Nonlocal strain gradient theory for the magneto-electro-elastic vibration response of a porous FG-core sandwich nanoplate with piezomagnetic face sheets resting on an elastic foundation

Abstract

The free vibration analysis of a nonlocal strain gradient elastic sandwich nanoplate with porous graded core and piezomagnetic face sheets is presented in this paper. The rectangular elastic sandwich nanoplate is resting on Pasternak's foundation. Porosities are distributed evenly and unevenly through the thickness of the core. The gradation of material properties having porosities is described using a modified power-law function. A nonlocal parameter and a strain gradient parameter are employed to describe both stiffness reduction and stiffness enhancement of nanoplates. The governing equations of the motion are derived from Hamilton’s principle based on the first order shear deformation theory. In addition, Eringen’s nonlocal strain gradient piezo-magneto-elasticity theory is used to consider nanoscale effects. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually, the natural frequency is surveyed for different side length ratios, nonlocal coefficient, porosity volume fraction, and parameters of foundation numerically with even and uneven porosity distributions.

Keywords

Vibration response sandwich nanoplate piezo-magneto-elasticity porous graded core nonlocal strain gradient theory first order shear deformation theory

Introduction

New theories have been developed to take into account nano- and microscale effects in small structures, including the strain gradient theory, nonlocal theory, nonlocal strain gradient theory, and coupled stress theory. The present research employed the nonlocal strain gradient elasticity theory to investigate the vibration response of a graded sandwich nanoplate based on the first order shear deformation theory (FSDT). A comprehensive literature survey is presented as:

The porous materials are combined of two elements as solid (body) and liquid or gas that wood, stone, sponge, … are examples of these materials in the nature. One can conclude that the porous media are appeared in both natural and synthetic forms such as wood, stone, cement, and ceramic. A literature review on the scientific reports indicate that the concept of porous media is used in many areas of applied science and engineering such as filtration, acoustics, geo-mechanics, soil mechanics, rock mechanics, petroleum engineering, hydrogeology, and biology. Addition of piezoelectric layers to structures made from porous materials leads to a nano-electro-mechanical system that is applicable in nano-engineering as a feedback control system.

Studies on the vibration response of porous FG structures, especially for beams, are still limited in number. For porous plates, the linear and nonlinear dynamic stability of a circular porous plate has been investigated to determine the critical loads in two distinct studies by Magnucka-Blandzi [1]. In another study, she also presented the problem of axi-symmetrical deflection and buckling of circular porous plates [2]. Over the past few years, the static and dynamic properties of sandwich porous beams and plates have been extensively investigated by various theoretical and experimental studies [3 –6]. Chen et al. [7 –9] obtained the numerical results of buckling, bending, linear, and nonlinear vibrations of FG porous beams made of open-cell metal foams and compared the influences from different porosity distributions. Magnucka-Blandzi [10] proposed the mathematical modeling of a simply supported rectangular sandwich porous plate with differential equations formulated by using the principle of stationary of the total potential energy. Arefi and Rahimi [11] studied the electro-elastic analysis of functionally graded piezoelectric cylinder with various boundary conditions based on FSDT and principle of minimum potential energy.

The nonlocal elasticity theory was used as proposed and developed by Eringen [12,13]; nonlocal theory of Eringen is based on the assumption that the stress at a point is considered as a function of the strain field at all neighbor points in the continuum body. The inter-atomic forces and atomic length scales directly come to the constitutive equations as material parameters. Moreover, in recent years, a few works are conducted to examine nanostructures based on the strain gradient theory [14 –19]. According to the strain gradient theory, the strain energy is a function of strain, strain gradients, and material length scale parameter. The nonlocal strain gradient calibration of nanostructures via experiments and molecular dynamic simulation shows that their mechanical characteristics can be described using two scale parameters [20 –22]. In fact, these two scale parameters consider the stiffness-softening and stiffness-hardening effects due to the nonlocal stress field and strain gradients on the mechanical behavior of nanostructures. Based on the nonlocal strain gradient theory, Li and Hu [23] examined the post-buckling analysis of size-dependent beams. Li et al. [24] studied the size-scaled effect on the wave propagation in functionally graded beams via the nonlocal strain gradient theory. Arefi and Khoshgoftar [25] studied the influence of the in-homogeneous index of the piezoelectric material and mechanical and electrical boundary conditions on the electro-elastic results of functionally graded piezoelectric spherical shell. The influence of electric and magnetic fields on the responses of micro- and nanobeams and plates was studied by Arefi and Zenkour [26 –30]. Barati [31] studied the vibration of porous FG nanoshells with even and uneven porosity distributions using the nonlocal strain gradient elasticity. It was observed that increasing the nonlocal parameter results in reduction in the vibration frequencies. However, an inverse trend was observed when considering strain gradient effects. An increase in the porosity volume fraction gave smaller natural frequencies. However, the uneven porosity distribution provided larger frequencies compared with the even porosity distribution. Increasing the length-to-thickness and radius-to-thickness ratios led to a more flexible nanoshell and smaller frequencies, while increasing the foundation coefficients gave larger frequencies. Ghorbanpour Arani and Zamani [32] presented investigation of the electric field effect on a size-dependent bending analysis of a functionally graded porous shear and normal deformable sandwich nanoplate on silica aerogel foundation. They reported the following results as: the porosity index has an important effect on dimensionless deflection and stresses of a sandwich nanoplate. The numerical results reveal that with the increase of porosity index, dimensionless deflection and value of dimensionless normal and shear stresses are increased. The investigation on the effect of the plate aspect ratio indicates dimensionless deflection of a sandwich nanoplate is increased significantly as this ratio increases. The influence of applied voltage is considerable in the numerical results. The higher applied voltage provides larger dimensionless normal strain in z-direction and deflection in an FG porous sandwich nanoplate. Ait Atmane et al. [33] used an efficient beam theory to the study bending, free vibration, and buckling analysis of porous FG beams on the elastic foundations. Literature search in the area of the vibration analysis of FG porous beams indicated that there is no report which considered the thermal environment effects on the vibration characteristics of porous FG beams and the material properties were assumed temperature independent. While one of the most important features of FGMs is thermal insulations so there is scientific need to be familiar with the thermo-mechanical behavior of FG porous structures subjected to thermal loadings. Shafiei et al. [34] investigated the size-dependent nonlinear vibration of porous and imperfect FG tapered microbeams. They declared that in low amounts of the porosity volume fractions, the normalized frequencies for even and uneven distributions of porosities overlap. Rahmat Talabi and Saidi [35] studied the free vibration analysis of a three-layer functionally graded circular/annular plate including an FG core and two piezoelectric layers based on Reddy’s plate theory. They discussed on the influence of the inhomogeneous index and various boundary conditions on the vibration responses of the sandwich plates. Wang et al. [36 –38] investigated the effect of the nonlocal elasticity theory and nonlinear geometric strains as well as shear stress and moment of inertia on the free vibration and wave propagation characteristics of carbon nanotubes based on various shear deformation theories.

