Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core

Abstract

This research addresses analytical solutions for vibration performance of simply supported sandwich plate subjected to coupled mechanical-electric-magnetic-thermal loads. The plate is made of graphene reinforced metal matrix composite (GRMMC) core and magneto-electro-elastic (MEE) face sheets. Three graphene distribution patterns, FG-X, FG-O, and UD are considered for evaluating the reinforcement efficiency of graphene. The Reddy’s higher order shear deformation plate theory and Hamilton’s principle are employed to formulate the equations of motion of the sandwich plate. Then, using double trigonometric functions for variables and applying Bubnov–Galerkin procedure, the system of nonlinear second-order differential equations is constructed to reveal the natural frequency, frequency ratio, and dynamic response of the sandwich plate. The influences of graphene distribution patterns and volume fraction, magnetic and electric potentials, temperature change, and volume fraction of piezoelectric phase are discussed in details. This research provides basis for the design of sandwich composite structures, especially smart structures and devices.

Keywords

vibration analysis sandwich plate magneto-electro-elastic graphene reinforced metal matrix composite negative Poisson’s ratio

1. Introduction

Magneto-electro-elastic (MEE) materials, which combine piezoelectric and piezomagnetic phases, are one of the most potential smart materials due to its superior coupling of magnetic, electric, and mechanical properties. MEE materials are used widely in various engineering applications such as sensors, actuators, and structural health monitoring systems for aerospace platforms (Hui et al., 2015; Yang and Yang, 2016). Numerous works have been published on the mechanical performance of MEE structures. Zhou et al. (2022) proposed the inhomogeneous MEE coupling element-free Galerkin method to investigate the statics and dynamics behaviors of MEE structures. Recently, Vinyas (2023, 2022a, 2022b, 2020) introduced the works on the static behavior of piezo-magneto-thermo-elastic nanocomposite sandwich plate with CNT agglomeration and dynamic response and vibration of MEE sandwich plate using three-dimensional finite element methods. Based on the first order shear deformation hypothesis, Zhao et al. (2022) developed a geometrically nonlinear finite element formulation for static and dynamic analysis of carbon nanotube reinforced MEE plates. For MEE sandwich plates, various computational techniques such as the first order shear deformation theory (Abdolhoseyni and Danesh, 2023), Timoshenko beam theory (Bamdad et al., 2019), four variable tangential–exponential refined theory (Ebrahimi et al., 2021), Reddy’s higher order shear deformation plate theory (Quang et al., 2022) and non-local elasticity theory (Sirimontree et al., 2023) were also used to determine the solutions for mechanical problems. Chu et al. (2023) evaluated the impacts of moving load and the use of a piezoelectric patch on the level of energy harvesting and dynamic behavior of a nano conical panel made from shape memory alloy located on a frictional substrate using the first order shear deformation theory. Based on Reddy's third order shear deformation theory, Vinyas et al. (2019) conducted a study on how the thickness of the piezoelectric inter-phase affects the coupled frequency response of three-phase smart MEE plates.

Graphene, with outstanding characteristics such as flexible, lightweight, high mechanical strength, high thermal, and electrical conductivity (Balandin, 2011; Khadem et al., 2022; Li et al., 2019), has been receiving considerable attention. Recently, graphene has become an ideal reinforcement for composite with polymer matrix in order to enhance the mechanical, thermal, and electrical properties. Plenty of theoretical and numerical investigations on the mechanical behaviors of graphene reinforced polymer matrix composite (GRPMC) have been conducted in recent years. In addition, Masoud et al. (2019) investigated the buckling and vibration characteristics of a nano-shell reinforced with graphene nanoplatelets under uniform axial loading using Hamilton’s principle. In 2022, Phuong et al. (2022) analytically examined the nonlinear post-buckling analysis of functionally graded GRPMC plates taking into account the nonlinear effect of elastic foundation subjected to external pressure and axial compression load. Furthermore, Fan et al. (2019, 2018a, 2018b) studied the nonlinear dynamic and low-velocity impact responses of functionally graded GRPMC laminated plate and beam resting on visco-elastic foundations subjected to a transverse impact loading. Al-Furjan et al. (2022, 2023) investigated the nonlinear mechanics of porous graphene platelets reinforced sandwich nanoplates based on various theories and wave propagation of a nano supercapacitor made from corrugated graphene layer covered by nano piezoelectric face sheets based on mathematical modeling.

Besides the pre-eminent properties mentioned above, the biggest drawback of GRPMC is inability to withstand high temperature. The replacement of polymer matrix by metal matrix enhances the temperature resistance of the graphene reinforced composite (Chen et al., 2020; Safina et al., 2022; Zhao et al., 2020). Due to variety of engineering applications, many studies are available in literature related to mechanical behaviors of graphene reinforced metal matrix composite (GRMMC) structures. Chu and Jia (2014) reported the use of graphene nanoplatelets to strengthen the bulk Cu-matrix composites. One of the special properties of GRMMC is the negative Poisson’s ratio, which creates structures with great shock and sound absorption. The research team of Shen published investigations on the effect of negative Poisson’s ratio on the post-buckling behavior of GRMMC laminated plates and shells in thermal environments based on the Reddy’s third order shear deformation plate theory (Shen et al., 2017, 2020, 2021). Mokhalingam et al. (2017) investigated the mechanical properties of graphene sheets reinforced Al nanocomposite under uni-axial loading using molecular dynamics.

From the literature survey done, no work has been conducted on the mathematical modeling of sandwich plate with MEE face sheets and GRMMC core. To fill this gap, this paper introduces the first investigation on the nonlinear vibration characteristics of MEE sandwich plate with GRMMC core layer on elastic foundations using the Reddy’s higher order shear deformation plate theory and analytical approach. The displayed expressions resulting from the analytical method not only serve as the foundation for designers when selecting geometric and material parameters but also serve as the objective functions in optimization problems. Three graphene distribution patterns and three values of volume fraction of piezoelectric phase are considered. Variety of numerical efforts indicates the effect of di;erent parameters on the natural frequency, frequency ratio, and dynamic response of the MEE sandwich plate.

2. Modelling and materials

As shown in Figure 1, a rectangular sandwich plate with MEE face sheets locating in the Cartesian coordinate system $(x, y, z)$ is considered. The core layer of sandwich plate is assumed to have 10 plies and each ply is made of GRMMC with the same thickness. The width and length of the sandwich plate are $a$ and $b$ , respectively. The thickness of GRMMC core layer and MEE face sheet are $h_{c}$ and $h_{f}$ , respectively. The MEE sandwich plate is rested on Pasternak-type elastic foundations with two stiffness $k_{1}, k_{2}$ and subjected to temperature change $Δ T$ , external pressure uniformly distributed on the surface of the plate $q$ , electric and magnetic potentials $\bar{Φ}$ and $\bar{Ψ}$ .

For Cu based GRMMC core layer, three different graphene distribution patterns, namely, UD, FG-X, and FG-O, as shown in Figure 2, are considered in this study. Five values of 0.05 (5%), 0.07 (7%), 0.09 (9%), 0.11 (11%), and 0.13 (13%) are chosen for graphene volume fraction $V_{G}$ . For FG-X type, the large concentration of graphene is shown on the top and bottom plies and the graphene volume fraction of ten plies is arranged to be ${[0.13 / 0.11 / 0.09 / 0.07 / 0.05]}_{S}$ . For FG-O type, the largest volume fraction of graphene is distributed at the middle plies and the ply arrangement is ${[0.05 / 0.07 / 0.09 / 0.11 / 0.13]}_{S}$ . Graphene volume fraction $V_{G}$ is unchanged for all ten plies of GRMMC core layer with uniformly distributed (UD) type of graphene reinforcement. The mechanical properties of GRMMC core layer depending on four values of temperature and five values of graphene volume fraction are expressed in Tables 1 and 2 (Fan et al., 2018a, 2018b, 2019).

Table 1.

Young’s and shear modulus of GRMMC core layer.

$T (K)$	$V_{G}$ (%)	$E_{11} (G P a)$	$E_{22} (G P a)$	$G_{12} (G P a)$	$G_{13} (G P a)$	$G_{23} (G P a)$
300	5	207.55	196.69	66.389	33.617	32.327
	7	249.62	235.67	80.214	26.881	26.895
	9	275.44	256.48	91.706	25.517	24.949
	11	307.06	286.97	108.42	23.616	22.802
	13	319.77	304.99	120.34	20.001	19.445
500	5	193.15	183.94	62.092	31.536	31.009
	7	235.50	221.18	77.036	25.845	25.809
	9	258.90	243.71	87.200	24.257	23.893
	11	293.37	273.66	100.92	21.853	20.786
	13	296.39	288.72	112.29	19.283	17.660
700	5	180.50	171.58	58.314	28.928	28.200
	7	219.61	206.17	72.686	24.115	23.587
	9	242.91	229.60	82.040	23.627	22.989
	11	271.55	254.96	97.670	17.003	16.290
	13	284.59	275.48	110.79	18.057	17.396
1000	5	155.35	148.86	53.053	24.109	23.701
	7	220.43	199.26	76.439	21.923	21.839
	9	197.96	178.74	65.355	22.997	22.394
	11	248.34	210.12	93.537	16.211	15.926
	13	253.87	223.40	99.681	16.731	15.988

Table 2.

Poisson’s ratio and thermal expansion coefficients of GRMMC core layer.

