An isogeometric approach for nonlocal bending and free oscillation of magneto-electro-elastic functionally graded nanobeam with elastic constraints

Abstract

This work uses isogeometric analysis (IGA), which is based on nonlocal hypothesis and higher-order shear beam hypothesis, to investigate the static bending and free oscillation of a magneto-electro-elastic functionally graded (MEE-FG) nanobeam subject to elastic boundary constraints (BCs). The magneto-electric boundary condition and the Maxwell equation are used to calculate the variation of electric and magnetic potentials along the thickness direction of the nanobeam. This study is innovative since it does not use the conventional boundary conditions. Rather, an elastic system of straight and torsion springs with controllable stiffness is used to support nanobeams’ beginning and end positions, creating customizable BCs. The governing equations of motion of nanobeams are established by applying Hamilton’s principle and IGA is used to determine deflections and natural frequency values. Verification studies were performed to evaluate the convergence and accuracy of the proposed method. Aside from this, the impact of the input parameters on the static bending and free oscillation of the MEE-FG nanobeam is examined in detail. These findings could be valuable for analyzing and designing innovative structures constructed of functionally graded MEE materials.

Graphical abstract

Keywords

elastic boundary conditions / isogeometric analysis / nanobeam via nonlocal theory / grading of magneto-electro-elastic functions

1 Introduction

Magneto-electro-elasticity has garnered significant attention in various industrial fields, particularly in nuclear devices, where a primary magnetic field is present. In magneto-electro thermoelastic materials, the application of thermal or a magnetic/electric field leads to mechanical deformation, owing to their distinctive ability to convert energy between three distinct forms: magnetic, electric, and mechanical. Numerous studies have explored the interplay between magnetic, thermal, and strain fields in recent years. Vinyas and Kattimani [1] developed the finite element (FE) formulation for static analysis of a magneto-electro-elastic (MEE) beam under different thermal loading and boundary conditions. Zhang et al. [2] investigated the static bending and free oscillation analysis of a transversely isotropic MEE beam lying on an elastic medium based on the Timoshenko beam hypothesis and the Navier method. Zhang et al. [3] analyzed MEE laminated beams’ nonlinear bending and oscillation issues in temperature environments based on the higher-order shear beam hypothesis in conjunction with the von Kármán type. For the free oscillation of multilayered MEE plates under simply supported boundary conditions, Xin and Hu [4] employed semi-analytical method. A semi-analytical technique based on the scaled boundary FE method (FEM) for studying the deformation of a MEE plate is carried out by Liu et al. [5]. Based on von Karman’s nonlinear strain-displacement hypothesis and the high-order shear deformation hypothesis, Xu et al. [6] examined MEE composite plates’ nonlinear free oscillation behavior.

Researchers’ interest in functionally graded materials (FGMs), a unique class of structural materials, has grown recently. These materials are nonhomogeneous, with their composition gradually varying along spatial coordinates. Originally, FGMs were primarily developed for aerospace and fusion reactors as thermal barrier materials. However, they have found use in a variety of industries, including as the military, automotive, biomedical, semiconductor, manufacturing, and general structural components in high-temperature environments. FGMs are engineered to combine the desirable properties of their constituents. For instance, thermal protection structures employ ceramic/metal FGMs to provide heat and corrosion resistance on the ceramic side while preserving the structural integrity and rigidity of the metal surface.

Additionally, MEE composites are useful for making hydrophones, electric packing, sensors, actuators, magnetic field probes, and medical ultrasonic imaging. Numerous works have been conducted to understand the mechanical behavior of structures composed of magneto-electroelastic and compositionally graded materials. Kattimani and Ray [7] investigated geometrically nonlinear vibrations of functionally graded MEE plates using the layer-wise shear deformation hypothesis and a three-dimensional FE method. Vinyas and Kattimani [8] used the FE approach to compute the multiphysics response of MEE plates in the thermal environment. Ebrahimi et al. [9] developed a four-variable shear deformation refined plate hypothesis for analyzing the free oscillation response of MEE-FG porous plates with various boundary constraints (BCs) lying on an elastic medium. Sh et al. [10] examined the geometrically nonlinear free oscillation and transient response of porous functionally graded MEE plates using the first-order shear deformation hypothesis and von Karman’s nonlinear strain. Zhang et al. [11] utilized the eight-node quadrilateral plate/shell elements based on the Mindlin hypothesis for the static and transient response of FG-MEE plates and shells.

To comprehend the mechanical response of nanostructural elements when employed as components in Nano-electromechanical Systems (NEMS), the development of size-dependent continuum theories such as couple stress hypothesis [12], strain gradient elasticity hypothesis [13], the nonlocal elasticity hypothesis [14–17], the nonlocal strain gradient hypothesis [18], surface effect [19–20], and flexoelectric effect [21] is essential for analyzing the mechanical response of nanostructures. Rabczuk et al. [22] developed the novel nonlocal operator hypothesis based on the variational principle proposed to solve partial differential equations. Ren et al. [23,24] proposed the higher-order nonlocal operator method to solve boundary value problems. Micro- and nanoscale devices and systems like biosensors, atomic force microscopes, micro-electromechanical systems, and NEMS frequently employ the MEE-FG nanobeams. The behavior of MEE-FG nanostructures using the nonlocal hypothesis is studied by several researchers, such as Arefi and Zenkour [25] proposed the Timoshenko beam model for analyzing wave propagation of a nanobeam made of MEE-FG resting on the Visco-Pasternak foundation. Xiao et al. [26] used Eringen’s nonlocal elasticity theory using a refined beam model to study MEE-FG porous nanobeams’ nonlinear thermal stability and post-stability. Żur et al. [27] presented the analytical solutions based on the nonlocal refined sinusoidal shear deformation plate hypothesis to compute MEE-FG nanoplates’ buckling and free vibration responses. Lyu and Ma [28] developed the differential quadrature method (DQM) for solving the nonlinear dynamic response of a MEE FG nanobeam. The thermomechanical behavior of nanoplates with magneto-electro-elastic face layers and a functionally graded porous core nanoplate is investigated by Koç et al. [29]. Zhuang et al. [30] utilized the nonlocal operator method to facilitate dynamic fracture by utilizing an explicit phase field model.

The isogeometric analysis (IGA) approach serves as a link between computer-aided design and FE analysis. The fundamental tenet of IGA is that unknown fields will be approximated using the same basis functions as are strictly utilized to characterize the geometry domain. When compared to other numerical approaches like the meshfree method, finite trip method, FEM, and others, it is anticipated that IGA would produce more accurate results with lower computing costs for high speed data transmission detail and refined plate theory issues since geometry is correctly modeled and the number of unknown terms is not increased. In particular, the IGA has recently been used in many structures since it can satisfy higher-order NURBS function derivatives and is appropriate for applying the elasticity nonlocal hypothesis. For the topological optimization of piezoelectric/flexoelectric materials, Ghasemi et al. [31–33] provided a design methodology based on a combination of IGA, level set, and point-wise density mapping techniques. To efficiently reuse numerical simulation on a group of topology-consistent models, Wang et al. [34] introduced IGA-Reuse-Net. Phase-field IGA was developed by Nguyen et al. [35] to analyze fracture propagation in porous functionally graded structures. Zhuang et al. [36] provides an overview of the theoretical and computer implementation components of phase field models of fracture.