The main purpose of this research is to present the free vibration response of a sandwich porous functionally graded nanoplate resting on Pasternak's foundation based on FSDT and nonlocal strain gradient piezo-magneto-elasticity relations. Seven governing equations of motion are derived using Hamilton's principle. The analytical solution approach is proposed for a simply supported nanoplate. The influence of important parameters of the problem such as scale parameters, ratio of side lengths, the ratio of thicknesses, and two parameters of foundation is considered on the numerical results.

Formulation

In this section, formulation of the problem is presented. Our model is a sandwich nanoplate including a porous FG-core and two piezomagnetic face sheets with lengths a and width b. The thickness of core and face sheets are considered as h and h_p (Figure 1). The material properties of core are changed based on the modified power-law along the thickness direction of nanoplate. The material properties of the core are assumed as [31]

\begin{array}{l} P (z) = (P_{c} - P_{m}) {(\frac{z}{h} + \frac{1}{2})}^{p} + P_{m} - (P_{c} + P_{m}) \frac{ξ}{2} even porosity distribution \\ P (z) = (P_{c} - P_{m}) {(\frac{z}{h} + \frac{1}{2})}^{p} + P_{m} - \frac{ξ}{2} (P_{c} + P_{m}) (1 - \frac{2 | z |}{h}) uneven porosity distribution \end{array}

(1a-b)

where $P (z)$ describes material properties such as Young’s Modulus and density along the thickness direction, $p$ is the material gradation index, $m$ and $c$ are corresponding to the material properties of the top and bottom of core layer respectively, and $ξ$ is the porosity volume fraction.

After definition of gradation of material properties, the basic governing equations can be implemented. In this paper, the nonlocal piezo-magneto-elasticity theory is used in accordance with the nonlocal strain gradient theory.

The nonlocal strain gradient theory is a well-known theory that considers small effects. Based on the nonlocal strain gradient theory, the stress field takes into account the effects of nonlocal elastic stress field and strain gradient stress field. This theory has generalized Eringen’s nonlocal elasticity theory by introducing a higher order strain tensor with nonlocality into the stored energy function. The developed theory is distinctive because Eringen’s nonlocal elasticity does not include nonlocality of higher order stresses while common strain gradient theories only consider local higher order strain gradients without nonlocal effects in global sense [22,39,40].

In the nonlocal strain gradient theory, a new stress tensor is introduced as follows:

So, the stress can be written as [31]

σ_{i j} = σ_{i j}^{(0)} - \nabla σ_{i j}^{(1)}

(2a)

where the stresses

σ_{i j}^{(0)}

and

σ_{i j}^{(1)}

are related to strain

ε_{i j}

and strain gradient

\nabla ε_{i j}

respectively by [31]

σ_{i j}^{(0)} = \int_{v} c_{i j k l} α_{0} (x, x^{'}, e_{0} a) {ε^{'}}_{k l} (x^{'}) d x^{'}

(2b)

σ_{i j}^{(1)} = l^{2} \int_{v} c_{i j k l} α_{1} (x, x^{'}, e_{1} a) \nabla {ε^{'}}_{k l} (x^{'}) d x^{'}

(2c)

where

e_{0} a

and

e_{1} a

are called lower and higher order nonlocal parameters; l denotes the length of the nanoplate that is equivalent to a = b = l;

c_{i j k l}

denotes the elastic coefficients. Also

α_{0} (x, x^{'}, e_{0} a)

and

α_{1} (x, x^{'}, e_{1} a)

denote nonlocal Kernel functions. As discussed in previous investigations, selection of appropriate forms of these functions results in the following expression of constitutive equation for the nonlocal strain gradient theory [22,31,39,40]

[1 - {(e_{1} a)}^{2} \nabla^{2}] [1 - {(e_{0} a)}^{2} \nabla^{2}] σ_{i j} = c_{i j k l} [1 - {(e_{1} a)}^{2} \nabla^{2}] ε_{k l} - c_{i j k l} l^{2} [1 - {(e_{0} a)}^{2} \nabla^{2}] \nabla^{2} ε_{k l}

(3a)

where

\nabla^{2}

denotes the Laplacian operator. For simplicity, supposing

e_{1} = e_{0} = e

in equation (3) yields the following equation

[1 - {(e a)}^{2} \nabla^{2}] σ_{i j} = c_{i j k l} [1 - l^{2} \nabla^{2}] ε_{k l}