$T (K)$	$V_{G}$ (%)	$v_{12}$	$α_{11} (\times 1 0^{- 6} /^{0} C)$	$α_{22} (\times 1 0^{- 6} /^{0} C)$
300	5	−0.0649	1.4224	1.4194
	7	−0.0594	1.0812	1.0813
	9	−0.0486	0.9976	1.0021
	11	−0.0364	0.8204	0.8204
	13	−0.0380	0.7984	0.8014
500	5	−0.0721	1.5037	1.5006
	7	−0.0718	1.3848	1.3957
	9	−0.0540	1.3174	1.3046
	11	−0.0532	1.2199	1.2053
	13	−0.0489	1.0232	1.0288
700	5	−0.0751	1.6222	1.6278
	7	−0.0736	1.4324	1.4286
	9	−0.0525	1.3589	1.3489
	11	−0.0581	1.2069	1.2066
	13	−0.0271	1.1367	1.1422
1000	5	−0.1118	1.7022	1.7331
	7	−0.0994	1.7205	1.6899
	9	−0.0875	1.5242	1.5416
	11	−0.0676	1.3028	1.3016
	13	−0.0415	1.3256	1.3083

The MEE face sheet is made of Cobalt Ferric

(C o F e_{2} O_{4})

and Barium Titanate

(B a T i O_{3})

which are transversely isotropic materials. The mechanical, electrical, magnetic, and thermal properties of MEE face sheet are assumed to change according to five values of volume fraction of piezoelectric phase

B a T i O_{3} (V_{B a})

as Table 3 (Vinyas, 2020, 2022a, 2022b; Vinyas et al., 2019).

Table 3.

Mechanical, electrical, magnetic, and thermal properties of MEE face sheet.

Material properties	Notation	$V_{B a}$
Material properties	Notation	0	0.2	0.5	0.8	1
Elastic constants $(G P a)$	$C_{11}^{f} = C_{22}^{f}$	286	250	220	175	166
	$C_{12}^{f}$	173	146	120	100	77
	$C_{13}^{f} = C_{23}^{f}$	170	145	120	100	78
	$C_{33}^{f}$	269.5	240	215	170	162
	$C_{44}^{f} = C_{55}^{f}$	45.3	45	45	50	43
	$C_{66}^{f}$	56.5	52	50	37.5	44.5
Piezoelectric constants $(C / m^{2})$	$e_{31}^{f}$	0	−2	−3.5	−4	−4.4
	$e_{33}^{f}$	0	4	9	14	18.6
	$e_{15}^{f}$	$0$	$0$	$0$	0	11.6
Dielectric constants $(1 0^{- 9} C / N m^{2})$	$η_{11}^{f} = η_{22}^{f}$	0.08	0.33	0.85	1	1.2
Dielectric constants $(1 0^{- 9} C / N m^{2})$	$η_{33}^{f}$	0.093	2.5	6.3	10	12.6
Magnetic permeability $(1 0^{- 4} {N s}^{2} / C^{2})$	$μ_{11}^{f} = μ_{22}^{f}$	−5.9	−3.9	−2.0	−0.8	0.05
Magnetic permeability $(1 0^{- 4} {N s}^{2} / C^{2})$	$μ_{33}^{f}$	1.57	1.33	0.9	0.5	0.1
Piezomagnetic constants $(N / A m)$	$q_{31}^{f}$	580	410	350	100	0
	$q_{33}^{f}$	700	550	320	120	0
	$q_{15}^{f}$	560	340	200	80	0
Magneto-electric constants $(1 0^{- 12} N s / V C ())$	$m_{11}^{f} = m_{22}^{f}$	0	2.8	5.5	6.8	0
Magneto-electric constants $(1 0^{- 12} N s / V C ())$	$m_{33}^{f}$	0	2000	2600	1500	0
Pyroelectric constant $(1 0^{- 7} C / m^{2} K)$	$p_{3}^{f}$	0	−3.5	−7.8	−10.8	0
Pyromagnetic constant $(1 0^{- 5} C / m^{2} K)$	$λ_{3}^{f}$	0	−36	−23	−8.5	0
Thermal expansion coefficients $(1 0^{- 6} K^{- 1})$	$α_{1}^{f} = α_{2}^{f}$	0	10.8	12.3	14.1	15.7
Thermal expansion coefficients $(1 0^{- 6} K^{- 1})$	$α_{3}^{f}$	0	9.3	8.2	7.2	6.4
Density $(k g / m^{3})$	$ρ_{f}$	5300	5400	5550	5700	5800

3. Basic formulations

By applying Hamilton’s principle and Reddy’s higher order shear deformation plate theory (Brush and Almroth, 1975; Reddy, 2004) the equations of motion of the MEE sandwich plate can be expressed by

\frac{\partial N_{x}}{\partial x} + \frac{\partial N_{x y}}{\partial y} = \bar{χ_{1}} \frac{\partial^{2} u}{\partial t^{2}} + \bar{χ_{2}} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} - \bar{χ_{3}} \frac{\partial^{3} w}{\partial t^{2} \partial x},

(1a)

\frac{\partial N_{x y}}{\partial x} + \frac{\partial N_{y}}{\partial y} = \bar{χ_{1}^{*}} \frac{\partial^{2} v}{\partial t^{2}} + \bar{χ_{2}^{*}} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} - \bar{χ_{3}^{*}} \frac{\partial^{3} w}{\partial t^{2} \partial y},

(1b)

\frac{\partial Q_{x}}{\partial x} + \frac{\partial Q_{y}}{\partial y} - 3 c_{1} (\frac{\partial R_{x}}{\partial x} + \frac{\partial R_{y}}{\partial y}) + c_{1} (\frac{\partial^{2} P_{x}}{\partial x^{2}} + 2 \frac{\partial^{2} P_{y}}{\partial x \partial y} + \frac{\partial^{2} P_{y}}{\partial y^{2}}) + q + N_{x} \frac{\partial^{2} w}{\partial x^{2}} + 2 N_{x y} \frac{\partial^{2} w}{\partial x \partial y} + N_{y} \frac{\partial^{2} w}{\partial y^{2}} - k_{1} w + k_{2} \nabla^{2} w = χ_{1} \frac{\partial^{2} w}{\partial t^{2}} + 2 ε χ_{1} \frac{\partial w}{\partial t} + \bar{χ_{3}} \frac{\partial^{3} u}{\partial t^{2} \partial x} + \bar{χ_{5}} \frac{\partial^{3} ϕ_{x}}{\partial t^{2} \partial x} + \bar{χ_{3}^{*}} \frac{\partial^{3} v}{\partial t^{2} \partial y} + \bar{χ_{5}^{*}} \frac{\partial^{3} ϕ_{y}}{\partial t^{2} \partial y} - c_{1}^{2} χ_{7} (\frac{\partial^{4} w}{\partial t^{2} \partial x^{2}} + \frac{\partial^{4} w}{\partial t^{2} \partial y^{2}}),

(1c)

\frac{\partial M_{x}}{\partial x} + \frac{\partial M_{x y}}{\partial y} - Q_{x} + 3 c_{1} R_{x} - c_{1} (\frac{\partial P_{x}}{\partial x} + \frac{\partial P_{x y}}{\partial y}) = \bar{χ_{2}} \frac{\partial^{2} u}{\partial t^{2}} + \bar{χ_{4}} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} - \bar{χ_{5}} \frac{\partial^{3} w}{\partial t^{2} \partial x},

(1d)

\frac{\partial M_{x y}}{\partial x} + \frac{\partial M_{y}}{\partial y} - Q_{y} + 3 c_{1} R_{y} - c_{1} (\frac{\partial P_{x y}}{\partial x} + \frac{\partial P_{y}}{\partial y}) = \bar{χ_{2}^{*}} \frac{\partial^{2} v}{\partial t^{2}} + \bar{χ_{4}^{*}} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} - \bar{χ_{5}^{*}} \frac{\partial^{3} w}{\partial t^{2} \partial y},

(1e)

\int_{- h_{c} / 2 - h_{f}}^{h_{c} / 2} (\frac{\partial D_{x}}{\partial x} \cos (γ z) + \frac{\partial D_{y}}{\partial y} \cos (γ z) + D_{z} γ \sin (γ z)) d z + \int_{h_{c} / 2}^{h_{c} / 2 + h_{f}} (\frac{\partial D_{x}}{\partial x} \cos (γ z) + \frac{\partial D_{y}}{\partial y} \cos (γ z) + D_{z} γ \sin (γ z)) d z = 0,

(1f)

\int_{- h_{c} / 2 - h_{f}}^{h_{c} / 2} (\frac{\partial B_{x}}{\partial x} \cos (γ z) + \frac{\partial B_{y}}{\partial y} \cos (γ z) + B_{z} γ \sin (γ z)) d z + \int_{h_{c} / 2}^{h_{c} / 2 + h_{f}} (\frac{\partial B_{x}}{\partial x} \cos (γ z) + \frac{\partial B_{y}}{\partial y} \cos (γ z) + B_{z} γ \sin (γ z)) d z = 0

(1g)

where

c_{1} = 4 / (3 h^{2}); u

and

v

are displacement terms of mid-plane along the

x

and

y

axes, respectively;

w

is the transverse deflection along the

z

axis;

ϕ_{x}

and

ϕ_{y}

are used to denote the rotations around

y

and

x

axes, respectively, and

\bar{χ_{1}} = χ_{1}, \bar{χ_{2}} = χ_{2} - c_{1} χ_{4}, \bar{χ_{3}} = c_{1} χ_{4}, \bar{χ_{4}} = χ_{3} - 2 c_{1} χ_{5} + c_{1}^{2} χ_{7}, \bar{χ_{5}} = c_{1} χ_{5} - c_{1}^{2} χ_{7}, (χ_{1}, χ_{2}, χ_{3}, χ_{4}, χ_{5}, χ_{7}) = \int_{- h_{f} - \frac{h_{c}}{2}}^{- \frac{h_{c}}{2}} ρ_{f} (z) (1, z, z^{2}, z^{3}, z^{4}, z^{6}) d z + \sum_{k = 1}^{10} \int_{h_{k - 1}}^{h_{k}} ρ_{c} (z) (1, z, z^{2}, z^{3}, z^{4}, z^{6}) d z + \int_{\frac{h_{c}}{2}}^{\frac{h_{c}}{2} + h_{f}} ρ_{f} (z) (1, z, z^{2}, z^{3}, z^{4}, z^{6}) d z