Surveying the literature reveals that few works deal with multi-physics coupled MEE-FG nanobeam using IGA and nonlocal theory. This research aims to propose an IGA based on a refined higher-order shear beam and nonlocal theories applied to investigate the free oscillation and bending of MEE-FG nanobeam with elastic BCs resting on the Pasternak elastic foundation. The Maxwell equation and magneto-electric boundary condition are used to determine the variation of electric and magnetic potentials along the thickness direction of the nanobeam. The novelty of this work is that the traditional boundary conditions are not used, but instead, the beginning and the end positions of the MEE-FG nanobeam are supported by an elastic system consisting of straight springs and torsion springs of high strength which its stiffness is controlled to give arbitrary BCs. The accuracy and convergence of the present method are verified through various numerical examples. The results obtained from this study can provide valuable insights into the design and optimization of MEE-FG nanobeams for various applications.

2 Modeling of magneto-electro-elastic functionally graded nanobeam with elastic boundary condition

Fig.1 shows that a MEE-FG nanobeam has proportions of a rectangular cross-section. The MEE-FG nanobeam has length L, width b, and thickness h. Mechanical properties of nanobeam materials vary with nanobeam thickness with an exponential law. The MEE-FG nanobeam is lying on elastic Pasternak’s medium with two stiffness factors, including spring layer stiffness k_{\mathrm{w}} and shear layer stiffness k_{\mathrm{s}}. Both of the positions of the MEE-FG nanobeam: the beginning (x=0) and the end of MEE-FG nanobeam (x=L) place a system of two groups of linear springs \left(K_{\mathrm{u}0},K_{\mathrm{b}0},K_{\mathrm{s}0};K_{\mathrm{u}\mathrm{L}},K_{\mathrm{b}\mathrm{L}},K_{\mathrm{s}\mathrm{L}}\right) and one groups of rotational springs \left(K_{\mathrm{b}x0},K_{\mathrm{s}x0};K_{\mathrm{b}x\mathrm{L}},K_{\mathrm{s}x\mathrm{L}}\right). The BCs can be easily created by assigning appropriate stiffness values to the arranged springs.

The following formulas illustrate how the mechanical characteristics of nanobeam materials change with thickness and porosity [37]:

(1)\mathrm{\Gamma }\left(\mathit{\text{z}}\right)={\mathrm{\Gamma }}_m{e}^{p_{\mathit{\text{z}}}\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)},

where \mathrm{\Gamma } denotes for an effective material property, such as Poisson’s ratio \upsilon, mass density \rho, or Young’s modulus E; parameters of thermal, humidity, electric and magnetic fields; m is the symbol for material properties at position \mathit{\text{z}}/h=-0.5, p_{\mathit{\text{z}}} represent the control coefficient for the material’s mechanical properties that vary via the nanobeam thickness.

The xy plane located on the midplane of the undeformed nanobeam and the nanobeam along the beam thickness direction is described by the Cartesian Oxyz coordinate system. Between the upper and bottom surfaces of the MEE-FG nanobeam, there is an electric potential V_0 and a magnetic potential {\Psi }_0 that affect the magneto-electro-elastic field, which is poled along the z direction. The nanobeam surface BCs are presented as the following Eq. (2) [38]:

(2)\phi \left(h/2\right)=V_0,\varphi \left(h/2\right)={\Psi }_0,\phi \left(-h/2\right)=0,\varphi \left(-h/2\right)=0,

where \varphi and \phi are the magnetic and electric potentials.

3 Mathematical foundations and solution methods

In this paper, the nonlocal theory to establish an MEE-FG nanobeam model incorporating sine high-order shear nanobeam hypothesis and the generalized strain−displacement relation can be written as Eq. (3) [39–42]:

(3)\mathit{u}=\{\begin{array}{l}{u}_x\left(x,\mathit{\text{z}},t\right)=u_0-\mathit{\text{z}}{\displaystyle \frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x}}-f\left(\mathit{\text{z}}\right){\displaystyle \frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}},\\ u_{\mathit{\text{z}}}\left(x,\mathit{\text{z}},t\right)=w^{\mathrm{b}}+w^{\mathrm{s}},\end{array}

where the displacements in the x and z directions are denoted by the variables u_x,u_{\mathit{\text{z}}}. u_0, w^{\mathrm{b}}, and w^{\mathrm{s}} are three unknowns that need to be determined.

Based on the displacement fields in Eq. (3), the linear nonzero strain components of MEE-FG nanobeams are made up of the following (Eqs. (4) and (5)):

(4){\epsilon }_{xx}=\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-\mathit{\text{z}}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-f\left(\mathit{\text{z}}\right)\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2},{\gamma }_{x\mathit{\text{z}}}=g\left(\mathit{\text{z}}\right).

(5)f\left(\mathit{\text{z}}\right)=\mathit{\text{z}}-\frac{h}{\text{π}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right),g\left(\mathit{\text{z}}\right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right).

3.1 Nonlocal magneto-electro-elastic fields

The stress state at a reference point in a beam body called x is believed to depend on the strain state there as well as the strain states at all other sites designated x^{\prime } in the beam body, per Eringen’s nonlocal elastic theory [17,43]. The overall form of the constitutive relation is integral over the whole region in the nonlocal elasticity-type representation. A nonlocal kernel function that describes how strains at several sites influence the stress at a given point is included in the integral.

The basic constitutive equations for a nonlocal isotropic material in the absence of body force can be expressed as Eq. (6):

(6){\sigma }_{ij}^{\mathrm{n}\mathrm{l}}={\int }_{\psi }^{}\alpha \left(\left[x^{\prime }-x\right],\tau \right){\sigma }_{ij}^{\mathrm{l}}\left(x^{\prime }\right)\mathrm{d}\psi \left(x^{\prime }\right),

where \alpha \left(\left[x^{\prime }-x\right],\tau \right) is the nonlocal kernel function, {\sigma }_{ij}^{\mathrm{n}\mathrm{l}} and {\sigma }_{ij}^{\mathrm{l}} are the nonlocal stress tensor and local stress tensor, respectively, and \tau is a length-scale factor. In Eq. (6), the volume integral is over the area \psi that the body occupies.

The form of Eq. (6) for nonlocal stress for the magnetoelectroelastic solid can be expanded similarly to Eqs. (7) and (8) for nonlocal electric displacement and magnetic induction.

(7)D_{ij}^{\mathrm{n}\mathrm{l}}={\int }_{\psi }^{}\alpha \left(\left[x^{\prime }-x\right],\tau \right)D_{ij}^{\mathrm{l}}\left(x^{\prime }\right)\mathrm{d}\psi \left(x^{\prime }\right),

(8)B_{ij}^{\mathrm{n}\mathrm{l}}={\int }_{\psi }^{}\alpha \left(\left[x^{\prime }-x\right],\tau \right)B_{ij}^{\mathrm{l}}\left(x^{\prime }\right)\mathrm{d}\psi \left(x^{\prime }\right),

where D_{ij}^{\mathrm{l}} and D_{ij}^{\mathrm{n}\mathrm{l}} are the local and nonlocal electric displacement components, correspondingly, and B_{ij}^{\mathrm{l}} and B_{ij}^{\mathrm{n}\mathrm{l}} are the local and nonlocal magnetic induction components, correspondingly.