(3b)

Based on the nonlocal strain gradient theory, electro-magneto-elastic relations, the stress, electric displacement, and magnetic induction are defined as

\begin{array}{l} [1 - {(e a)}^{2} \nabla^{2}] σ_{i j} = c_{i j k l} [1 - l^{2} \nabla^{2}] ε_{k l} - e_{k i j} E_{k} - f_{k i j} H_{k} \\ [1 - {(e a)}^{2} \nabla^{2}] D_{i} = e_{i j k l} [1 - l^{2} \nabla^{2}] ε_{k l} + κ_{i k} E_{k} + g_{i k} H_{k} \\ [1 - {(e a)}^{2} \nabla^{2}] B_{i} = f_{i k l} [1 - l^{2} \nabla^{2}] ε_{k l} + g_{i k} E_{k} + μ_{i k} H_{k} \end{array}

(4)

where

σ_{i j}

D_{i}

, and

B_{i}

are dedicated to the nonlocal stress, electric displacement, and magnetic induction, respectively. Also

c_{i j k l}

e_{k i j}

f_{k i j}

κ_{i k}

_, and

μ_{i k}

represent the elastic constant, piezoelectric, piezomagnetic, dielectric, magneto-electric, and magnetic constants correspondingly.

ε_{i j}

u_{i}

E_{i}

_, and

H_{i}

are strain, displacement, electric, and magnetic fields. Electric and magnetic potentials are described by

Φ

and

Ψ

, respectively. For the nonlocal strain gradient theory, the basic equations can be expressed as [41]

ε_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}), E_{i} = - \frac{\partial Φ}{\partial x_{i}}, H_{i} = - \frac{\partial Ψ}{\partial x_{i}}

(5)

\begin{array}{l} (1 - μ \nabla^{2}) {\begin{cases} σ_{x x} \\ σ_{y y} \\ τ_{y z} \\ τ_{x z} \\ τ_{x y} \end{cases}} = (1 - λ \nabla^{2}) [\begin{matrix} c_{11} & c_{12} & 0 & 0 & 0 \\ c_{12} & c_{22} & 0 & 0 & 0 \\ 0 & 0 & c_{44} & 0 & 0 \\ 0 & 0 & 0 & c_{55} & 0 \\ 0 & 0 & 0 & 0 & c_{66} \end{matrix}] {\begin{cases} ε_{x x} \\ ε_{y y} \\ 2 ε_{y z} \\ 2 ε_{x z} \\ 2 ε_{x y} \end{cases}} \\ - [\begin{matrix} 0 & 0 & e_{31} \\ 0 & 0 & e_{32} \\ 0 & e_{24} & 0 \\ e_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{cases} E_{x} \\ E_{y} \\ E_{z} \end{cases}} - [\begin{matrix} 0 & 0 & f_{31} \\ 0 & 0 & f_{32} \\ 0 & f_{24} & 0 \\ f_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{cases} H_{x} \\ H_{y} \\ H_{z} \end{cases}} \end{array}

(6)

\begin{matrix} (1 - μ \nabla^{2}) {\begin{cases} D_{x} \\ D_{y} \\ D_{z} \end{cases}} = (1 - λ \nabla^{2}) [\begin{matrix} 0 & 0 & e_{15} & 0 & 0 \\ 0 & 0 & e_{24} & 0 & 0 \\ e_{31} & e_{32} & e_{33} & 0 & 0 \end{matrix}] {\begin{cases} ε_{x x} \\ ε_{y y} \\ 2 ε_{y z} \\ 2 ε_{x z} \\ 2 ε_{x y} \end{cases}} \\ + [\begin{matrix} κ_{11} & 0 & 0 \\ 0 & κ_{22} & 0 \\ 0 & 0 & κ_{33} \end{matrix}] {\begin{cases} E_{x} \\ E_{y} \\ E_{z} \end{cases}} + [\begin{matrix} g_{11} & 0 & 0 \\ 0 & g_{22} & 0 \\ 0 & 0 & g_{33} \end{matrix}] {\begin{cases} H_{x} \\ H_{y} \\ H_{z} \end{cases}} \end{matrix}

(7)

\begin{matrix} (1 - μ \nabla^{2}) {\begin{cases} B_{x} \\ B_{y} \\ B_{z} \end{cases}} = (1 - λ \nabla^{2}) [\begin{matrix} 0 & 0 & f_{15} & 0 & 0 \\ 0 & 0 & f_{24} & 0 & 0 \\ f_{31} & f_{32} & f_{33} & 0 & 0 \end{matrix}] {\begin{cases} ε_{x x} \\ ε_{y y} \\ 2 ε_{y z} \\ 2 ε_{x z} \\ 2 ε_{x y} \end{cases}} \\ + [\begin{matrix} g_{11} & 0 & 0 \\ 0 & g_{22} & 0 \\ 0 & 0 & g_{33} \end{matrix}] {\begin{cases} E_{x} \\ E_{y} \\ E_{z} \end{cases}} + [\begin{matrix} μ_{11} & 0 & 0 \\ 0 & μ_{22} & 0 \\ 0 & 0 & μ_{33} \end{matrix}] {\begin{cases} H_{x} \\ H_{y} \\ H_{z} \end{cases}} \end{matrix}

(8)

In the above equations, μ = (ea)² and λ = l² are nonlocal and length scale parameters, respectively. (a) is an internal characteristic length (lattice parameter, granular size, or molecular diameters) and e0 is a constant appropriate to each material for adjusting the model to match some reliable results from experiments or other theories. The value of e is experimentally estimated by comparing the scattering curves of plane waves and atomistic dynamics.