(2)

The relationships between strain field and displacement components for a MEE sandwich plate are described as (Brush and Almroth, 1975; Reddy, 2004)

[\begin{array}{l} ε_{x} \\ ε_{y} \\ γ_{x y} \end{array}] = [\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{array}] + z [\begin{array}{l} k_{x}^{1} \\ k_{y}^{1} \\ k_{x y}^{1} \end{array}] + z^{3} [\begin{array}{l} k_{x}^{3} \\ k_{y}^{3} \\ k_{x y}^{3} \end{array}], [\begin{array}{l} γ_{x z} \\ γ_{y z} \end{array}] = [\begin{array}{l} γ_{x z}^{0} \\ γ_{y z}^{0} \end{array}] + z^{2} [\begin{array}{l} k_{x z}^{2} \\ k_{y z}^{2} \end{array}]

(3)

where

[\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{array}] = [\begin{array}{l} \frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} \\ \frac{\partial v}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} \end{array}], [\begin{array}{l} γ_{x z}^{0} \\ γ_{y z}^{0} \end{array}] = [\begin{array}{l} \frac{\partial w}{\partial x} + ϕ_{x} \\ \frac{\partial w}{\partial y} + ϕ_{y} \end{array}],

(4)

[\begin{array}{l} k_{x}^{1} \\ k_{y}^{1} \\ k_{x y}^{1} \end{array}] = [\begin{array}{l} \frac{\partial ϕ_{x}}{\partial x} \\ \frac{\partial ϕ_{y}}{\partial y} \\ \frac{\partial ϕ_{x}}{\partial y} + \frac{\partial ϕ_{y}}{\partial y} \end{array}], [\begin{array}{l} k_{x}^{3} \\ k_{y}^{3} \\ k_{x y}^{3} \end{array}] = - c_{1} [\begin{array}{l} \frac{\partial ϕ_{x}}{\partial x} + \frac{\partial^{2} w}{\partial x^{2}} \\ \frac{\partial ϕ_{y}}{\partial y} + \frac{\partial^{2} w}{\partial y^{2}} \\ \frac{\partial ϕ_{x}}{\partial y} + \frac{\partial ϕ_{y}}{\partial y} + 2 \frac{\partial^{2} w}{\partial x \partial y} \end{array}], [\begin{array}{l} k_{x z}^{2} \\ k_{y z}^{2} \end{array}] = - 3 c_{1} [\begin{array}{l} \frac{\partial w}{\partial x} + ϕ_{x} \\ \frac{\partial w}{\partial y} + ϕ_{y} \end{array}]

From equation (4), the relationship between strain components and deflection is expressed by the following geometrical compatibility equation (Brush and Almroth, 1975; Reddy, 2004)

\frac{\partial^{2} ε_{x}^{0}}{\partial y^{2}} + \frac{\partial^{2} ε_{y}^{0}}{\partial x^{2}} - \frac{\partial^{2} γ_{x y}^{0}}{\partial x \partial y} = {\frac{\partial^{2} w}{\partial x \partial y}}^{2} - \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}}

(5)

Based on Maxwell’s equation, the electric and magnetic potentials are defined by the combination of cosine and linear functions as follows (Vinyas, 2020, 2022a, 2022b; Vinyas et al., 2019)

\bar{Φ} (x, y, z, t) = - \cos (γ z) Φ (x, y, t) + \frac{2 z ϕ_{0}}{h}, \bar{Ψ} (x, y, z, t) = - \cos (γ z) Ψ (x, y, t) + \frac{2 z ψ_{0}}{h}

(6)

where

γ = \frac{π}{h}; ϕ_{0}

and

ψ_{0}

denote the applied electric and magnetic potentials, respectively;

Φ (x, y, t), Ψ (x, y, t)

are time-dependent distributions of electric and magnetic potentials on reference surface.

The components of electric and magnetic fields are presented as the negative gradient of electric and magnetic potentials, that is

{\begin{cases} E_{x} \\ E_{y} \\ E_{z} \end{cases}} = {\begin{cases} - \frac{\partial \bar{Φ}}{\partial x} \\ - \frac{\partial \bar{Φ}}{\partial y} \\ - \frac{\partial \bar{Φ}}{\partial z} \end{cases}} = {\begin{cases} \cos (γ z) \frac{\partial Φ}{\partial x} \\ \cos (γ z) \frac{\partial Φ}{\partial y} \\ - β \sin (γ z) Φ - \frac{2 ϕ_{0}}{h} \end{cases}}, {\begin{cases} H_{x} \\ H_{y} \\ H_{z} \end{cases}} = {\begin{cases} - \frac{\partial \bar{Ψ}}{\partial x} \\ - \frac{\partial \bar{Ψ}}{\partial y} \\ - \frac{\partial \bar{Ψ}}{\partial z} \end{cases}} = {\begin{cases} \cos (γ z) \frac{\partial Ψ}{\partial x} \\ \cos (γ z) \frac{\partial Ψ}{\partial y} \\ - β \sin (γ z) Ψ - \frac{2 ψ_{0}}{h} \end{cases}}

(7)

The relations between stress and strain tensors for kth layer of GRMMC core are presented as

{[\begin{array}{l} σ_{x x} \\ σ_{y y} \\ σ_{x y} \\ σ_{x z} \\ σ_{y z} \end{array}]}_{k} = [\begin{array}{l} Q_{11}^{c} & Q_{12}^{c} & 0 & 0 & 0 \\ Q_{12}^{c} & Q_{22}^{c} & 0 & 0 & 0 \\ 0 & 0 & Q_{66}^{c} & 0 & 0 \\ 0 & 0 & 0 & Q_{44}^{c} & 0 \\ 0 & 0 & 0 & 0 & Q_{55}^{c} \end{array}] [{[\begin{array}{l} ε_{x x} \\ ε_{y y} \\ ε_{x y} \\ ε_{x z} \\ ε_{y z} \end{array}]}_{k} - [\begin{array}{l} α_{11}^{c} Δ T \\ α_{22}^{c} Δ T \\ 0 \\ 0 \\ 0 \end{array}]]

(8)

in which

Δ T

is temperature change in the environment and

Q_{11}^{c} = \frac{E_{11}^{c}}{1 - ν_{12}^{c} ν_{21}^{c}}, Q_{12}^{c} = \frac{ν_{21}^{c} E_{11}^{c}}{1 - ν_{12}^{c} ν_{21}^{c}}, Q_{22}^{c} = \frac{E_{22}^{c}}{1 - ν_{12}^{c} ν_{21}^{c}}, Q_{44}^{c} = G_{23}^{c}, Q_{55}^{c} = G_{13}^{c}, Q_{66}^{c} = G_{12}^{c}

(9)

The symbols of $B_{x}, B_{y}, B_{z}$ and $D_{x}, D_{y}, D_{z}$ are the magneticux density and electric displacement components, respectively. For MEE face sheets, the constitutive equations can be written as (Vinyas, 2020, 2022a, 2022b; Vinyas et al., 2019)

\begin{array}{l} σ_{x x}^{f} = \bar{C_{11}} ε_{x x}^{f} + \bar{C_{12}} ε_{y y}^{f} - \bar{e_{31}} E_{z} - \bar{q_{31}} H_{z} - \bar{α_{1}} Δ T, \\ σ_{y y}^{f} = \bar{C_{11}} ε_{x x}^{f} + \bar{C_{22}} ε_{y y}^{f} - \bar{e_{32}} E_{z} - \bar{q_{32}} H_{z} - \bar{α_{2}} Δ T, \\ τ_{x y}^{f} = \bar{C_{66}} γ_{x y}^{f}, \\ τ_{y z}^{f} = \bar{C_{44}} γ_{y z}^{f} - \bar{e_{24}} E_{y} - \bar{q_{24}} H_{y}, \\ τ_{x z}^{f} = \bar{C_{55}} γ_{x z}^{f} - \bar{e_{15}} E_{x} - \bar{q_{15}} H_{x}, \\ D_{x}^{f} = \bar{e_{15}} γ_{x z}^{f} + \bar{η_{11}} E_{x} + \bar{m_{11}} H_{x} + \bar{p_{1}} Δ T, \\ D_{y}^{f} = \bar{e_{24}} γ_{y z}^{f} + \bar{η_{22}} E_{y} + \bar{m_{22}} H_{y} + \bar{p_{2}} Δ T, D_{z}^{f} = \bar{e_{31}} ε_{x x}^{f} + \bar{e_{32}} ε_{y y}^{f} + \bar{η_{33}} E_{z} + \bar{m_{33}} H_{z} + \bar{p_{3}} Δ T, \\ B_{x}^{f} = \bar{q_{15}} γ_{x z}^{f} + \bar{m_{11}} E_{x} + \bar{μ_{11}} H_{x} + \bar{λ_{1}} Δ T, \\ B_{y}^{f} = \bar{q_{24}} γ_{y z}^{f} + \bar{m_{22}} E_{y} + \bar{μ_{22}} H_{y} + \bar{λ_{2}} Δ T, \\ B_{z}^{f} = \bar{q_{31}} ε_{x x}^{f} + \bar{q_{32}} ε_{y y}^{f} + \bar{m_{33}} E_{z} + \bar{μ_{33}} H_{z} + \bar{λ_{3}} Δ T \end{array}

(10)

where the components of

\bar{C_{i j}}, \bar{e_{k l}}, \bar{q_{k l}}, \bar{α_{i}}, \bar{η_{i j}}, \bar{m_{i j}}, \bar{μ_{i j}}, \bar{p_{i}}

and

\bar{λ_{i}}

are expressed in Appendix A.