The integral-partial differential equations of nonlocal elasticity reduce to partial differential equations in accordance with Eringen’s nonlocal hypothesis. Consequently, the following differential constitutive relations are utilized to represent the nonlocal constitutive behavior of a Hookean solid.

(9){\sigma }_{ij}^{\mathrm{n}\mathrm{l}}-{\eta }^2{\mathrm{\nabla }}^2{\sigma }_{ij}^{\mathrm{n}\mathrm{l}}={\sigma }_{ij}^{\mathrm{l}},

(10)D_{ij}^{\mathrm{n}\mathrm{l}}-{\eta }^2{\mathrm{\nabla }}^2{D}_{ij}^{\mathrm{n}\mathrm{l}}=D_{ij}^{\mathrm{l}},

(11)B_{ij}^{\mathrm{n}\mathrm{l}}-{\eta }^2{\mathrm{\nabla }}^2{B}_{ij}^{\mathrm{n}\mathrm{l}}=B_{ij}^{\mathrm{l}},

where {\mathrm{\nabla }}^2 = {\mathrm{\partial }}^2/\mathrm{\partial }{x}^2 + {\mathrm{\partial }}^2/\mathrm{\partial }{y}^2 is Laplacian operator. \eta \left(\mathrm{n}\mathrm{m}\right) is the nonlocal coefficient, and the nonlocal factor for carbon nanotubes should be less than 4 nm², according to Wang et al. [39]. In the study of MEE-FG nanobeam, the nonlocal coefficient \eta is considered to vary according to the thickness direction via the law of change as other properties of the material, and these predicted results match with reality.

(12)\eta \left(\mathit{\text{z}}\right)={\eta }_{\mathrm{m}}{e}^{p_{\mathit{\text{z}}}\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)},

where {\eta }_{\mathrm{m}} is the nonlocal coefficient of the material at \mathit{\text{z}}=-h/2.

For MEE-FG nanobeam, below is a calculation of the stress component, electric displacement, and magnetic displacement:

(13)\left(1-{\eta }^2\left(\mathit{\text{z}}\right)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right){\sigma }_{xx}=C_{11}{\epsilon }_{xx}-e_{31}{E}_{\mathit{\text{z}}}-f_{31}{H}_{\mathit{\text{z}}},

(14)\left(1-{\eta }^2\left(\mathit{\text{z}}\right)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right){\sigma }_{xz}=C_{44}{\epsilon }_{x\mathit{\text{z}}}-e_{15}{E}_x-f_{15}{H}_x,

(15)\left(1-{\eta }^2\left(\mathit{\text{z}}\right)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)D_{\mathit{\text{z}}}=e_{31}{\epsilon }_{xx}+h_{33}{E}_{\mathit{\text{z}}}+g_{33}{H}_{\mathit{\text{z}}},

(16)\left(1-{\eta }^2\left(\mathit{\text{z}}\right)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)B_{\mathit{\text{z}}}=f_{31}{\epsilon }_{xx}+g_{33}{E}_{\mathit{\text{z}}}+{\mu }_{33}{H}_{\mathit{\text{z}}},

where E_i and H_i represent the electric and magnetic field intensities, respectively; h_{ij} and {\mu }_{ij} are the dielectric and magnetic permeability coefficients; and C_{ij},e_{ij},f_{ij}, and g_{ij} stand for the elastic, piezoelectric, piezomagnetic, and magneto-electric constants.

Maxwell’s vector equations in the quasi-static are met when the magnetic intensity and electric vector are given as gradients of magnetic potential \varphi and electric potential \phi, respectively [38]:

(17)\begin{array}{rl}& E_i=-\frac{\mathrm{\partial }\phi }{\mathrm{\partial }i},\\ & H_i=-\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }i}.\end{array}

3.2 General equation of motion

The equations of motion of the MEE-FG nanobeam with arbitrary BCs are derived using Hamilton’s principle [1,44–45]:

(18){\int }_{t_1}^{t_2}\left(\delta U^{\mathrm{b}}+\delta U^{\mathrm{B}\mathrm{C}}+\delta V-\delta \overline{K}\right)\mathrm{d}t=0,

where \delta U^{\mathrm{b}} is the variations of strain energy, \delta U^{\mathrm{B}\mathrm{C}} is the variations of potential energy function stored in the springs, \delta V is the variations of potential energy, and \delta \overline{K} is the variations of kinetic energy.

The variation of the MEE-FG nanobeam strain energy \delta U^{\mathrm{b}} may be expressed as

(19)\begin{array}{rl}\delta U^{\mathrm{b}}=& {\int }_0^L{\int }_{-h/2}^{h/2}\left({\sigma }_{xx}\delta {\epsilon }_{xx}+{\sigma }_{x\mathit{\text{z}}}\delta {\epsilon }_{x\mathit{\text{z}}}-D_x\delta E_x-D_y\delta E_y-D_{\mathit{\text{z}}}\delta E_{\mathit{\text{z}}}-B_x\delta H_x-B_y\delta H_y-B_z\delta H_{\mathit{\text{z}}}\right)\mathrm{d}\mathit{\text{z}}\mathrm{d}x\\ =& {\int }_0^L\left({\mathit{N}}_x\frac{{\mathrm{\partial }\delta u}_0}{\mathrm{\partial }x}-{\mathit{M}}_x\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{b}}}{{\mathrm{\partial }x}^2}-{\mathit{L}}_x\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{s}}}{{\mathrm{\partial }x}^2}+{\mathit{Q}}_{x\mathit{\text{z}}}\frac{{\mathrm{\partial }\delta w}^{\mathrm{s}}}{\mathrm{\partial }x}-{\mathit{D}}_{\mathit{\text{z}}}\frac{\mathrm{\partial }\delta \varphi }{\mathrm{\partial }\mathit{\text{z}}}-{\mathit{B}}_{\mathit{\text{z}}}\frac{\mathrm{\partial }\delta \phi }{\mathrm{\partial }\mathit{\text{z}}}\right)\mathrm{d}x,\end{array}

The below contentions operate to the resultants: {\mathit{N}}_x,{\mathit{M}}_x,{\mathit{L}}_x, and {\mathit{Q}}_{x\mathit{\text{z}}}.

{\mathit{N}}_x={\int }_{-h/2}^{h/2}{\sigma }_{xx}\mathrm{d}\mathit{\text{z}},

{\mathit{M}}_x={\int }_{-h/2}^{h/2}\mathit{\text{z}}{\sigma }_{xx}\mathrm{d}\mathit{\text{z}},

{\mathit{L}}_x={\int }_{-h/2}^{h/2}f\left(\mathit{\text{z}}\right){\sigma }_{xx}\mathrm{d}\mathit{\text{z}},

(20){\mathit{Q}}_{xz}={\int }_{-h/2}^{h/2}g\left(\mathit{\text{z}}\right){\sigma }_{x\mathit{\text{z}}}\mathrm{d}\mathit{\text{z}}.

The representation of the change in the applied external forces potential energy \delta V is as

(21)\begin{array}{rl}\delta V=& -{\int }_0^L(q\left(x,y\right)+k_w\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)-(k_s+N_m+N_e+N_a\\ & +N_T+N_H)\frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }{x}^2})\delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\mathrm{d}x.\end{array}

The following are the induced normal forces on a MEE-FG nanobeam caused by an external electric and magnetic potential [38]:

N_{\mathrm{m}}=N_0,

(22)N_{\mathrm{e}}={\int }_{-h/2}^{h/2}\frac{e_{31}{V}_0}{h}\mathrm{d}\mathit{\text{z}},

N_{\mathrm{a}}={\int }_{-h/2}^{h/2}\frac{f_{31}{\mathrm{\Omega }}_0}{h}\mathrm{d}\mathit{\text{z}}.