An FSDT is utilized in this research for displacement components [42,43]

\begin{array}{l} u (x, y, z, t) = u_{0} (x, y, t) + z ϕ_{x} (x, y, t) \\ v (x, y, z, t) = v_{0} (x, y, t) + z ϕ_{y} (x, y, t) \\ w (x, y, z, t) = w_{0} (x, y, t) \end{array}

(9)

where

(u, v, w)

are displacement along the

(x, y, z)

direction and

(u_{0}, v_{0}, w_{0})

are mid-surface.

ϕ_{x}

and

ϕ_{y}

represents rotation around the

y

and

x

axis, respectively. Based on the displacement field defined in equation (9) and using linear strain–displacement relations, five in-plane strain components are derived as

\begin{array}{l} ε_{x x} = \frac{\partial u_{0}}{\partial x} + z \frac{\partial ϕ_{x}}{\partial x} \\ ε_{y y} = \frac{\partial v_{0}}{\partial y} + z \frac{\partial ϕ_{y}}{\partial y} \end{array} \begin{array}{l} ε_{y z} = \frac{1}{2} (\frac{\partial w_{0}}{\partial y} + ϕ_{y}) \\ ε_{x z} = \frac{1}{2} (\frac{\partial w_{0}}{\partial x} + ϕ_{x}) \\ ε_{x y} = \frac{1}{2} (\frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} + z \frac{\partial ϕ_{x}}{\partial y} + z \frac{\partial ϕ_{y}}{\partial x}) \end{array}

(10)

For the completion of basic relations, the electric and magnetic fields and potentials are required. Electric and magnetic potentials are assumed as a combination of a cosine function along the planar directions and a linear function along the thickness direction. The second term implies applied electric and magnetic potentials and the first term imposes homogeneous boundary conditions. The distributions of electric and magnetic potentials are assumed as [44 –46]

Φ (x, y, z, t) = - \cos (\frac{π \tilde{z}}{h_{p}}) ϕ (x, y, t) + \frac{2 \tilde{z} ϕ_{0}}{h_{p}}

(11)

Ψ (x, y, z, t) = - \cos (\frac{π \tilde{z}}{h_{p}}) ψ (x, y, t) + \frac{2 \tilde{z} ψ_{0}}{h_{p}}

(12)

where

ϕ (x, y, t)

and

ψ (x, y, t)

express planar distribution of electric and magnetic potentials and

ψ 0 and ϕ 0

are applied electric and magnetic potentials. The new variable

\tilde{z} = z \pm \frac{h}{2} \pm \frac{h_{p}}{2}

is defined for piezomagnetic face sheets. For the top layer,

\tilde{z} = \overset{b}{z} = z - \frac{h}{2} - \frac{h_{p}}{2}

and for the bottom layer,

\tilde{z} = \overset{t}{z} = z + \frac{h}{2} + \frac{h_{p}}{2}

. Based on electric and magnetic potentials defined in equations (11) and (12), the electric and magnetic fields are derived as

E_{x} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ϕ}{\partial x}, E_{y} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ϕ}{\partial y}, E_{z} = - \frac{π}{h} \sin (\frac{π \tilde{z}}{h}) ϕ - \frac{2 ϕ_{0}}{h}

(13)

H_{x} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ψ}{\partial x}, H_{y} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ψ}{\partial y}, H_{z} = - \frac{π}{h} \sin (\frac{π \tilde{z}}{h}) ψ - \frac{2 ψ_{0}}{h},

(14)

The governing equations of motion are derived by employing Hamilton’s principle as follows

δ ∏ = \int (δ K_{e} - δ U_{s} - δ W_{l}) d t

(15)

where

(K_{e}, U_{s}, W_{l})

are kinetic energy, strain energy, and energy due to foundation reaction which can be expressed as

K_{e} = \frac{1}{2} \iint_{v} \int ρ [{(\frac{\partial u x}{\partial t})}^{2} + {(\frac{\partial u y}{\partial t})}^{2} + {(\frac{\partial u z}{\partial t})}^{2}] d V

(16)

\begin{matrix} U_{s} = \frac{1}{2} \int \int \int ((σ_{x x} ε_{x x} + σ_{y y} ε_{y y} + 2 σ_{x y} ε_{x y} + 2 σ_{x z} ε_{x z} + 2 σ_{y z} ε_{y z}) \\ - (D_{x} E_{x} + D_{y} E_{y} + D_{z} E_{z}) - (B_{x} H_{x} + B_{y} H_{y} + B_{z} H_{z})) d V \end{matrix}

(17)

W_{l} = - \frac{1}{2} \int \int F (x, y) w - (N_{E} + N_{M}) [{(\frac{\partial w}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}] d A

(18)

where

F (x, y)

is transverse loading that can be calculated as [42,43]

F (x, y) = k_{w} w - k_{G} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}})

(19)

where

k_{w} and k_{G}

are Pasternak foundation parameters related to Winkler and shear constants, respectively.