The internal force and moment terms of MEE sandwich plate are defined by

(N_{i}, M_{i}, P_{i}) = \int_{- \frac{h_{c}}{2} - h_{f}}^{- \frac{h_{c}}{2}} σ_{i}^{f} (1, z, z^{3}) d z + \sum_{k = 1}^{10} \int_{h_{k - 1}}^{h_{k}} σ_{i}^{C} (1, z, z^{3}) d z + \int_{\frac{h_{c}}{2}}^{\frac{h_{c}}{2} + h_{f}} σ_{i}^{f} (1, z, z^{3}) d z, i = x x, y y, x y, (Q_{i}, R_{i}) = \int_{- \frac{h_{c}}{2} - h_{f}}^{- \frac{h_{c}}{2}} σ_{i z}^{f} (1, z^{2}) d z + + \sum_{k = 1}^{10} \int_{h_{k - 1}}^{h_{k}} σ_{i z}^{C} (1, z^{2}) d z + \int_{\frac{h_{c}}{2}}^{\frac{h_{c}}{2} + h_{f}} σ_{i z}^{f} (1, z^{2}) d z, i = x, y

(11)

Substitution of equation (3) into equations (8) and (10) then substituting results into equation (11), the procedure for deriving the equations for internal force and moment is as follows

\begin{array}{l} N_{x} = a_{11} ε_{x}^{0} + a_{12} ε_{y}^{0} + b_{11} k_{x}^{1} + b_{12} k_{y}^{1} + e_{11} k_{x}^{3} + e_{12} k_{y}^{3} - Φ_{1} E_{z} - Γ_{1} H_{z} - α_{1} Δ T, \\ N_{y} = a_{12} ε_{x}^{0} + a_{22} ε_{y}^{0} + b_{12} k_{x}^{1} + b_{22} k_{y}^{1} + e_{12} k_{x}^{3} + e_{22} k_{y}^{3} - Φ_{2} E_{z} - Γ_{2} H_{z} - α_{2} Δ T, \\ N_{x y} = a_{66} γ_{x y}^{0} + b_{66} k_{x y}^{1} + e_{66} k_{x y}^{3}, \\ M_{x} = b_{11} ε_{x}^{0} + b_{12} ε_{y}^{0} + d_{11} k_{x}^{1} + d_{12} k_{y}^{1} + f_{11} k_{x}^{3} + f_{12} k_{y}^{3} - Φ_{3} E_{z} - Γ_{3} H_{z} - α_{3} Δ T, \\ M_{y} = b_{12} ε_{x}^{0} + b_{22} ε_{y}^{0} + d_{12} k_{x}^{1} + d_{22} k_{y}^{1} + f_{12} k_{x}^{3} + f_{22} k_{y}^{3} - Φ_{4} E_{z} - Γ_{4} H_{z} - α_{4} Δ T, \\ M_{x y} = b_{66} γ_{x y}^{0} + d_{66} k_{x y}^{1} + f_{66} k_{x y}^{3}, \\ P_{x} = e_{11} ε_{x}^{0} + e_{12} ε_{y}^{0} + f_{11} k_{x}^{1} + f_{12} k_{y}^{1} + h_{11} k_{x}^{3} + h_{12} k_{y}^{3} - Φ_{5} E_{z} - Γ_{5} H_{z} - α_{5} Δ T, \\ P_{y} = e_{12} ε_{x}^{0} + e_{22} ε_{y}^{0} + f_{12} k_{x}^{1} + f_{22} k_{y}^{1} + h_{12} k_{x}^{3} + h_{22} k_{y}^{3} - Φ_{6} E_{z} - Γ_{6} H_{z} - α_{6} Δ T, \\ P_{x y} = e_{66} γ_{x y}^{0} + f_{66} k_{x y}^{1} + h_{66} k_{x y}^{3}, \\ Q_{x} = a_{44} γ_{x z}^{0} + d_{44} k_{x z}^{2} - Φ_{7} E_{x} - Γ_{7} H_{x}, \\ Q_{y} = a_{55} γ_{y z}^{0} + d_{55} k_{y z}^{2} - Φ_{8} E_{y} - Γ_{8} H_{y}, \\ R_{x} = d_{44} γ_{x z}^{0} + f_{44} k_{x z}^{2} - Φ_{9} E_{x} - Γ_{9} H_{x}, \\ R_{y} = d_{55} γ_{y z}^{0} + f_{55} k_{y z}^{2} - Φ_{10} E_{y} - Γ_{10} H_{y} \end{array}

(12)

with the expressions of coefficients

a_{i j}, b_{i j}, d_{i j}, e_{i j}, f_{i j}, h_{i j} (i j = 11,12,22,44,55,66); Γ_{i}, Φ_{i} (i = \bar{1,10}); α_{j} (j = \bar{1,6})

may be found in Appendix B.

To reduce the number of unknowns and equations, the Airy’s stress function $I (x, y, t)$ is introduced as

N_{x} = \frac{\partial^{2} I}{\partial y^{2}}, N_{y} = \frac{\partial^{2} I}{\partial x^{2}}, N_{x y} = - \frac{\partial^{2} I}{\partial x \partial y}

(13)

From equation (12), the following expressions are obtained as

ε_{x}^{0} = g_{11} \frac{\partial^{2} f}{\partial y^{2}} - g_{12} \frac{\partial^{2} f}{\partial x^{2}} + g_{13} \frac{\partial ϕ_{x}}{\partial x} + g_{14} \frac{\partial ϕ_{y}}{\partial y} - c_{1} g_{15} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ϕ_{x}}{\partial x}) - c_{1} g_{16} (\frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial ϕ_{y}}{\partial y}) + g_{17} ϕ_{0} + g_{18} ψ_{0} + g_{19} Δ T,

(14)

ε_{y}^{0} = g_{21} \frac{\partial^{2} f}{\partial x^{2}} - g_{12} \frac{\partial^{2} f}{\partial y^{2}} + g_{23} \frac{\partial ϕ_{x}}{\partial x} + g_{24} \frac{\partial ϕ_{y}}{\partial y} - c_{1} g_{25} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ϕ_{x}}{\partial x}) - c_{1} g_{26} (\frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial ϕ_{y}}{\partial y}) + g_{27} ϕ_{0} + g_{28} ψ_{0} + g_{29} Δ T, γ_{x y}^{0} = - g_{31} \frac{\partial^{2} f}{\partial x \partial y} + g_{32} (\frac{\partial ϕ_{x}}{\partial y} + \frac{\partial ϕ_{y}}{\partial x}) - c_{1} g_{33} (2 \frac{\partial^{2} w}{\partial x \partial y} + \frac{\partial ϕ_{x}}{\partial y} + \frac{\partial ϕ_{y}}{\partial x})

in which

\begin{array}{l} Δ = a_{11} a_{22} - a_{12}^{2}, g_{11} = \frac{a_{22}}{Δ}, g_{12} = \frac{a_{12}}{Δ}, g_{13} = \frac{b_{12} a_{12} - b_{11} a_{22}}{Δ}, g_{14} = \frac{b_{22} a_{12} - b_{12} a_{22}}{Δ}, \\ g_{15} = \frac{e_{12} a_{12} - e_{11} a_{22}}{Δ}, g_{16} = \frac{e_{22} a_{12} - e_{12} a_{22}}{Δ}, g_{17} = 2 \frac{2 e_{32} a_{12} - 2 e_{31} a_{22}}{Δ}, \\ g_{18} = 2 \frac{2 q_{32} a_{12} - 2 q_{31} a_{22}}{Δ}, g_{19} = \frac{- α_{2} a_{12} + α_{1} a_{22}}{Δ}, g_{21} = \frac{a_{11}}{Δ}, I_{23} = \frac{b_{11} a_{12} - b_{12} a_{11}}{Δ}, \\ g_{24} = \frac{b_{12} a_{12} - b_{22} a_{11}}{Δ}, g_{25} = \frac{e_{11} a_{12} - e_{12} a_{11}}{Δ}, g_{26} = \frac{e_{12} a_{12} - e_{22} a_{11}}{Δ}, \\ g_{27} = 2 \frac{2 e_{31} a_{12} - 2 e_{32} a_{11}}{Δ}, g_{28} = 2 \frac{2 q_{31} a_{12} - 2 q_{32} a_{11}}{Δ}, \\ I_{29} = \frac{- α_{1} a_{12} + α_{2} a_{11}}{Δ}, I_{31} = \frac{1}{a_{66}}, I_{32} = - \frac{b_{66}}{a_{66}}, I_{33} = - \frac{e_{66}}{a_{66}} \end{array}

(15)

Substituting stress function from equation (13) into equations (1a) and (1b), one has

\frac{\partial^{2} u}{\partial t^{2}} = - \frac{\bar{χ_{2}}}{\bar{χ_{1}}} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} + \frac{\bar{χ_{3}}}{\bar{χ_{1}}} \frac{\partial^{3} w}{\partial t^{2} \partial x}, \frac{\partial^{2} v}{\partial t^{2}} = - \frac{\bar{χ_{2}^{*}}}{\bar{χ_{1}^{*}}} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} + \frac{\bar{χ_{3}^{*}}}{\bar{χ_{1}^{*}}} \frac{\partial^{3} w}{\partial t^{2} \partial y}

(16)