The force caused by temperature and moisture is determined as follows [46–47]

N_T={\int }_{-h/2}^{h/2}{\alpha }_{11}\mathrm{\Delta }T\mathrm{d}\mathit{\text{z}},

(23)N_H={\int }_{-h/2}^{h/2}{\beta }_{11}\mathrm{\Delta }C\mathrm{d}\mathit{\text{z}},

where \mathrm{\Delta }T and \mathrm{\Delta }C are the thermal and moisture changes, respectively. {\alpha }_{11} and {\beta }_{11} are the temperature and moisture modulus of the material, respectively. The variation of kinetic energy \delta \overline{K} is given by

(24)\begin{array}{rl}\delta \overline{K}=& {\int }_0^L{\int }_{-h/2}^{h/2}\rho \left(\mathit{\text{z}}\right)\left(\left(\frac{{\mathrm{\partial }u}_0}{\mathrm{\partial }t}-\overline{\mathit{\text{z}}}\frac{{\mathrm{\partial }}^2{w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}-\overline{f}\frac{{\mathrm{\partial }}^2{w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\right)\left(\frac{{\mathrm{\partial }\delta u}_0}{\mathrm{\partial }t}-\overline{\mathit{\text{z}}}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}-\overline{f}\frac{{\mathrm{\partial }}^2\delta w^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\right)+\frac{\mathrm{\partial }\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }t}\frac{\mathrm{\partial }\delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }t}\right)\mathrm{d}\mathit{\text{z}}\mathrm{d}x\\ =& {\int }_0^L(I_1\left(\frac{{\mathrm{\partial }u}_0}{\mathrm{\partial }t}\frac{\mathrm{\partial }\delta u_0}{\mathrm{\partial }t}+\frac{\mathrm{\partial }\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }t}\frac{\mathrm{\partial }\delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }t}\right)+I_2\left(\frac{{\mathrm{\partial }u}_0}{\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^2{w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }\delta u}_0}{\mathrm{\partial }t}\right)+I_3\left(\frac{{\mathrm{\partial }u}_0}{\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^2{w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }\delta u}_0}{\mathrm{\partial }t}\right)\\ & +I_4\frac{{\mathrm{\partial }}^2{w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}+I_5\left(\frac{{\mathrm{\partial }}^2{w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^2{w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\right)+I_6\frac{{\mathrm{\partial }}^2{w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2{\delta w}^{\mathrm{s}}}{\mathrm{\partial }x\mathrm{\partial }t})\mathrm{d}x,\end{array}

where

(25)\left(I_1,I_2,I_3,I_4,I_5,I_6\right)={\int }_{-h/2}^{h/2}\rho \left(\mathit{\text{z}}\right)(1,-\overline{\mathit{\text{z}}},-\overline{f},{\overline{\mathit{\text{z}}}}^2,\overline{\mathit{\text{z}}}\overline{f},{\overline{f}}^2)\mathrm{d}\mathit{\text{z}},

The boundary springs’ stored variations in the potential energy function \delta U^{\mathrm{B}\mathrm{C}}:

(26)\begin{array}{rl}\delta U^{\mathrm{B}\mathrm{C}}=& {\left(K_{\mathrm{u}0}{u}_0\delta u_0+K_{\mathrm{b}0}{w}^{\mathrm{b}}\delta w^{\mathrm{b}}+K_{\mathrm{s}0}{w}^{\mathrm{s}}\delta w^{\mathrm{s}}+K_{\mathrm{b}x0}\frac{\mathrm{\partial }{w}^{\mathrm{b}}}{\mathrm{\partial }x}\frac{\mathrm{\partial }\delta w^{\mathrm{b}}}{\mathrm{\partial }x}+K_{\mathrm{s}x0}\frac{\mathrm{\partial }{w}^{\mathrm{s}}}{\mathrm{\partial }x}\frac{\mathrm{\partial }\delta w^{\mathrm{s}}}{\mathrm{\partial }x}\right)}_{x=0}\\ & +{\left(K_{\mathrm{u}\mathrm{L}}{u}_0\delta u_0+K_{\mathrm{b}\mathrm{L}}{w}^{\mathrm{b}}\delta w^{\mathrm{b}}+K{}_{\mathrm{s}\mathrm{L}}{w}^{\mathrm{s}}\delta w^{\mathrm{s}}+K_{\mathrm{b}x\mathrm{L}}\frac{\mathrm{\partial }{w}^{\mathrm{b}}}{\mathrm{\partial }x}\frac{\mathrm{\partial }\delta w^{\mathrm{b}}}{\mathrm{\partial }x}+K_{\mathrm{s}x\mathrm{L}}\frac{\mathrm{\partial }{w}^{\mathrm{s}}}{\mathrm{\partial }x}\frac{\mathrm{\partial }\delta w^{\mathrm{s}}}{\mathrm{\partial }x}\right)}_{x=L}.\end{array}

The equations of motion of MEE-FG nanobeam with elastic BCs may be derived by substituting Eqs. (19), (41), (21), (24), and (26) into Eq. (18), and integrating by parts and collecting the coefficient of \delta u_0,{\delta w}^{\mathrm{b}},{\delta w}^{\mathrm{s}},\delta \phi , and \delta \varphi, respectively shown as follow:

(27)\delta u_0:\frac{\mathrm{\partial }{\mathit{N}}_x}{\mathrm{\partial }x}=I_1\frac{{\mathrm{\partial }}^2{u}_0}{\mathrm{\partial }{t}^2}+I_2\frac{{\mathrm{\partial }}^3{w}^{\mathrm{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }x}+I_3\frac{{\mathrm{\partial }}^3{w}^{\mathrm{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }x},

(28)\begin{array}{rl}\delta w^{\mathrm{b}}:\frac{{\mathrm{\partial }}^2{\mathit{M}}_x}{\mathrm{\partial }{x}^2}=& I_1\frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }{t}^2}-I_2\frac{{\mathrm{\partial }}^3{U}_0}{\mathrm{\partial }x\mathrm{\partial }{t}^2}-I_4\frac{{\mathrm{\partial }}^4{w}^{\mathrm{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}\\ & -I_5\frac{{\mathrm{\partial }}^4{w}^{\mathrm{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-k_{\mathrm{w}}\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)+(k_{\mathrm{s}}+N_{\mathrm{m}}\\ & +N{}_{\mathrm{e}}+N_{\mathrm{a}}+N_{\mathrm{T}}+N{}_{\mathrm{H}})\frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }{x}^2},\end{array}

(29)\begin{array}{rl}& \delta w^{\mathrm{b}}:\frac{{\mathrm{\partial }}^2{\mathit{L}}_x}{\mathrm{\partial }{x}^2}+\frac{\mathrm{\partial }{\mathit{Q}}_{xz}}{\mathrm{\partial }x}=I_1\frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }{t}^2}-I_3\frac{{\mathrm{\partial }}^3{u}_0}{\mathrm{\partial }x\mathrm{\partial }{t}^2}\\ & -I_5\frac{{\mathrm{\partial }}^4{w}^{\mathrm{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-I_6\frac{{\mathrm{\partial }}^4{w}^{\mathrm{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-k_{\mathrm{w}}\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)+(k{}_{\mathrm{s}}+N_{\mathrm{m}}\\ & +N_{\mathrm{e}}+N_{\mathrm{a}}+N{}_{\mathrm{T}}+N{}_{\mathrm{H}})\frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }{x}^2}\mathrm{c}\end{array}

(30)\delta \varphi :\frac{\mathrm{\partial }{D}_{\mathit{\text{z}}}}{\mathrm{\partial }\mathit{\text{z}}}=0,

(31)\delta \phi :\frac{\mathrm{\partial }{B}_{\mathit{\text{z}}}}{\mathrm{\partial }\mathit{\text{z}}}=0.

