	(8.14)	(9.94)	(12.57)	(12.55)

Table 9.

The effect of piezoelectric layers on the natural frequency (Hz) of plates under SSSC and SCSC boundary conditions for both closed and open circuit conditions (e₀=0.3, Steel foam – PZT4).

				Natural Frequency (Hz) ( $β$ )
Porosity Distribution				Symm.	Asym.	Unif.	Symm.	Asym.	Unif.
BC’s	a/b	2h/a	h_p/2h	Closed			Open
SSSC	1	0.05	0.1	295.422	286.115	284.357	304.117	295.165	293.419
			0.1	(3.56)	(4.72)	(4.89)	(6.60)	(8.03)	(8.23)
			0.2	312.895	305.657	304.197	328.909	322.230	320.717
			0.2	(9.68)	(11.87)	(12.21)	(15.30)	(17.94)	(18.30)
		0.1	0.1	559.114	543.024	539.828	573.923	558.682	555.404
			0.1	(2.58)	(3.69)	(3.85)	(5.30)	(6.68)	(6.84)
			0.2	585.271	573.116	570.487	611.087	600.633	597.319
			0.2	(7.38)	(9.43)	(9.74)	(12.12)	(14.69)	(14.91)
	2	0.05	0.1	829.957	805.679	800.892	852.395	829.276	824.457
			0.1	(2.81)	(3.93)	(4.09)	(5.59)	(6.97)	(7.15)
			0.2	870.989	852.568	848.623	910.567	894.275	889.706
			0.2	(7.89)	(9.97)	(10.29)	(12.79)	(15.35)	(15.63)
		0.1	0.1	1432.753	1399.957	1392.460	1461.275	1431.384	1423.098
			0.1	(0.68)	(1.63)	(1.77)	(2.68)	(3.91)	(4.01)
			0.2	1467.151	1443.725	1437.615	1507.453	1493.177	1479.716
			0.2	(3.10)	(4.81)	(5.07)	(5.93)	(8.40)	(8.14)
SCSC	1	0.05	0.1	357.784	346.739	344.626	368.092	357.397	355.382
			0.1	(3.37)	(4.53)	(4.69)	(6.35)	(7.74)	(7.96)
			0.2	378.071	369.540	367.789	396.865	388.710	387.212
			0.2	(9.24)	(11.40)	(11.73)	(14.67)	(17.18)	(17.63)
		0.1	0.1	660.008	642.239	638.556	676.192	659.052	655.688
			0.1	(2.06)	(3.13)	(3.28)	(4.56)	(5.83)	(6.05)
			0.2	686.544	673.363	670.343	713.889	701.361	698.883
			0.2	(6.17)	(8.13)	(8.42)	(10.39)	(12.62)	(13.04)
	2	0.05	0.1	1096.922	1067.113	1060.954	1124.209	1095.359	1089.823
			0.1	(2.22)	(3.30)	(3.46)	(4.77)	(6.04)	(6.27)
			0.2	1142.928	1120.802	1115.745	1189.466	1167.969	1164.306
			0.2	(6.51)	(8.50)	(8.80)	(10.85)	(13.07)	(13.54)
		0.1	0.1	1770.357	1737.363	1728.633	1797.626	1766.428	1758.387
			0.1	(−0.40)	(0.43)	(0.55)	(1.13)	(2.11)	(2.28)
			0.2	1790.366	1767.748	1760.615	1821.283	1807.736	1792.678
			0.2	(0.72)	(2.18)	(2.41)	(2.46)	(4.49)	(4.27)

Conclusion

The linear free vibration characteristics of porous-cellular rectangular plates integrated with piezoelectric layers has been examined via some boundary layer functions. Mindlin plate theory has been applied to study Levy-type boundary conditions. The material properties have been assumed to vary gradually through the plate thickness according to three different functions. Numerical results show that trend for variation of natural frequency is highly dependent upon the type of porosity distribution. It is found that porosity variations increase the natural frequency in case of symmetric porosity distribution while there is decrease in the fundamental frequency when pores are uniformly/asymmetrically distributed. Furthermore, for similar plates with equal masses, it is observed that plates with symmetrically distributed pores possess the largest frequencies followed by asymmetric and uniform porosity distributions, respectively. Moreover, the growth in natural frequency (due to adding piezoelectric layers) of plates with uniformly and asymmetrically distributed pores is more significant as compared with the corresponding results for symmetric porosity distribution. Also, the natural frequencies of plates with open circuit piezoelectric layers are larger than the corresponding ones for closed circuit condition. In addition, neglecting the electrical effect of piezoelectric layers can give inaccurate for plates with open circuit condition unlike the other electrical condition.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

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