And $N_{E} and N_{M}$ are in-plane applied electric and magnetic loads, respectively, and are defined as

N_{E} = - \int \frac{2 ϕ_{0}}{h_{p}} e_{31} d z, N_{M} = - \int \frac{2 ψ_{0}}{h_{p}} f_{31} d z

(20)

By substitution of variation forms of kinetic energy, strain energy, and energy due to external works into Hamilton's principle and equating the coefficients of $δ u_{0}, δ v_{0}, δ w_{0}, δ ϕ_{x}, δ ϕ_{y}, δ ϕ$ , and $δ ψ$ to zero, the following equations of motion can be obtained

\begin{array}{l} δ u : \frac{\partial N_{x x}}{\partial x} + \frac{\partial N_{x y}}{\partial y} = I_{0} \frac{\partial^{2} u}{\partial t^{2}} + I_{1} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} \\ δ v : \frac{\partial N_{y y}}{\partial y} + \frac{\partial N_{x y}}{\partial x} = I_{0} \frac{\partial^{2} v}{\partial t^{2}} + I_{1} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} \\ δ w : \frac{\partial N_{x z}}{\partial x} + \frac{\partial N_{y z}}{\partial y} - k_{w} w + k_{G} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) - (N_{E} + N_{M}) (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) = I_{0} \frac{\partial^{2} w}{\partial t^{2}} \end{array} \begin{array}{l} δ ϕ_{x} : + \frac{\partial M_{x x}}{\partial x} + \frac{\partial M_{x y}}{\partial y} - N_{x z} = I_{1} (\frac{\partial^{2} u}{\partial t^{2}}) + I_{2} (\frac{\partial^{2} ϕ_{x}}{\partial t^{2}}) \\ δ ϕ_{y} : + \frac{\partial M_{y y}}{\partial y} + \frac{\partial M_{x y}}{\partial x} - N_{y z} = I_{1} (\frac{\partial^{2} v}{\partial t^{2}}) + I_{2} (\frac{\partial^{2} ϕ_{y}}{\partial t^{2}}) \\ δ ϕ : \int_{- h / 2}^{h / 2} [\cos (\frac{π z}{h}) \frac{\partial D_{x}}{\partial x} + \cos (\frac{π z}{h}) \frac{\partial D_{y}}{\partial y} + \frac{π}{h} \sin (\frac{π z}{h}) D_{z}] d z \\ δ ψ : \int_{- h / 2}^{h / 2} [\cos (\frac{π z}{h}) \frac{\partial B_{x}}{\partial x} + \cos (\frac{π z}{h}) \frac{\partial B_{y}}{\partial y} + \frac{π}{h} \sin (\frac{π z}{h}) B_{z}] d z \end{array}

(21)

where

(I_{0}, I_{1}, I_{2})

are defined in Appendix 1.

$N_{i j}$ and $M_{i j}$ indicate the in-plane stress resultants and bending moments, respectively. They are described by the following

\begin{array}{l} N_{x x} = \int_{- h / 2}^{h / 2} σ_{x x} d z, N_{y y} = \int_{- h / 2}^{h / 2} σ_{y y} d z, N_{x y} = \int_{- h / 2}^{h / 2} σ_{x y} d z \\ N_{x z} = k_{s} \int_{- h / 2}^{h / 2} σ_{x z} d z, N_{y z} = k_{s} \int_{- h / 2}^{h / 2} σ_{y z} d z, \\ M_{x x} = \int_{- h / 2}^{h / 2} σ_{x x} z d z, M_{y y} = \int_{- h / 2}^{h / 2} σ_{y y} z d z, M_{x y} = \int_{- h / 2}^{h / 2} σ_{x y} z d z \end{array}

(22)

where

k_{s}

denotes the shear stress correction factor and consider equal to 5/6 for the plate.

By the substitution of equations (10), (13), and (14) into equations (6) to (8), then using equation (22), the seven equations of motion, equation (21), can be derived as

\begin{array}{l} δ u : (1 - λ \nabla^{2}) (\begin{array}{l} A_{66} (u_{0, y y} + v_{0, x y}) + B_{66} (ϕ_{x, y y} + ϕ_{y, x y}) \\ + (B_{11} ϕ_{x, x x} + A_{11} u_{0, x x} + B_{12} ϕ_{y, x y} + A_{12} v_{0, x y}) \end{array}) \\ - (1 - μ \nabla^{2}) (I_{0} {\ddot{u}}_{0} + I_{1} {\ddot{ϕ}}_{x}) + ϕ_{, x} Ξ_{31} + ψ_{, x} β_{31} = 0 \end{array}

(23)

\begin{array}{l} δ v : (1 - λ \nabla^{2}) (\begin{array}{l} A_{66} (v_{0, x x} + u_{0, x y}) + B_{66} (ϕ_{y, x x} + ϕ_{x, x y}) \\ + A_{22} v_{0, y y} + B_{22} ϕ_{y, y y} + A_{12} u_{0, x y} + B_{12} ϕ_{x, x y} \end{array}) \\ - (1 - μ \nabla^{2}) (I_{0} {\ddot{v}}_{0} + I_{1} {\ddot{ϕ}}_{y}) + ϕ_{, y} Ξ_{32} + ψ_{, y} β_{32} = 0 \end{array}

(24)

\begin{array}{l} δ w : - k_{s} (k_{15} ψ_{, x x} + M_{15} ϕ_{, x x} + M_{24} ϕ_{, y y} + k_{24} . ψ_{, y y}) \\ + (1 - λ \nabla^{2}) (k_{s} A_{44} (w_{0, y y} + ϕ_{y, y}) + k_{s} A_{55} (w_{0, x x} + ϕ_{x, x}) + k_{G} (w_{0, x x} + w_{0, y y}) - k_{w} w_{0}) \\ + (1 - μ \nabla^{2}) (k_{w} w_{0} - I_{0} {\ddot{w}}_{0} - k_{G} w_{0}) + (N_{E} + N_{M}) \nabla^{2} w_{0} \\ + \frac{2 k_{s}}{h} (T_{15} ϕ_{0, x x} + T_{24} ϕ_{0, y y} + P_{15} ψ_{0, x x} + P_{24} ψ_{0, y y}) = 0 \end{array}