The following definitions apply to the general boundary conditions:

\begin{array}{}\text{(32a)}& \begin{array}{rl}& x=0:{\mathit{N}}_x=K_{\mathrm{u}0}{u}_0,{\mathit{Q}}_{x\mathit{\text{z}}}=K_{\mathrm{b}0}{w}^{\mathrm{b}}+K_{\mathrm{s}0}{w}^{\mathrm{s}},\\ & {\mathit{M}}_x=K_{\mathrm{b}x0}\frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x},{\mathit{L}}_x=K_{\mathrm{s}x0}\frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x},\end{array}\end{array}

\begin{array}{}\text{(32b)}& \begin{array}{rl}& x=L:-{\mathit{N}}_x=K_{\mathrm{u}\mathrm{L}}{u}_0,-{\mathit{Q}}_{x\mathit{\text{z}}}=K_{\mathrm{b}\mathrm{L}}{w}^{\mathrm{b}}+K_{\mathrm{s}\mathrm{L}}{w}^{\mathrm{s}},\\ & -{\mathit{M}}_x=K_{\mathrm{b}x\mathrm{L}}\frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x},-{\mathit{L}}_x=K_{\mathrm{s}x\mathrm{L}}\frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}.\end{array}\end{array}

Equations (15) and (16) can be simply substituted into Eqs. (30) and (31) to obtain the following equations:

(33)e_{31}\frac{\mathrm{\partial }{\epsilon }_{xx}}{\mathrm{\partial }\mathit{\text{z}}}-h_{33}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{\mathit{\text{z}}}^2}-g_{33}\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\mathit{\text{z}}}^2}=0,

(34)f_{31}\frac{\mathrm{\partial }{\epsilon }_{xx}}{\mathrm{\partial }\mathit{\text{z}}}-g_{33}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{\mathit{\text{z}}}^2}-{\mu }_{33}\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\mathit{\text{z}}}^2}=0.

By adopting Crammer’s rule, one obtained:

(35)\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{\mathit{\text{z}}}^2}=\frac{{\mu }_{33m}{e}_{31m}-g_{33m}{f}_{31m}}{h_{33m}{\mu }_{33m}-g_{33m}^2}\frac{\mathrm{\partial }{\epsilon }_{xx}}{\mathrm{\partial }\mathit{\text{z}}}={\lambda }_1\frac{\mathrm{\partial }{\epsilon }_{xx}}{\mathrm{\partial }\mathit{\text{z}}},

(36)\frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{\mathit{\text{z}}}^2}=\frac{h_{33m}{f}_{31m}-g_{33m}{e}_{31m}}{h_{33m}{\mu }_{33m}-g_{33m}^2}\frac{\mathrm{\partial }{\epsilon }_{xx}}{\mathrm{\partial }\mathit{\text{z}}}={\lambda }_2\frac{\mathrm{\partial }{\epsilon }_{xx}}{\mathrm{\partial }\mathit{\text{z}}}.

Integrating Eqs. (35) and (36) with the electric and magnetic boundary conditions given by Eq. (2), one gets:

(37)\begin{array}{rl}\phi =& {\lambda }_1(\mathit{\text{z}}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-\frac{1}{2}\left({\mathit{\text{z}}}^2-\frac{h^2}{4}\right)\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-(\frac{{\mathit{\text{z}}}^2}{2}-\frac{h^2}{8}\\ & +\frac{h^2}{{\text{π}}^2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right))\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2})+\frac{V_0}{h}\mathit{\text{z}}+\frac{V_0}{2},\end{array}

(38)\begin{array}{rl}\varphi =& {\lambda }_2(\mathit{\text{z}}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-\frac{1}{2}\left({\mathit{\text{z}}}^2-\frac{h^2}{4}\right)\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-(\frac{{\mathit{\text{z}}}^2}{2}-\frac{h^2}{8}\\ & +\frac{h^2}{{\text{π}}^2}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right))\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2})+\frac{{\mathrm{\Psi }}_0}{h}\mathit{\text{z}}+\frac{{\mathrm{\Psi }}_0}{2},\end{array}

(39)E_{\mathit{\text{z}}}=-{\lambda }_1\left(\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-\mathit{\text{z}}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-\left(\mathit{\text{z}}-\frac{h}{\text{π}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right)\right)\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2}\right)-\frac{V_0}{2},

(40)H_{\mathit{\text{z}}}=-{\lambda }_2\left(\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-\mathit{\text{z}}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-\left(\mathit{\text{z}}-\frac{h}{\text{π}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right)\right)\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2}\right)-\frac{{\mathrm{\Psi }}_0}{2}.

From Eqs. (13), (14), and (20), the internal force components are rewritten as follows:

\begin{array}{}\text{(41a)}& {\mathit{N}}_x-{\eta }^2\left(\mathit{\text{z}}\right){\mathit{N}}_{x,xx}=A_{11}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-B_{11}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-B_{11}^{\mathrm{s}}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2},\end{array}

\begin{array}{}\text{(41b)}& {\mathit{M}}_x-{\eta }^2\left(\mathit{\text{z}}\right){\mathit{M}}_{x,xx}=B_{11}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-F_{11}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-F_{11}^{\mathrm{s}}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2},\end{array}

\begin{array}{}\text{(41c)}& {\mathit{L}}_x-{\eta }^2\left(\mathit{\text{z}}\right){\mathit{L}}_{x,xx}=B_{11}^{\mathrm{s}}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}-F_{11}^{\mathrm{s}}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}-H_{11}\frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2},\end{array}

\begin{array}{}\text{(41d)}& {\mathit{Q}}_{x\mathit{\text{z}}}-{\eta }^2\left(\mathit{\text{z}}\right){\mathit{Q}}_{x\mathit{\text{z}},xx}=A_{11}^{\mathrm{s}}\frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x},\end{array}

where

\left(A_{11},B_{11},F_{11},B_{11}^{\mathrm{s}},F_{11}^{\mathrm{s}},H_{11}\right)={\int }_{-h/2}^{h/2}{\overline{C}}_{11}\left(\mathit{\text{z}}\right)\left(1,\overline{\mathit{\text{z}}},{\overline{\mathit{\text{z}}}}^2,\overline{f},\overline{\mathit{\text{z}}}\overline{f},{\overline{f}}^2\right)\mathrm{d}\mathit{\text{z}},

(42)A_{11}^s={\int }_{-h/2}^{h/2}{\overline{C}}_{44}\left(\mathit{\text{z}}\right){\overline{g}}^2\mathrm{d}\mathit{\text{z}},

{\overline{C}}_{11}\left(\mathit{\text{z}}\right)=C_{11}+e_{31}{M}_1+f_{31}{M}_2,

(43){\overline{C}}_{44}\left(\mathit{\text{z}}\right)=C_{44},

where \overline{\mathit{\text{z}}}=\mathit{\text{z}}-t_0,\overline{f}=f\left(\overline{\mathit{\text{z}}}\right),\overline{g}=g\left(\overline{\mathit{\text{z}}}\right), and t_0 is the MEE-FG nanobeam’s mean surface-to-neutral surface distance, which is described as follows:

(44)t_0={\int }_{-h/2}^{h/2}{\overline{C}}_{11}\left(\mathit{\text{z}}\right)\mathit{\text{z}}\mathrm{d}\mathit{\text{z}}/{\int }_{-h/2}^{h/2}{\overline{C}}_{11}\left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}.