Substitution of equation (16) into equations (1c)–(1g), the motion equations can be rearranged as

\frac{\partial Q_{x}}{\partial x} + \frac{\partial Q_{y}}{\partial y} - 3 c_{1} (\frac{\partial R_{x}}{\partial x} + \frac{\partial R_{y}}{\partial y}) + c_{1} (\frac{\partial^{2} P_{x}}{\partial x^{2}} + 2 \frac{\partial^{2} P_{x y}}{\partial x \partial y} + \frac{\partial^{2} P_{y}}{\partial y^{2}}) + \frac{\partial^{2} I}{\partial y^{2}} \frac{\partial^{2} w}{\partial x^{2}} - 2 \frac{\partial^{2} I}{\partial x \partial y} \frac{\partial^{2} w}{\partial x \partial y} + \frac{\partial^{2} I}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} + q - k_{1} w + k_{2} \nabla^{2} w = χ_{1} \frac{\partial^{2} w}{\partial t^{2}} + 2 ε χ_{1} \frac{\partial w}{\partial t} + \bar{χ_{5}} \frac{\partial^{3} ϕ_{x}}{\partial t^{2} \partial x} + \bar{χ_{5}^{*}} \frac{\partial^{3} ϕ_{y}}{\partial t^{2} \partial y} + \bar{χ_{7}} \frac{\partial^{4} w}{\partial t^{2} \partial x^{2}} + \bar{χ_{7}^{*}} \frac{\partial^{4} w}{\partial t^{2} \partial y^{2}},

(17a)

\frac{\partial M_{x}}{\partial x} + \frac{\partial M_{x y}}{\partial y} - Q_{x} + 3 c_{1} R_{x} - c_{1} (\frac{\partial P_{x}}{\partial x} + \frac{\partial P_{x y}}{\partial y}) = \bar{χ_{3}} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} - \bar{χ_{5}} \frac{\partial^{3} w}{\partial t^{2} \partial x},

(17b)

\frac{\partial M_{x y}}{\partial x} + \frac{\partial M_{y}}{\partial y} - Q_{y} + 3 c_{1} R_{y} - c_{1} (\frac{\partial P_{x y}}{\partial x} + \frac{\partial P_{y}}{\partial y}) = \bar{χ_{3}^{*}} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} - \bar{χ_{5}^{*}} \frac{\partial^{3} w}{\partial t^{2} \partial y},

(17c)

\int_{- h_{c} / 2 - h_{f}}^{h_{c} / 2} (\frac{\partial D_{x}}{\partial x} \cos (γ z) + \frac{\partial D_{y}}{\partial y} \cos (γ z) + D_{z} γ \sin (γ z)) d z + \int_{h_{c} / 2}^{h_{c} / 2 + h_{f}} (\frac{\partial D_{x}}{\partial x} \cos (γ z) + \frac{\partial D_{y}}{\partial y} \cos (γ z) + D_{z} γ \sin (γ z)) d z = 0,

(17d)

\int_{- h_{c} / 2 - h_{f}}^{h_{c} / 2} (\frac{\partial B_{x}}{\partial x} \cos (γ z) + \frac{\partial B_{y}}{\partial y} \cos (γ z) + B_{z} γ \sin (γ z)) d z + \int_{h_{c} / 2}^{h_{c} / 2 + h_{f}} (\frac{\partial B_{x}}{\partial x} \cos (γ z) + \frac{\partial B_{y}}{\partial y} \cos (γ z) + B_{z} γ \sin (γ z)) d z = 0

(17e)

in which

\bar{χ_{3}} = \bar{χ_{4}} - \frac{{(\bar{χ_{2}})}^{2}}{\bar{χ_{1}}}, \bar{χ_{3}^{*}} = \bar{χ_{4}^{*}} - \frac{{(\bar{χ_{2}^{*}})}^{2}}{\bar{χ_{1}^{*}}}, \bar{χ_{3}} = \bar{χ_{5}} - \frac{\bar{χ_{2}} \bar{χ_{3}}}{\bar{χ_{1}}}, \bar{χ_{3}^{*}} = \bar{χ_{5}^{*}} - \frac{\bar{χ_{2}^{*}} \bar{χ_{3}^{*}}}{\bar{χ_{1}^{*}}}, \bar{χ_{3}} = {(\bar{χ_{3}})}^{2} / \bar{χ_{1}} - c_{1}^{2} χ_{7}, \bar{j_{3}^{*}} = {(\bar{χ_{3}^{*}})}^{2} / \bar{χ_{1}^{*}} - c_{1}^{2} χ_{7}

(18)

Inserting equation (14) into equation (12) then the results into equations (17a)–(17e) results in

\begin{array}{l} A_{11} (w) + A_{12} (ϕ_{x}) + A_{13} (ϕ_{y}) + A_{14} (I) + A_{15} (Φ) + A_{16} (Ψ) + B (w, I) + q = χ_{1} \frac{\partial^{2} w}{\partial t^{2}} + 2 ε χ_{1} \frac{\partial w}{\partial t} + \bar{χ_{5}} \frac{\partial^{3} ϕ_{x}}{\partial t^{2} \partial x} + \bar{χ_{5}^{*}} \frac{\partial^{3} ϕ_{y}}{\partial t^{2} \partial y} + \bar{χ_{7}} \frac{\partial^{4} w}{\partial t^{2} \partial x^{2}} + \bar{χ_{7}^{*}} \frac{\partial^{4} w}{\partial t^{2} \partial y^{2}}, \\ A_{21} (w) + A_{22} (ϕ_{x}) + A_{23} (ϕ_{y}) + A_{24} (I) + A_{25} (Φ) + A_{26} (Ψ) = \bar{χ_{3}} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} - \bar{χ_{5}} \frac{\partial^{3} w}{\partial t^{2} \partial x}, \\ A_{31} (w) + A_{32} (ϕ_{x}) + A_{33} (ϕ_{y}) + A_{34} (I) + A_{35} (Φ) + A_{36} (Ψ) = \bar{χ_{3}^{*}} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} - \bar{χ_{5}^{*}} \frac{\partial^{3} w}{\partial t^{2} \partial y}, \\ \int_{- h_{c} / 2 - h_{f}}^{h_{c} / 2} (\frac{\partial D_{x}}{\partial x} \cos (γ z) + \frac{\partial D_{y}}{\partial y} \cos (γ z) + D_{z} γ \sin (γ z)) d z + \int_{h_{c} / 2}^{h_{c} / 2 + h_{f}} (\frac{\partial D_{x}}{\partial x} \cos (γ z) + \frac{\partial D_{y}}{\partial y} \cos (γ z) + D_{z} γ \sin (γ z)) d z = 0, \\ \int_{- h_{c} / 2 - h_{f}}^{h_{c} / 2} (\frac{\partial B_{x}}{\partial x} \cos (γ z) + \frac{\partial B_{y}}{\partial y} \cos (γ z) + B_{z} γ \sin (γ z)) d z + \int_{h_{c} / 2}^{h_{c} / 2 + h_{f}} (\frac{\partial B_{x}}{\partial x} \cos (γ z) + \frac{\partial B_{y}}{\partial y} \cos (γ z) + B_{z} γ \sin (γ z)) d z = 0 \end{array}

(19)

in which

A_{1 i}, A_{2 i}, A_{3 i} (i = \bar{1,6}), B

are given in Appendix C.

The compatibility equation is rewritten in terms of stress function, deflection and rotations by substituting equation (14) into equation (5) as

g_{21} \frac{\partial^{4} I}{\partial x^{4}} + g_{11} \frac{\partial^{4} I}{\partial y^{4}} + F_{1} \frac{\partial^{4} I}{\partial x^{2} \partial y^{2}} + F_{2} \frac{\partial^{3} ϕ_{x}}{\partial x^{3}} + F_{3} \frac{\partial^{3} ϕ_{x}}{\partial x \partial y^{2}} + F_{4} \frac{\partial^{3} ϕ_{y}}{\partial y^{3}} + F_{5} \frac{\partial^{3} ϕ_{y}}{\partial y \partial x^{2}} - c_{1} g_{25} \frac{\partial^{4} w}{\partial x^{4}} - c_{1} g_{16} \frac{\partial^{4} w}{\partial y^{4}} + F_{6} \frac{\partial^{4} w}{\partial x^{2} \partial y^{2}} - ({\frac{\partial^{2} w}{\partial x \partial y}}^{2} - \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}}) = 0

(20)

where

F_{1} = g_{31} - 2 g_{12}, F_{2} = g_{23} - c_{1} g_{25}, F_{3} = g_{13} - c_{1} g_{15} - g_{32} + c_{1} g_{33}, F_{4} = g_{14} - c_{1} g_{16}, F_{5} = g_{24} - c_{1} g_{26} - g_{32} + c_{1} I_{33}, F_{6} = - c_{1} g_{15} - c_{1} g_{26} + 2 c_{1} g_{33}

(21)

4. Solution procedures

In the current study, four edges of the MEE sandwich plate are assumed to be simply supported and the boundary conditions are

w = ϕ_{y} = M_{x} = N_{x y} = 0, N_{x} = N_{x 0} a t x = 0, a, w = ϕ_{x} = M_{y} = N_{x y} = 0, N_{y} = N_{y 0} a t y = 0, b

(22)

with

N_{x 0}, N_{y 0}

present the fictitious compressive loading at simply supported edges of MEE sandwich plate.