(25)

\begin{matrix} δ ϕ_{x} : (1 - λ \nabla^{2}) (\begin{array}{l} - k_{s} A_{55} (w_{0, x} + ϕ_{x}) + Q_{66} (ϕ_{x, y y} + ϕ_{y, x y}) + B_{11} u_{0, x x} \\ + B_{66} (u_{0, y y} + v_{0, x y}) + Q_{11} ϕ_{x, x x} + Q_{12} ϕ_{y, x y} + B_{12} v_{0, x y} \end{array}) \\ + k_{s} (M_{15} ϕ_{, x} + O_{15} ψ_{, x}) + E_{31} ϕ_{, x} + F_{31} ψ_{, x} - (1 - μ \nabla^{2}) (I_{1} {\ddot{u}}_{0} + I_{2} {\ddot{ϕ}}_{x}) = 0 \end{matrix}

(26)

\begin{matrix} δ ϕ_{y} : (1 - λ \nabla^{2}) (\begin{array}{l} - k_{s} A_{44} (w_{0, y} + ϕ_{y}) + Q_{66} (ϕ_{y, x x} + ϕ_{x, x y}) + B_{66} (v_{0, x x} + u_{0, x y}) \\ + Q_{12} ϕ_{x, x y} + Q_{22} ϕ_{y, y y} + Q_{22} ϕ_{y, y y} + B_{22} v_{0, y y} + B_{12} u_{0, x y} \end{array}) \\ + E_{32} ϕ_{, y} + F_{32} ψ_{, y} + k_{s} (M_{24} ϕ_{, y} + O_{24} . ψ_{, y}) - (1 - μ \nabla^{2}) (I_{1} {\ddot{v}}_{0} + I_{2} {\ddot{ϕ}}_{y}) = 0 \end{matrix}

(27)

\begin{array}{l} δ ϕ : k_{s} M_{15} (w_{0, x x} + ϕ_{x, x}) + k_{s} M_{24} (w_{0, y y} + ϕ_{y, y}) + X_{11} ϕ_{, x x} + X_{22} ϕ_{, y y} + Y_{11} ψ_{, x x} \\ + Y_{22} ψ_{, y y} + E_{31} ϕ_{x, x} + E_{32} ϕ_{y, y} - X_{33} ϕ - Y_{33} ψ + Ξ_{31} u_{0, x} + Ξ_{32} v_{0, y} = 0 \end{array}

(28)

\begin{array}{l} δ ψ : Y_{11} ϕ_{, x x} + Y_{22} ϕ_{, y y} + Z_{11} ψ_{, x x} + Z_{22} ψ_{, y y} + F_{31} ϕ_{x, x} + F_{32} ϕ_{y, y} + β_{31} . u_{0, x} \\ + β_{32} v_{0, y} - Y_{33} ϕ - Z_{33} ψ + k_{s} O_{15} (w_{0, x x} + ϕ_{x, x}) + k_{s} O_{24} (w_{0, y y} + ϕ_{y, y}) = 0 \end{array}

(29)

where

(A_{i j}, B_{i j}, E_{i j}, F_{i j}, O_{i j}, Q_{i j}, X_{i j}, Y_{i j}, Z_{i j}, β_{i j}, M_{i j}, Ξ_{i j})

are defined in Appendix 1.

Solution procedure

An analytical solution for a simply supported rectangular EGGS plate is obtained using Navier solution technique by [47]. The boundary conditions of a simply supported rectangular plate can be expressed as

\begin{array}{l} x = 0, a and y = 0, b \\ w = ϕ = ψ = ϕ_{x} = ϕ_{y} = 0 \\ w ″ = ϕ ″ = ψ ″ = ϕ_{x} ″ = ϕ_{y} ″ = 0 \end{array}

(30)

The displacements are assumed as the series of double trigonometric functions that satisfy boundary conditions

\begin{matrix} u (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{m n} \cos (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ v (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V_{m n} \sin (\frac{m π}{a} x) \cos (\frac{n π}{b} y) e^{i ω t} \\ w (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{m n} \sin (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ ϕ_{x} (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{x m n} \cos (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ ϕ_{y} (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{y m n} \sin (\frac{m π}{a} x) \cos (\frac{n π}{b} y) e^{i ω t} \\ ϕ (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{m n} \sin (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ ψ (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ψ_{m n} \sin (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \end{matrix}

(31)

By substituting the equation (31) into governing equations (23) to (29), the governing equations of motion are reduced to a characteristic equation including the mass and stiffness matrices. The characteristic equation and corresponding stiffness and mass matrices for finding the Eigenvalues can be defined as

\begin{array}{l} {[\begin{matrix} K_{11} & K_{12} & 0 & K_{14} & K_{15} & K_{16} & K_{17} \\ K_{21} & K_{22} & 0 & K_{24} & K_{25} & K_{26} & K_{27} \\ 0 & 0 & K_{33} & K_{34} & K_{35} & K_{36} & K_{37} \\ K_{41} & K_{42} & K_{43} & K_{44} & K_{45} & K_{46} & K_{47} \\ K_{51} & K_{52} & K_{53} & K_{54} & K_{55} & K_{56} & K_{57} \\ K_{61} & K_{62} & K_{63} & K_{64} & K_{65} & K_{66} & K_{67} \\ K_{71} & K_{72} & K_{73} & K_{74} & K_{75} & K_{76} & K_{77} \end{matrix}] - [\begin{matrix} I_{11} & 0 & 0 & I_{14} & 0 & 0 & 0 \\ 0 & I_{21} & 0 & 0 & I_{25} & 0 & 0 \\ 0 & 0 & I_{33} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{44} & 0 & 0 & 0 \\ I_{51} & I_{52} & 0 & 0 & I_{55} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] \times ω^{2}} \\ \times {\begin{matrix} U_{m n} \\ V_{m n} \\ W_{m n} \\ ϕ_{x m n} \\ ϕ_{y m n} \\ ϕ_{m n} \\ ψ_{m n} \end{matrix}} = {0} \end{array}

(32)

and

K_{i j}

are defined in Appendix 1. The nonlocal natural frequencies of a nanoplate are calculated using solution of the Eigenvalue equation

D e t (K - ω^{2} M) = 0

Numeric investigation

The numerical results of the problem are presented in this section. Before the presentation of numerical results, a comparison with literature for verification and validation is required.