Using the Galerkin solution to Eqs. (27)–(29) with associated weight functions (\delta u_0,{\delta w}^{\mathrm{b}},\mathrm{a}\mathrm{n}\mathrm{d}{\delta w}^{\mathrm{s}}). In present work, the traction on the Neumann boundary [48–49] is disregarded:

(45)\begin{array}{rl}& {\int }_0^L\left({\mathit{\epsilon }}_1^{\mathrm{T}}{\mathit{D}}_1\delta {\mathit{\epsilon }}_1+{\mathit{\epsilon }}_2^{\mathrm{T}}{A}_{11}^{\mathrm{s}}\delta {\mathit{\epsilon }}_2\right)\mathrm{d}x+k_{\mathrm{w}}{\int }_0^L{\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}^{\mathrm{T}}\delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\mathrm{d}x+(k_{\mathrm{w}}{\eta }^2\left(-h/2\right)+k_s+N_m+N_e+N_a+N_T+N_H)\\ & {\int }_0^L{\left(\frac{\mathrm{\partial }\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }x}\right)}^{\mathrm{T}}\delta \frac{\mathrm{\partial }\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{\mathrm{\partial }x}\mathrm{d}x+(k_{\mathrm{s}}{\eta }^2\left(-h/2\right)+N_{m\eta }+N_{e\eta }+N_{a\eta }+N_{T\eta }+N_{H\eta })\\ & {\int }_0^L{\left(\frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{{\mathrm{\partial }x}^2}\right)}^{\mathrm{T}}\delta \frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{{\mathrm{\partial }x}^2}\mathrm{d}x+K_{u0}{u_0\delta u_0|}_{x=0}+{K_{b0}{w}^{\mathrm{b}}\delta w^{\mathrm{b}}|}_{x=0}+{K_{s0}{w}^{\mathrm{s}}\delta w^{\mathrm{s}}|}_{x=0}+{K_{bx0}\frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x}\delta \frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x}|}_{x=0}+\\ & {K_{\mathrm{s}x0}\frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}\delta \frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}|}_{x=0}+K_{\mathrm{u}\mathrm{L}}{u_0\delta u_0|}_{x=L}+{K_{\mathrm{b}\mathrm{L}}{w}^{\mathrm{b}}\delta w^{\mathrm{b}}|}_{x=L}+{K_{\mathrm{s}\mathrm{L}}{w}^{\mathrm{s}}\delta w^{\mathrm{s}}|}_{x=L}+{K_{\mathrm{b}x\mathrm{L}}\frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x}\delta \frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x}|}_{x=L}+{K_{\mathrm{s}x\mathrm{L}}\frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}\delta \frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}|}_{x=L}=\\ & {\int }_0^L\left(\left(q\left(x\right)-{\eta }^2\left(h/2\right)q_{,xx}\right)\delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)-q\left(x\right){\eta }^2\left(h/2\right)\delta \frac{{\mathrm{\partial }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)}{{\mathrm{\partial }x}^2}\right)\mathrm{d}x+{\int }_0^L\left({\left(\frac{{\mathrm{\partial }}^2\mathit{u}}{\mathrm{\partial }{t}^2}\right)}^{\mathrm{T}}{\mathit{H}}_m\delta \mathit{u}+{\left(\frac{{\mathrm{\partial }}^3\mathit{u}}{\mathrm{\partial }x\mathrm{\partial }{t}^2}\right)}^{\mathrm{T}}{\mathit{H}}_{m\eta }\delta \frac{\mathrm{\partial }\mathit{u}}{\mathrm{\partial }x}\right)\mathrm{d}x,\end{array}

where

{\mathit{\epsilon }}_1={\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}}& -{\displaystyle \frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{b}}}{\mathrm{\partial }{x}^2}}& -{\displaystyle \frac{{{\mathrm{\partial }}^{2}w}^{\mathrm{s}}}{\mathrm{\partial }{x}^2}}\end{array}\right\}}^{\mathrm{T}},

{\mathit{\epsilon }}_2=\frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x},

(46){\mathit{D}}_1=\left[\begin{array}{ccc}{A}_{11}& B_{11}& B_1^{\mathrm{s}}\\ B_{11}& F_{11}& F_{11}^{\mathrm{s}}\\ B_{11}^{\mathrm{s}}& F_{11}^{\mathrm{s}}& H_{11}\end{array}\right],

\mathit{u}=\left\{\begin{array}{c}{u}_0\\ w^{\mathrm{b}}\\ w^{\mathrm{s}}\\ {\displaystyle \frac{{\mathrm{\partial }w}^{\mathrm{b}}}{\mathrm{\partial }x}}\\ {\displaystyle \frac{{\mathrm{\partial }w}^{\mathrm{s}}}{\mathrm{\partial }x}}\end{array}\right\},

{\mathit{H}}_m=\left[\begin{array}{ccccc}{I}_1& 0& 0& I_2& I_3\\ 0& I_1& I_1& 0& 0\\ 0& I_1& I_1& 0& 0\\ I_2& 0& 0& I_4& I_5\\ I_3& 0& 0& I_5& I_6\end{array}\right],

(47){\mathit{H}}_{m\eta }=\left[\begin{array}{ccccc}{J}_1& 0& 0& J_2& J_3\\ 0& J_1& J_1& 0& 0\\ 0& J_1& J_1& 0& 0\\ J_2& 0& 0& J_4& J_5\\ J_3& 0& 0& J_5& J_6\end{array}\right],

(48)\left(J_1,J_2,J_3,J_4,J_5,J_6\right)={\int }_{-h/2}^{h/2}\rho \left(\mathit{\text{z}}\right){\eta }^2\left(\mathit{\text{z}}\right)(1,-\overline{\mathit{\text{z}}},-\overline{f},{\overline{\mathit{\text{z}}}}^2,\overline{\mathit{\text{z}}}\overline{f},{\overline{f}}^2)\mathrm{d}\mathit{\text{z}},

N_{e\eta }={\int }_{-h/2}^{h/2}\frac{e_{31}{V}_0}{h}{\eta }^2\left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}},

(49)N_{a\eta }={\int }_{-h/2}^{h/2}\frac{f_{31}{\mathrm{\Omega }}_0}{h}{\eta }^2\left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}},

N_{T\eta }={\int }_{-h/2}^{h/2}{\eta }^2\left(\mathit{\text{z}}\right){\alpha }_{11}\mathrm{\Delta }T\mathrm{d}\mathit{\text{z}},

(50)N_{H\eta }={\int }_{-h/2}^{h/2}{\eta }^2\left(\mathit{\text{z}}\right){\beta }_{11}\mathrm{\Delta }C\mathrm{d}\mathit{\text{z}}.