Because the effect of temperature on the vibration of MEE sandwich plate is also considered, four edges are assumed to be immovable in the middle plane, and one has

\begin{array}{l} u = 0 a t x = 0, a, \\ v = 0 a t y = 0, b \end{array}

(23)

The condition (23) may be satisfied by applying following expressions (Quang et al., 2022)

\int_{0}^{b} \int_{0}^{a} \frac{\partial u}{\partial x} d x d y = 0, \int_{0}^{a} \int_{0}^{b} \frac{\partial v}{\partial x} d y d x = 0

(24)

in which the derivatives of displacement components are obtained from equations (4) and (14) as

\begin{array}{l} \frac{\partial u}{\partial x} = g_{11} \frac{\partial^{2} I}{\partial y^{2}} - g_{12} \frac{\partial^{2} I}{\partial x^{2}} + g_{13} \frac{\partial ϕ_{x}}{\partial x} + g_{14} \frac{\partial ϕ_{y}}{\partial y} - c_{1} g_{15} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ϕ_{x}}{\partial x}) - c_{1} g_{16} (\frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial ϕ_{y}}{\partial y}) + g_{17} ϕ_{0} + g_{18} ψ_{0} + g_{19} Δ T - \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}, \\ \frac{\partial v}{\partial y} = g_{21} \frac{\partial^{2} I}{\partial x^{2}} - g_{12} \frac{\partial^{2} I}{\partial y^{2}} + g_{23} \frac{\partial ϕ_{x}}{\partial x} + g_{24} \frac{\partial ϕ_{y}}{\partial y} - c_{1} g_{25} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ϕ_{x}}{\partial x}) - c_{1} g_{26} (\frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial ϕ_{y}}{\partial y}) + g_{27} ϕ_{0} + g_{28} ψ_{0} + g_{29} Δ T - \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2} \end{array}

(25)

In order to meet boundary conditions, the double trigonometric functions for five unknowns of the MEE sandwich plate are proposed as

\begin{array}{l} w (x, y, t) = W (t) {\sin λ}_{m} x {\sin δ}_{n} y, \\ Φ (x, y, t) = ϕ (t) {\sin λ}_{m} x {\sin δ}_{n} y, \\ Ψ (x, y, t) = ψ (t) {\sin λ}_{m} x {\sin δ}_{n} y, \\ ϕ_{x} (x, y, t) = Φ_{x} (t) {\cos λ}_{m} x {\sin δ}_{n} y, \\ ϕ_{y} (x, y, t) = Φ_{y} (t) {\sin λ}_{m} x {\cos δ}_{n} y \end{array}

(26)

where

λ_{m} = m π / a, δ_{n} = n π / b; W (t), Φ_{x} (t), Φ_{y} (t), ϕ (t)

and

ψ (t)

are the maximum values of the deflection, electric, and magnetic potential and rotations, respectively.

The form and coefficients of Airy’s stress function are determined by introducing equation (26) into equation (20) as

I (x, y, t) = J_{1} (t) \cos 2 λ_{m} x + J_{2} (t) \cos 2 δ_{n} y + J_{3} (t) {\sin λ}_{m} x {\sin δ}_{n} y + \frac{1}{2} N_{y 0} x^{2} + \frac{1}{2} N_{x 0} y^{2}

(27)

where

J_{1} = \frac{δ_{n}^{2}}{32 g_{21} λ_{m}^{2}} W^{2}, J_{2} = \frac{λ_{m}^{2}}{32 g_{11} δ_{n}^{2}} W^{2}, J_{3} = L_{1} W + L_{2} Φ_{x} + L_{3} Φ_{y}

(28)

with

\begin{array}{l} L_{1} = \frac{c_{1} g_{25} λ_{m}^{4} + c_{1} g_{16} δ_{n}^{4} - F_{6} λ_{m}^{2} δ_{n}^{2}}{g_{21} λ_{m}^{4} + F_{1} λ_{m}^{2} δ_{n}^{2} + g_{11} δ_{n}^{4}}, \\ L_{2} = \frac{- (F_{2} λ_{m}^{3} + F_{3} λ_{m} δ_{n}^{2})}{g_{21} λ_{m}^{4} + F_{1} λ_{m}^{2} δ_{n}^{2} + g_{11} δ_{n}^{4}}, L_{3} = \frac{- (F_{4} δ_{n}^{3} + F_{5} λ_{m}^{2} δ_{n})}{g_{21} λ_{m}^{4} + F_{1} λ_{m}^{2} δ_{n}^{2} + g_{11} δ_{n}^{4}} \end{array}

(29)

Substituting the form of variables in equations (26)–(29) into equation (19) and applying Galerkin procedure, the motion equations are transformed into the following forms

\begin{array}{l} j_{11} W + j_{12} Φ_{x} + j_{13} Φ_{y} + j_{14} W Φ_{x} + j_{15} W Φ_{y} + j_{16} ϕ + j_{17} ψ + [o_{1} - N_{x 0} λ_{m}^{2} - N_{y 0} δ_{n}^{2}] W + (o_{2} + o_{3}) W^{2} + o_{4} W^{4} + o_{5} q = χ_{0} \frac{\partial^{2} W}{\partial t^{2}} + 2 ε χ_{1} \frac{\partial W}{\partial t} - λ_{m} \bar{χ_{5}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - δ_{n} \bar{χ_{5}^{*}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}}, \\ (j_{21} + o_{6}) W + j_{22} Φ_{x} + j_{23} Φ_{y} + j_{24} ϕ + j_{25} ψ + o_{7} W^{2} = \bar{χ_{3}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - λ_{m} \bar{χ_{5}} \frac{\partial^{2} W}{\partial t^{2}}, \\ (j_{31} + o_{8}) W + j_{32} Φ_{x} + j_{33} Φ_{y} + j_{34} ϕ + j_{35} ψ + o_{9} W^{2} = \bar{χ_{3}^{*}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} - δ_{n} \bar{χ_{5}^{*}} \frac{\partial^{2} W}{\partial t^{2}}, \\ j_{41} W + j_{42} Φ_{x} + j_{43} Φ_{y} + j_{44} ϕ + j_{45} ψ = 0, \\ j_{51} W + j_{52} Φ_{x} + j_{53} Φ_{y} + j_{54} ϕ + j_{55} ψ = 0 \end{array}

(30)

in which the expressions of coefficients

j_{1 i} (i = \bar{1,7}), j_{m n} (m = \bar{2,3,4,5}, n = \bar{1,5}), o_{i} (i = \bar{1,9})

are highlighted in Appendix D.

The coefficients $N_{x 0}$ are $N_{y 0}$ of stress function are determined by replacing equations (26)–(28) into equation (25) then the results into equation (24) as

\begin{array}{l} N_{x 0} = l_{1} W + l_{4} W^{2} + l_{2} Φ_{x} + l_{3} Φ_{y} + l_{5} ϕ_{0} + l_{6} ψ_{0} + l_{7} Δ T, \\ N_{y 0} = m_{1} W + m_{4} W^{2} + m_{2} Φ_{x} + m_{3} Φ_{y} + m_{5} ϕ_{0} + m_{6} ψ_{0} + m_{7} Δ T \end{array}

(31)

with the expressions of coefficients

g_{i} (i = \bar{1,7})

and

f_{j} (j = \bar{1,7})

may be found in Appendix E.

Substitution of equation (31) into equation (30) gives

\begin{array}{l} (j_{11} + o_{1}^{1}) W + j_{12} Φ_{x} + j_{13} Φ_{y} + j_{14}^{1} W Φ_{x} + j_{15}^{1} W Φ_{y} + j_{16} ϕ + j_{17} ψ + (o_{2}^{1} + o_{3}) W^{2} + o_{4}^{1} W^{3} + o_{5} q = χ_{0} \frac{\partial^{2} W}{\partial t^{2}} + 2 ε χ_{1} \frac{\partial W}{\partial t} - λ_{m} \bar{χ_{5}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - δ_{n} \bar{χ_{5}^{*}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}}, \\ (j_{21} + o_{6}) W + j_{22} Φ_{x} + j_{23} Φ_{y} + j_{24} ϕ + j_{25} ψ + o_{7} W^{2} = \bar{χ_{3}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - λ_{m} \bar{χ_{5}} \frac{\partial^{2} W}{\partial t^{2}}, \\ (j_{31} + o_{8}) W + j_{32} Φ_{x} + j_{33} Φ_{y} + j_{34} ϕ + j_{35} ψ + o_{9} W^{2} = \bar{χ_{3}^{*}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} - δ_{n} \bar{χ_{5}^{*}} \frac{\partial^{2} W}{\partial t^{2}}, \\ j_{41} W + j_{42} Φ_{x} + j_{43} Φ_{y} + j_{44} ϕ + j_{45} ψ = 0, \\ j_{51} W + j_{52} Φ_{x} + j_{53} Φ_{y} + j_{54} ϕ + j_{55} ψ = 0 \end{array}

(32)

where

\begin{array}{l} j_{14}^{1} = (j_{14} - λ_{m}^{2} l_{2} - δ_{n}^{2} m_{2}), l_{15}^{1} = (j_{15} - λ_{m}^{2} l_{3} - δ_{n}^{2} m_{3}), \\ o_{1}^{1} = o_{1} - (λ_{m}^{2} l_{5} + δ_{n}^{2} m_{5}) ϕ_{0} - (λ_{m}^{2} l_{6} + δ_{n}^{2} m_{6}) ψ_{0} \\ - (λ_{m}^{2} l_{7} + δ_{n}^{2} m_{7}) Δ T, \\ o_{2}^{1} = (o_{2} - λ_{m}^{2} l_{1} - δ_{n}^{2} m_{1}), n_{4}^{1} = (n_{4} - λ_{m}^{2} l_{4} - δ_{n}^{2} m_{4}) \end{array}

(33)