Verification

In order to verify the present study, the current results are compared with corresponding literature. For the validation of results, the integrated piezomagnetic face sheets are removed and local non-dimensional natural frequencies of the isotropic plate are compared. The material properties are assumed based on Table 1 according to [25,31,48,49].

Table 1.

The material properties of core.

E_c (GPa)	ρ_c (Kg/m³)	ν_c
380	3800	0.3
E_m (GPa)70	ρ_m (Kg/m³) 2707	ν_m0.3

For comparison, the frequency ratio is considered as

Frequency ratio= \frac{Frequency using nonlocal theory}{Frequency using local theory}

(33)

Table 2 indicates the frequency ratio of a single-layer nanoplate for different nonlocal parameters. The comparison between the present results and Arani and Zamani [50], Pradhan and Kumar [51], Pradhan and Phadikar [52], and Liu et al. [53] reveals a good agreement with corresponding literature.

Table 2.

Frequency ratio of an isotropic single-layer nanoplate $(a=b=10 nm)$ .

Nonlocal parameter (nm²)	Present	Arani and Zamani [50]	Pradhan and Kumar[51]	Pradhan and Phadikar[52]	Liu et al.[53]
0	1.0000	1.0000	1.0000	1.0000	1.0000
1	0.9138	0.9138	0.9139	0.9139	0.9139
2	0.8467	0.8467	0.8468	0.8467	0.8467
3	0.7925	0.7925	0.7926	0.7925	0.7925

Table 3 demonstrates the comparison with higher order shear deformation theory by Reddy and Phan, exact elasticity solution by Srinivas et al. [54], quasi-3D hyperbolic shear deformation theory by Hebali et al. [55], FSDT by Whitney et al. [56], and sinusoidal shear and normal deformation theory by Arani and Zamani [50] for different modes. This comparison indicates that the present numerical results are in good agreement with the literature.

Table 3.

Comparison of numerical results of a single-layer nanoplate with literatures $Ω =ω h \sqrt{\frac{ρ}{G}}$ with $(ν =0 .3,a/h=10,a/b=1)$ .

Mode (m,n)	Present	Arani and Zamani [50]	Hebali et al. [55]	Srinivas et al. [54]	Whitney et al. [56]	Reddy and Phan [57]
(1,1)	0.0930	0.0933	0.0933	0.0932	0.0930	0.0931
(1,2)	0.2219	0.2230	0.2228	0.2226	0.2220	0.2222
(2,2)	0.3405	0.3428	0.3422	0.3421	0.3406	0.3411
(1,3)	0.4149	0.4182	0.4173	0.4171	0.4149	0.4158
(2,3)	0.5205	0.5253	0.5240	0.5239	0.5206	0.5221
(3,3)	0.6834	0.6911	0.6890	0.6889	0.6834	0.6862
(2,4)	0.7446	0.7537	0.7512	0.7511	0.7447	0.7481
(1,5)	0.9174	0.9304	0.9268	0.9268	0.9174	0.923
(4,4)	1.0764	1.0938	1.0890	1.0889	1.0764	1.0847

Another validation survey, for various modes of local non-dimensional, is offered in Table 4. The results of the current study are compared with Jha et al. [58], using higher order shear and normal deformation theory, Hebali et al. [55], using quasi-3D hyperbolic shear deformation theory, sinusoidal shear and normal deformation theory by Arani and Zamani [50], and Shahrjerdi et al. [59], using second-order shear deformation. The non-dimensional frequencies in the present work are a bit lower with respect to other literatures due to the displacement field assumptions. One can conclude that this is due to employing the FSDT instead of other theories applied in [50 –53].

Table 4.

Comparison of numerical results of a single-layer nanoplate with literatures $Ω =ω \frac{a^{2}}{h} \sqrt{\frac{ρ}{E}}$ with $(a/h=10,a/b=0 .5)$ .

Mode (m,n)	Present	Arani and Zamani [50]	Hebali et al. [55]	Jha et al. [58]	Shahrjerdi et al. [59]
(1,1)	3.6917	3.7048	3.6959	3.6911	3.6983
(1,2)	5.8298	5.8542	5.8392	5.8323	5.8498
(2,1)	11.9401	12.0111	11.9752	11.965	12.0345
(2,2)	13.8859	13.9760	13.9324	13.921	14.0144
(2,3)	17.0388	17.1641	17.1070	17.096	17.2325
(3,2)	25.9087	26.1598	26.0579	26.051	26.3462
(3,3)	28.6949	28.9933	28.8754	28.871	29.2257

Numerical results of a sandwich nanoplate

In this section, the numerical results of a sandwich nanoplate made of exponentially functionally graded materials are presented. The material properties of the core are assumed based on Table 1 and material properties of piezomagnetic face sheets are assumed by [42,44 –46,60 –63] in Table 5.

Table 5.