3.3 NURBS-based finite element formulation

The knot vector does not decrease and is always between 0 and 1 in a one-dimensional space. The representation of this can be found in the equation \mathit{k}\left(\zeta \right)=\{{\zeta }_1=0,\dots ,{\zeta }_i\dots ,{\zeta }_{n+p+1}=1\}, where n denotes the number of basis assignments, p is the polynomial order, and i is the knot index. Recursively written, the ith B-spline basis function of degree p, known as N_{i,p}\left(\zeta \right), is [50–54]:

(51)

(52)p\ge 1:N_{i,p}\left(\zeta \right)=\frac{\zeta -{\zeta }_i}{{\zeta }_{i+p}-{\zeta }_i}{N}_{i,p-1}\left(\zeta \right)+\frac{{\zeta }_{i+p+1}-\zeta }{{\zeta }_{i+p+1}-{\zeta }_{i+1}}{N}_{i+1,p-1}\left(\zeta \right).

It is evident that two (p-1) order basis functions can be combined linearly to find the p order B-spline basis function. The NURBS line and the ith univariate NURBS basis functions {\mathit{R}}_{i,p} of order p can then be found.

(53){\mathit{R}}_{i,p}=\frac{N_{i,p}\left(\zeta \right)w_i}{\sum _{i=1}^n{N}_{i,p}\left(\zeta \right)w_i},

where the weight associated with the ith control point is denoted by the nonnegative actual integer w_i, the set of control points is represented by p_i=x_i,i=1,2,...,n, and the NURBS basis function {\mathit{R}}_{i,p} is regulated by Eq. (53). When all weights equal 1, the NURBS basis function degenerates into a B-spline.

Based on IGA, the displacements {\mathit{d}}^h in the middle plane of the MEE-FG nanobeam is written as [51–52]:

(54){\mathit{d}}^h={\sum }_{i=1}^{ne}{\mathit{R}}_i{\mathit{d}}_i,

(55){\mathit{d}}_{\mathit{i}}={[u_{0i},w_i^{\mathrm{b}},w_i^{\mathrm{s}}]}^{\mathrm{T}},

where ne=(p+1) is the number of control points per physical element, and {\mathit{R}}_i and {\mathit{d}}_i stand for the shape functions and the unknown displacement tensor at control point i, respectively.

Replacing Eq. (54) into Eq. (3), the displacements u_0,w^{\mathrm{b}}, and w^{\mathrm{s}} can be rewritten by

u_0={\sum }_{i=1}^{ne}\left[\begin{array}{ccc}{\mathit{R}}_i& 0& 0\end{array}\right]{\mathit{d}}_i={\sum }_{i=1}^{ne}{\mathit{H}}_i^1{\mathit{d}}_i,

(56)w^{\mathrm{b}}={\sum }_{i=1}^{ne}\left[\begin{array}{ccc}0& {\mathit{R}}_i& 0\end{array}\right]{\mathit{d}}_i={\sum }_{i=1}^{ne}{\mathit{H}}_i^2{\mathit{d}}_i,

w^{\mathrm{s}}={\sum }_{i=1}^{ne}\left[\begin{array}{ccc}0& 0& {\mathit{R}}_i\end{array}\right]{\mathit{d}}_i={\sum }_{i=1}^{ne}{\mathit{H}}_i^3{\mathit{d}}_i.

Replace Eq. (56) into Eq. (45), the equations for static displacement natural frequency of the MEE-FG nanobeam are determined as follows:

Static bending problem:

(57)\left(\mathit{K}+{\mathit{K}}_{\mathrm{B}\mathrm{C}}\right)\mathit{d}=\overline{\mathit{F}}.

Specific oscillation problem:

(58)\left(\mathit{K}+{\mathit{K}}_{\mathrm{B}\mathrm{C}}-{\omega }^2\mathit{M}\right)\overline{\mathit{d}}=0,

where \mathit{d}=\overline{\mathit{d}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right), \omega is the natural frequency; \mathit{K}=\sum _{i=1}^{Tn}\left(K_{\mathrm{b}}^i+K_{\mathrm{f}}^i+K_{\mathrm{H}\mathrm{T}}^i+K_{\mathrm{M}\mathrm{E}}^i\right);\mathit{M}=\sum _{i=1}^{Tn}{M}_{\mathrm{b}}^i; K_{\mathrm{b}}^i, K_{\mathrm{f}}^i, K_{\mathrm{H}\mathrm{T}}^i, K_{\mathrm{b}}^i and \overline{\mathit{F}}=\sum _{i=1}^{Tn}{\overline{F}}_{\mathrm{b}}^i,{\overline{F}}_{\mathrm{b}}^i are beam stiffness matrix, foundation stiffness, hygro-thermo stiffness, electromagnetic stiffness, mass matrix, and load vector for the ith beam element of the MEE-FG nanobeam; Tn is the total number of beam elements, and {\mathit{K}}_{\mathrm{B}\mathrm{C}} is the stiffness matrix of the elastic BCs. These components are given in the Appendix. The displacement \mathit{d} in Eq. (57) inserting into Eqs. (13)–(16), which yields stress, electric displacement, and magnetic induction components.

4 Numerical results

From the theoretical formulas developed in Section 2 and the isogeometric method in Section 3, Matlab’s software is used to build a set of programs for the free oscillation and bending problems of MEE-FG nanobeam lying on Pasternak medium in the hydro-thermal environment with elastic constraints. After verifying the program’s convergence and correctness, a series of input parameter surveys were performed to evaluate the influence of these parameters on the displacement, stress and natural frequency of the MEE-FG nanobeam. The following dimensionless parameters are provided to make it easier to compare the findings of this paper with those of previous publications:

The dimensionless deflection, stresses and natural frequency:

w_1^{\ast \ast }=w\left(\frac{L}{2}\right)\cdot \frac{100C_{11m}{h}^3}{q_0{L}^4},

{\mathrm{\Omega }}_i={\omega }_i\frac{L^2}{h}\sqrt{\frac{{\rho }_m}{C_{11m}}},

{\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{h}{q_0{L}_x}{\sigma }_{xx}\left(\frac{L}{2},\mathit{\text{z}}\right),

{\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{h}{q_0{L}_x}{\sigma }_{x\mathit{\text{z}}}\left(0,\mathit{\text{z}}\right),

D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{{10}^{10}{D}_{\mathit{\text{z}}}\left({\displaystyle \frac{L}{2}},\mathit{\text{z}}\right)}{e_{31m}},

(59)B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{{10}^{10}{B}_{\mathit{\text{z}}}\left({\displaystyle \frac{L}{2}},\mathit{\text{z}}\right)}{f_{31m}}.