From the last two equations of the system equation (32), we can determine the dependence of the amplitudes of electric and magnetic potentials on the amplitudes of deflection and rotations. Substitution of these expressions into the first three equations, equation (32) becomes

\begin{array}{l} (j_{11}^{1} + o_{1}^{1}) W + j_{12}^{1} Φ_{x} + j_{13}^{1} Φ_{y} + j_{14}^{1} W Φ_{x} + j_{15}^{1} W Φ_{y} + (o_{2}^{1} + o_{3}) W^{2} \\ + o_{4}^{1} W^{3} + o_{5} q = χ_{0} \frac{\partial^{2} W}{\partial t^{2}} + 2 ε χ_{1} \frac{\partial W}{\partial t} - λ_{m} \bar{χ_{5}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - δ_{n} \bar{χ_{5}^{*}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}}, \\ (j_{21}^{1} + o_{6}) W + j_{22}^{1} Φ_{x} + j_{23}^{1} Φ_{y} + o_{7} W^{2} = \bar{χ_{3}} \frac{\partial^{2} Φ_{x}}{\partial t^{2}} - λ_{m} \bar{χ_{5}} \frac{\partial^{2} W}{\partial t^{2}}, \\ (j_{31}^{1} + o_{8}) W + j_{32}^{1} Φ_{x} + j_{33}^{1} Φ_{y} + o_{9} W^{2} = \bar{χ_{3}^{*}} \frac{\partial^{2} Φ_{y}}{\partial t^{2}} - δ_{n} \bar{χ_{5}^{*}} \frac{\partial^{2} W}{\partial t^{2}} \end{array}

(34)

in which the expressions of coefficients

j_{l m}^{1} (l = \bar{1,3}, m = \bar{1,3})

are expressed in Appendix F.

The dynamic response of MEE sandwich plate subjected to uniformly distribute transverse load $q = Q \sin Ω t$ can be expressed by applying the fourth-order Runge–Kutta method. The initial conditions are assumed to be $W (0) = 0, \frac{d W}{d t} (0) = 0, Φ_{x} (0) = 0, \frac{d Φ_{x}}{d t} (0) = 0, Φ_{y} (0) = 0, \frac{d Φ_{y}}{d t} (0) = 0 .$

The natural frequency of the MEE sandwich plate is the smallest positive solution of the following equation

| \begin{array}{l} j_{11}^{1} + o_{1}^{1} + χ_{0} ω^{2} & j_{12}^{1} - λ_{m} \bar{χ_{5}} ω^{2} & j_{13}^{1} - δ_{n} \bar{χ_{5}^{*}} ω^{2} \\ j_{21}^{1} + o_{6} - λ_{m} \bar{χ_{5}} ω^{2} & j_{22}^{1} + \bar{χ_{3}} ω^{2} & j_{23}^{1} \\ j_{31}^{1} + o_{8} - δ_{n} \bar{χ_{5}^{*}} ω^{2} & j_{32}^{1} & j_{33}^{1} + \bar{χ_{3}^{*}} ω^{2} \end{array} | = 0

(35)

5. Results and discussion

5.1. Verification studies

The accuracy and effectiveness of the proposed model and method are verified by comparing present results with those from accessible literature. The functionally graded graphene reinforced composite laminated plates with Poly (methyl methacrylate) matrix is considered. The laminated plate has ten layers in which the thickness of each layer is

0.2 m m

. The length and width of the plate are set as

a = b = 20 m m

. The material properties of matrix and graphene are assumed to depend on temperature at

T = 300 K

for this case. The dimensionless natural frequencies which are defined as

\bar{Ω} = Ω (a^{2} / h) \sqrt{ρ_{m} / E_{m}}

are used to compare with obtained results of Shen et al. (2017) based on Reddy’s higher order shear deformation plate theory and two-step perturbation technique.

ρ_{m}

and

E_{m}

are the density and Young’s modulus of the matrix. Three values of vibration modes

(m, n) = (1,1), (1,2), (1,3)

and three graphene distribution patterns FG-X, FG-O, and UD are considered. The arrangement of graphene volume fractions of ten layers is

{[0.11 / 0.09 / 0.07 / 0.05 / 0.03]}_{S}

for FG-X,

{[0.04 / 0.05 / 0.07 / 0.09 / 0.11]}_{S}

for FG-O, and

{[0.07]}_{10}

for UD. The comparison results from Table 4 demonstrate that present results and those obtained by Shen et al. (2017) match very well.

Table 4.

Comparison of dimensionless natural frequency $\bar{Ω} = Ω (a^{2} / h) \sqrt{ρ_{m} / E_{m}}$ of graphene reinforced composite laminated plates.

Source	$(m, n)$	Graphene distribution pattern
Source	$(m, n)$	UD	FG-X	FG-O
Shen et al. (2017)	(1,1)	28.0982	29.5212	23.5769
Present	(1,1)	27.4161	28.4721	23.2696
Shen et al. (2017)	(1,2)	64.8785	65.6449	56.0228
Present	(1,2)	64.0284	64.1537	55.7595
Shen et al. (2017)	(1,3)	116.0301	113.4456	103.7875
Present	(1,3)	115.3143	111.8775	103.9921

5.2. Parametric studies

5.2.1. Natural frequency

Table 5 and Figure 3 demonstrate the variation of the natural frequencies

(\times 1 0^{3} s^{- 1})

of the simply supported MEE sandwich plate having different elastic foundations coefficients

k_{1} (M P a / m), k_{2} (M P a . m),

temperature change

Δ T (K)

and graphene distribution pattern. It is seen that the natural frequency becomes higher as the temperature change increases. The reasonable reason is, according to Tables 1 and 2, the existence of temperature will reduce the elastic modulus of core layer, which results in the stiffness reduction of MEE sandwich plate. Moreover, one may see that the downward trend of natural frequency caused by temperature change will be more significant with high values of temperature change. It is because of the nonlinear effect of temperature on material properties. Results from Table 5 and Figure 3 also indicate that MEE sandwich plate with FG-X graphene distribution pattern exhibits the largest natural frequencies followed by UD, and FG-O has the weakest natural frequencies. The reason to arise this trend is higher graphene is concentrated on the middle plane of MEE sandwich plate with the FG-O distribution pattern. Contrary, the MEE sandwich plate with the FG-X distribution pattern has the largest stiffness due to the large concentration of graphene on the top and bottom faces. For elastic foundations, the natural frequency of MEE sandwich plate on elastic foundations is higher than plate without elastic foundations.

Table 5.

Variation of the natural frequencies $(\times 1 0^{3} s^{- 1})$ of the MEE sandwich plate having different elastic foundations coefficients $k_{1} (M P a / m), k_{2} (M P a . m)$ , temperature change $Δ T (K)$ , and graphene distribution pattern.

$(k_{1}, k_{2})$	$Δ T (K)$	Graphene distribution pattern
$(k_{1}, k_{2})$	$Δ T (K)$	FG-X	FG-O	UD
(0, 0)	0	2.4165	2.1952	2.3068
	200	2.2040	2.0277	2.0946
	400	1.9986	1.7391	1.8654
(0.1, 0.02)	0	2.4259	2.2055	2.3166
	200	2.2143	2.0388	2.1054
	400	2.1814	1.7520	2.0634

Table 6, Figures 4 and 5 display the variation of the natural frequencies

(\times 1 0^{3} s^{- 1})

of the MEE sandwich plate depending on electric potential

ϕ_{0} (V)

, magnetic potentials

ψ_{0} (A)

, and volume fraction of piezoelectric phase

B a T i O_{3}

of MEE face sheets

V_{B a}

. The FG-O is chosen for graphene distribution pattern. For electric and magnetic potentials, both of positive and negative values are considered in this research. It is seen that the natural frequencies of MEE sandwich plate will decrease as electric potential becomes lower and magnetic potential gets larger. The reason may be attributed to the increment in the mid-plane maximum electric and magnetic potentials with the lower value of

ψ_{0} (A)

and higher value of

ϕ_{0} (V)

. It also can be seen from Table 6, Figures 4 and 5 that the natural frequencies of the MEE sandwich plate significantly reduces as the volume fraction of piezoelectric phase increases. The reason for this trend is that, as the proportion of piezoelectric phase increase and the proportion of piezomagnetic phase reduces, the elastic stiffness coefficient remarkably decreases. In addition, it is completely understandable that the natural frequency is unchanged with different values of electric potential when

V_{B a} = 0

. Similarly, the natural frequency is unchanged with different values of magnetic potential when

V_{B a} = 1

(only piezoelectric phase).

Table 6.

Variation of the natural frequencies $(\times 1 0^{3} s^{- 1})$ of the MEE sandwich plate depending on electric potential $ϕ_{0} (V)$ , magnetic potential $ψ_{0} (A)$ , and volume fraction of piezoelectric phase $B a T i O_{3}$ of MEE face sheets $V_{B a}$ .

$ϕ_{0} (V)$	$ψ_{0} (A)$	$V_{B a}$
$ϕ_{0} (V)$	$ψ_{0} (A)$	0	0.2	0.5	0.8	1
−400	−200	2.0294	1.9990	1.8969	1.8534	1.8715
	0	2.0906	2.0340	1.9770	1.8676	1.8715
	200	2.1500	2.0684	2.0539	1.8818	1.8715
0	−200	2.0294	1.9950	1.8887	1.8414	1.8586
	0	2.0906	2.0301	1.9691	1.8558	1.8586
	200	2.1500	2.0645	2.0464	1.8700	1.8586
400	−200	2.0294	1.9910	1.8805	1.8294	1.8456
	0	2.0906	2.0261	1.9613	1.8438	1.8456
	200	2.1500	2.0606	2.0388	1.8582	1.8456

5.2.2. Dynamic response

Figures 6 and 7, respectively, show the variation of dynamic response of the MEE sandwich plate depending on magnetic potential and electric potential, respectively. The effect of temperature on the mechanical properties of core layer is considered in this research. Obviously, the harmonic form of the uniformly distributed external pressure causes the harmonic form of the dynamic response. In addition, the results from those figures display the opposite effect of magnetic and electric potentials on the deflection amplitude of the sandwich plate. The rate of change of the maximum deflection amplitude is 32.47% for the change of $400 A$ in magnetic potential. For electric potential, the change of maximum deflection amplitude is negligible. Increasing the electric potential from $- 400 V$ to $400 V$ causes the augment of 5% in maximum deflection amplitude.