The material properties of piezomagnetic face-sheets.

c₁₁(GPa)	c₁₂(GPa)	c₁₃(GPa)	c₂₂(GPa)	c₃₃(GPa)	c₄₄(GPa)
226	125	124	226	216	44.2
c₅₅(GPa)	c₆₆(GPa)	e₃₁(C/m²)	e₁₅(C/m²)	e₃₂(C/m²)	e₂₄(C/m²)
44.2	50.5	−2.2	5.8	−2.2	5.8
e₃₃(C/m²)	f₃₁(N/Am)	f₁₅(N/Am)	f₃₂(N/Am)	f₂₄(N/Am)	f₃₃(N/Am)
9.3	290.1	275	290.1	275	349.9
κ₁₁(C/Vm)	κ₂₂(C/Vm)	κ₃₃(C/Vm)	g₁₁(Ns/CV)	g₂₂(Ns/CV)	g₃₃(Ns/CV)
5.64 × 10⁻⁹	5.64 × 10⁻⁰⁹	6.35 × 10⁻⁹	5.367 × 10⁻¹²	5.367 × 10⁻¹²	2737.5 × 10⁻¹²
	μ₁₁(Ns²/CV)	μ₂₂(Ns²/CV)	μ₃₃(Ns²/CV)	ρ(Kg/m³)
	−0.000297	−0.000297	83.5 × 10⁻⁶	5.55 × 10³

Shown in Figure 2 is the influence of the non-dimensional side length ratio (a/b) on the fundamental natural frequencies for even and uneven porosities. It is observed that with increase of the ratio of side lengths $\frac{a}{b}$ , the natural frequencies are decreased where $\frac{a}{b} \leq 1$ . For the increase of $\frac{a}{b}$ , the natural frequencies are increased where $\frac{a}{b} > 1$ .

Shown in Figure 3 are the variation of fundamental natural frequencies of a nanoplate in terms of ratio $\frac{h}{h_{p}}$ for cases the total thickness ${h+2h}_{p}$ of a nanoplate is assumed fixed for even and uneven porosities. We see that with the increase of ratio $\frac{h}{h_{p}}$ , all natural frequencies are increased. It can be discussed that with increase of ratio $\frac{h}{h_{p}}$ , the portion of core rather than piezomagnetic face sheets are increased. Since the stiffness of core is more than the stiffness of face sheets, increase of ratio $\frac{h}{h_{p}}$ leads to a stiffer plate and finally reaches to greater natural frequencies. As you can see in the fixed amount h/h_p, natural frequencies for the even porosity are greater than the uneven porosity.

The effect of the shear parameter of foundation on the fundamental natural frequencies for even and uneven porosities is studied in Figure 4. We can conclude that with increase in the shear parameter, the stiffness of foundation is increased and consequently the natural frequencies are increased. Figure 5 shows variation of natural frequencies in terms of the spring parameter of foundation ( $k_{w}$ ) for even and uneven porosities. One can conclude that increase of the spring parameter of foundation leads to increase of the stiffness of foundation and consequently leads to increase of natural frequencies of a nanoplate. By comparing Figures 4 and 5, it can be concluded that Winkler foundation parameter has a lower effect than Pasternak foundation parameter.

Figure 6 examines the effects of a strain gradient parameter on the fundamental natural vibration frequencies of an FG nanoplate. It is possible to obtain the frequency results of the nonlocal elasticity theory without nonlocal by setting μ = 0. Based on the nonlocal strain gradient theory, increasing strain gradient parameter results in larger vibration frequency for every values of the nonlocal parameter. On the other hand, with increase in the material gradation index (p), natural vibration frequencies are reduced. Figure 7 indicates effect of the nonlocal parameter (μ) of the nanoplate on the fundamental natural vibration frequencies of the FG nanoplate. We can conclude that with the increase in the nonlocal (μ), fundamental natural vibration frequencies are decreased. Figures 8 and 9 show the effects of nonlocal and strain gradient parameters under different values of even porosities.

Conclusion

The free vibration analysis of a shear deformable sandwich functionally graded porous nanoplate with piezomagnetic face sheets resting on Pasternak's foundation was studied in this paper. The surrounding medium was described by Pasternak model including both direct and shear effects. Eringen’s nonlocal piezo-magneto-elasticity theory was used to consider size effects. The nonlocal constitutive relations were developed for the graded porous core and two piezomagnetic face sheets. The governing equations of motion were derived using Hamilton’s principle based on FSDT. The analytical method was proposed for solution of the governing equations of motion to study the influence of parameters of the nanostructure such as the in-homogeneous index, nonlocal parameter, two parameters of foundation, and non-dimensional geometric parameters on the free vibration responses. The main results of our analysis are expressed as:

Increase porosity volume fractions of porous graded core leads to a stiffer core and consequently increases the natural frequencies of a nanoplate.

Discussion on the influence of the nonlocal parameter indicates that with increase of this parameter, all natural frequencies are decreased. One can conclude that this decrease is due to decrease of the stiffness of nanomaterial with increase of the nonlocal parameter.

The numerical results indicate that with increase of the non-dimensional thickness ratio $\frac{h}{h_{p}}$ , the natural frequencies are increased. This is due to increase of the bending stiffness of the sandwich structure. The material properties indicate that the core is much stiffer than face sheets and consequently leads to a stiffer structure.

The non-dimensional side length ratio (a/b) has a significant influence on the vibration characteristics of the sandwich nanoplate. The numerical results indicate that the vibration behavior of a nanoplate is depending on the value of (a/b). For $\frac{a}{b} < 1$ , the natural frequencies are decreased with the increase of $\frac{a}{b}$ and for $\frac{a}{b} > 1$ the natural frequencies are increased with the increase of $\frac{a}{b}$ .

Two parameters of the foundation can significantly change the natural frequencies of a nanoplate. One can conclude that increase of both parameters of foundation increases the natural frequencies significantly.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the University of Kashan (grant number: 574613/026). The first author would like to thank the Iranian Nanotechnology Development Committee for their financial support.

Appendix 1

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