Figures 8 and 9 show the variation of the dynamic response of the MEE sandwich plate depending on graphene distribution pattern (FG-X, FG-O, UD) and volume fraction of graphene $V_{G} = 7 %, 9 %$ and $11 %$ for UD graphene distribution patterns, respectively. It is seen that the MEE plate with FG-O graphene distribution pattern has the largest deflection amplitude and FG-X graphene distribution pattern exhibits the lowest deflection amplitude. The differences may be caused by the different degrees of the stiffness enhancement reinforced by the three types of graphene distribution patterns. Besides, the deflection amplitude is significantly reduced by increasing the graphene volume fraction. This is because the elastic modulus of core layer will improve by adding graphene to the metal matrix, which depicts that higher stiffness is generated.

6. Conclusions

In this research work, the analytical approach for the vibration characteristics of sandwich plate made of GRMMC core layer and MEE face sheets is presented. The framework of Hamilton’s principle is employed to derive the basic equations then these equations are solved by applying Galerkin procedure. The influences of temperature change, electric, and magnetic potentials, graphene distribution pattern and volume fraction of MEE phase are investigated through parametric studies. The main conclusions are drawn below:

(1) The reinforcement of graphene to metal matrix enhances remarkably the mechanical properties of the GRMMC core layer. Therefore, the natural frequency of sandwich plate increases whereas the deflection amplitude reduces significantly as the graphene volume fraction increases.

(2) Among three graphene distribution patterns, The FG-X sandwich plate has the highest natural frequency whereas the FG-O sandwich plate has the highest deflection amplitude.

(3) Increasing the magnetic potential or reducing the electric potential raises the mechanical properties of sandwich plate and this increase of mechanical properties leads to the rise of natural frequency.

(4) The elastic modulus and stiffness of the sandwich plate reduce significantly as the temperature change increases, which results in the reduction of natural frequency.

Supplemental Material

Supplemental Material - Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core

Supplemental Material for Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core by Quan Tran Quoc, Dat Ngo Dinh, and Dinh Duc Nguyen in Journal of Vibration and Control

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 research is funded by the Project number QG.23.65 of Vietnam National University, Hanoi. The authors are grateful for this support.

ORCID iDs

Quoc Tran Quan

Dinh Duc Nguyen

Supplemental Material

Supplemental material for this article is available online.

References

Abdolhoseyni

Danesh

(2023) Sound transmission loss of a sandwich functionally graded cylindrical shell integrated with magneto-electro-elastic patches. Journal of Sound and Vibration 543: 117350.

Al-Furjan

MSH

Shan

Shen

, et al. (2022) Combination of FEM-DQM for nonlinear mechanics of porous GPL-reinforced sandwich nanoplates based on various theories. Thin-Walled Structures 178: 109495.

Al-Furjan

MSH

Shan

, et al. (2023) Nano supercapacitors with practical application in aerospace technology: vibration and wave propagation analysis. Aerospace Science and Technology 133: 108082.

Balandin

(2011) Thermal properties of graphene and nanostructured carbon materials. Nature Materials 10: 569–581.

Bamdad

Mohammadimehr

Alambeigi

(2019) Analysis of sandwich Timoshenko porous beam with temperature-dependent material properties: magneto-electro-elastic vibration and buckling solution. JVC/Journal of Vibration and Control 25: 2875–2893.

Brush

Almroth

(1975) Buckling of Bars, Plates and Shells. New York, NY: Graw-Hill.

Chen

Yang

Dong

, et al. (2020) Advances in graphene reinforced metal matrix nanocomposites: Mechanisms, processing, modelling, properties and applications. Nanotechnology and Precision Engineering 3: 189–210.

Chu

Jia

(2014) Enhanced strength in bulk graphene–copper composites. Physica Status Solidi (A) Applications and Materials Science 190: 184–190.

Chu

Al-Furjan

MSH

Kolahchi

(2023) Energy harvesting and dynamic response of SMA nano conical panels with nanocomposite piezoelectric patch under moving load. Engineering Structures 292: 116538.

10.

Ebrahimi

Farazmandnia

Kokaba

, et al. (2021) Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory. Engineering with Computers 37: 921–936.

11.

Fan

Xiang

Shen

, et al. (2018a) Nonlinear low-velocity impact response of FG-GRC laminated plates resting on visco-elastic foundations. Composites, Part B: Engineering 144: 184–194.

12.

Fan

Xiang

Shen

, et al. (2018b) Low-velocity impact response of FG-GRC laminated beams resting on visco-elastic foundations. International Journal of Mechanical Sciences 141: 117–126.

13.

Fan

Xiang

Shen

(2019) Nonlinear forced vibration of FG-GRC laminated plates resting on visco-Pasternak foundations. Composite Structures 209: 443–452.

14.

Hui

Nan

Sun

, et al. (2015) High resolution magnetometer based on a high frequency magnetoelectric MEMS-CMOS oscillator. Journal of Microelectromechanical Systems 24: 134–143.

15.

Khadem

Hasan

Rahman

ANMM

, et al. (2022) Fabrication, properties, and performance of graphene-based textile fabrics for supercapacitor applications: a review. Journal of Energy Storage 56: 105988.

16.

Wang

(2019) A review on enhancement of mechanical and tribological properties of polymer composites reinforced by carbon nanotubes and graphene sheet: molecular dynamics simulations. Composites, Part B: Engineering 160: 348–361.

17.

Masoud

Ali

Mostafa

, et al. (2019) Buckling and frequency analysis of the nonlocal strain–stress gradient shell reinforced with graphene nanoplatelets. JVC/Journal of Vibration and Control 25: 2627–2640.

18.

Mokhalingam

Kumar

Srivastava

(2017) Mechanical behaviour of graphene reinforced aluminum nano composites. Materials Today Proceedings 4: 3952–3958.

19.

Phuong

Dong

Van Doan

, et al. (2022) Nonlinear buckling of higher-order shear deformable stiffened FG-GRC laminated plates with nonlinear elastic foundation subjected to combined loads. Aerospace Science and Technology 127: 107736.

20.

Quang

Quan

Phuong

(2022) Static buckling analysis and geometrical optimization of magneto-electro-elastic sandwich plate with auxetic honeycomb core. Thin-Walled Structures 173: 108935.

21.

Reddy

(2004) Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. Boca Raton: CRC Press.

22.

Safina

Krylova

Baimova

(2022) Molecular dynamics study of the mechanical properties and deformation behavior of graphene/metal composites. Materials Today Physics 28: 100851.

23.

Shen

Xiang

Lin

(2017) Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments. Computer Methods in Applied Mechanics and Engineering 319: 175–193.

24.

Shen

Xiang

Reddy

(2020) Effect of negative Poisson’s ratio on the post-buckling behavior of FG-GRMMC laminated plates in thermal environments. Composite Structures 253: 112731.

25.

Shen

Xiang

Reddy

(2021) Assessment of the effect of negative Poisson’s ratio on the thermal postbuckling of temperature dependent FG-GRMMC laminated cylindrical shells. Computer Methods in Applied Mechanics and Engineering 376: 113664.

26.

Sirimontree

Thongchom

Saffari

, et al. (2023) Effects of thermal environment and external mean flow on sound transmission loss of sandwich functionally graded magneto-electro-elastic cylindrical nanoshell. European Journal of Mechanics - A: Solids 97: 104774.

27.

Vinyas

(2020) On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT. Composite Structures 240: 112044.

28.

Vinyas

(2022a) Active control of nonlinear coupled transient vibrations of multifunctional sandwich plates with agglomerated FG-CNTs core/magneto-electro-elastic facesheets. Thin-Walled Structures 179: 109547.

29.

Vinyas

(2022b) Nonlinear damping of auxetic sandwich plates with functionally graded magneto-electro-elastic facings under multiphysics loads and electromagnetic circuits. Composite Structures 290: 115523.

30.

Vinyas

(2023) Artificial neural network (ANN) based investigation on the static behaviour of piezo-magneto-thermo-elastic nanocomposite sandwich plate with CNT agglomeration and porosity. International Journal of Non-linear Mechanics 153: 104406.

31.

Vinyas

Sunny

Harursampath

, et al. (2019) Influence of interphase on the multi-physics coupled frequency of three-phase smart magneto-electro-elastic composite plates. Composite Structures 226: 111254.

32.

Yang

(2016) A new angular velocity sensor with ultrahigh resolution using magnetoelectric effect under the principle of Coriolis force. Sensors and Actuators A: Physical 238: 234–239.

33.

Zhao

Bai

, et al. (2020) An overview of graphene and its derivatives reinforced metal matrix composites: preparation, properties and applications. Carbon 170: 302–326.

34.

Zhao

Zhang

Wang

, et al. (2022) Nonlinear analysis of carbon nanotube reinforced functionally graded plates with magneto-electro-elastic multiphase matrix. Composite Structures 297: 115969.

35.

Zhou

Yang

, et al. (2022) On the static analysis of inhomogeneous magneto-electro-elastic plates in thermal environment via element-free Galerkin method. Engineering Analysis with Boundary Elements 134: 539–552.

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