The dimensionless foundation stiffness coefficients:

K_{\mathrm{w}}=\frac{k_{\mathrm{w}}{L}^4}{D_c},

K_{\mathrm{s}}=\frac{k_{\mathrm{s}}{L}^2}{D_{\mathrm{c}}},

D_{\mathrm{c}}=\frac{h^3{C}_{11m}}{12},

where the initial data are given as follows: L=10\mathrm{n}\mathrm{m}, h=0.5\mathrm{n}\mathrm{m},b=h;{\eta }_m=1\mathrm{n}\mathrm{m},\mathrm{\Delta }T=50\mathrm{K},\mathrm{\Delta }C=0.5,K_{\mathrm{w}}=10, K_{\mathrm{s}}=1,V_0={10}^{-2}\mathrm{V},{\mathrm{\Psi }}_0={10}^{-4}\mathrm{A},p_{\mathit{\text{z}}}=1. The material properties of the {\mathrm{B}\mathrm{a}\mathrm{T}\mathrm{i}\mathrm{O}}_3\text{–}{\mathrm{C}\mathrm{o}\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_4 composite are reported in Ref. [55], in which C_11m= 226 GPa, C_44m= 44 GPa, e_31m= –2.2 C/m², f_31m = 290.2 N/(A·m), h_33m = 6.35 × 10^–9 C²/(Nm²) , g_{33m}=2737.5\times {10}^{-12} Ns/(Vm²), {\mu }_{33m}=83.5\times {10}^{-6} Ns²/C², {\alpha }_{11m}=4.74\times {10}^5 N/mK, {\beta }_{11m}=2.486\times {10}^7 Nm²/kg, {\rho }_m=5500 kg/m³. The beams are subjected to uniformly distributed loads q_0\left(\mathrm{n}\mathrm{N}\right).

4.1 Reliability and accuracy

The performance characteristics, including numerical verification and convergence analysis, are looked at in this subsection. Three sets of traditional restraint conditions are assessed, as previously mentioned: clamped (C), free (F), and simply supported (S). Furthermore, several forms of elastic (E) constraint conditions are considered.

First, as mentioned before, the artificial spring method approximates the border. Seven springs are used, and the constraint parameters are designated by K_{ri}, r=u,b,s,bx,sx; and i=0,L. As such, an analysis of their convergence properties is required. The changes in {\mathrm{\Omega }}_i (i=1,2,4,9) as a function of constraint parameters K_{ri}=k, where boundary and convective spring parameters are included, are shown in Fig.2. In every case, one set of springs at each end has been changed from 10⁻² to 10¹⁶, whereas the others stay the same (= 10¹⁶). After analyzing the data, it is clear that the natural frequency, or Ω_i, varies in three stages: in the first, it is nearly constant and reaches its minimum when . In the second stage, known as the recovery phase, the natural frequency increases from its minimum to its maximum when . During this phase, the beam’s two boundary positions remain deformed. Following this period, the natural frequency remains constant when k exceeds k>{4\times 10}^{11}, one calls this the stage of solidification of the beam ends, the beam ends at this time have no deformation. The reason for this physical phenomenon is that the beam’s overall stiffness is increased by the stiffness of the spring systems at both ends of the beam. Among the five values K_{\mathrm{u}i},K_{\mathrm{b}i},K_{\mathrm{s}i},K_{\mathrm{b}xi},K_{\mathrm{s}xi}, the change of K_{\mathrm{b}xi} makes the natural frequency converge the fastest, while the slowest is the value of the spring K_{\mathrm{u}i} and K_{\mathrm{b}i}. For straight beams and static bending problems, one can ignore u_0 displacement without changing the calculation results, but in this work, we do not ignore u_0 displacement for the purpose of as a basis for the calculations of both curved and non-straight beam structures in subsequent studies.

Through comparative research with normal boundary conditions and adding other elastic boundary conditions, this work lists out the stiffness coefficient to achieve the desired boundary conditions as follows:

C: K_{\mathrm{u}i}=K_{\mathrm{b}i}=K_{\mathrm{s}i}={10}^{14},K_{\mathrm{b}xi}={10}^{14},K_{\mathrm{s}xi}={10}^{10};

S: K_{\mathrm{b}i}=K_{\mathrm{s}i}={10}^{14},K_{\mathrm{u}i}=K_{\mathrm{b}xi}=K_{\mathrm{s}xi}=0;

F: K_{\mathrm{b}i}=K_{\mathrm{s}i}=K_{\mathrm{u}i}=K_{\mathrm{b}xi}=K_{\mathrm{s}xi}=0;

Elastic of first type (\mathrm{E}1): K_{\mathrm{u}i}={10}^8,K_{\mathrm{b}i}=K_{\mathrm{s}i}={10}^{14},K_{\mathrm{b}xi}={10}^{14},K_{\mathrm{s}xi}={10}^8;

Elastic of second type (\mathrm{E}2): K_{\mathrm{u}i}={10}^{14},K_{\mathrm{b}i}=K_{\mathrm{s}i}={10}^{10},K_{\mathrm{b}xi}={10}^8,K_{\mathrm{s}xi}={10}^8;

Elastic of third type (\mathrm{E}3): K_{\mathrm{u}i}={10}^{14},K_{\mathrm{b}i}=K_{\mathrm{s}i}=K_{\mathrm{b}xi}=K_{\mathrm{s}xi}={10}^8.

First, Tab.1 and Fig.3 present the convergence results for the dimensionless natural frequency of magneto electro elastic isotropic nanobeam with various numbers of elements. From this table, it can be found that the normalized natural frequency of the proposed method has an excellent convergence with the Ref. [55] using Timoshenko beam theory and Navier’s method. In this study, the order of NURBS basis functions is used: 2-order (p=2), 3-order (p=3) and 4-order (p=4). The rate of convergence is fast when the order of NURBS basis functions is larger than two. Consequently, the cubic NURBS basis functions are used in the following instances.

Secondly, the first three nondimensional natural frequencies {\mathrm{\Omega }}_i of MEE nanobeam subject to different electric loads and magnetic loads are presented in Tab.2 and Tab.3. The material and geometric factors of the structure can be found in Ref. [55]. These tables show that the suggested method’s natural frequency, {\mathrm{\Omega }}_i, agrees well with Ref. [55].

Thirdly, Tab.4 shows the nondimensional fundamental natural frequency {\mathrm{\Omega }}_1 of an SS-FG straight nanobeam under various hygro-thermal environments. The current work’s results are compared with those of Ref. [56] that uses Navier’s method and refined high-order shear deformation nanobeam theory. It can be seen that the numerical values obtained are fairly similar to those given in Ref. [56]. It can be concluded that the results proposed by the present work are completely satisfactory with the published results (the maximum error is not more than 1% due to the difference in the shear strain theory used).

4.2 Deflection and natural frequency

This subsection presents and discusses the effects of factors on vertical displacement and natural frequency in tables and graphs. The dimensionless deflection w_1 and first-two natural frequencies {\mathrm{\Omega }}_1, {\mathrm{\Omega }}_2 of MEE-FG nanobeam with electric V_{\text{0}}, magnetic load {\mathrm{\Psi }}_{\text{0}} and grading index p_{\mathit{\text{z}}} is described by Tab.5. In this table, seven elastic BCs were used, including CC, SS, E1E1, E2E2, E3E3, CE1, CE2; electric load varies from V_{\text{0}}=-{\text{10}}^{-\text{2}}\text{V} to V_{\text{0}}={\text{10}}^{-\text{2}}\text{V}; magnetic load varies from {\mathrm{\Psi }}_{\text{0}}=-{\text{10}}^{-\text{4}}\text{A} to {\mathrm{\Psi }}_{\text{0}}={\text{10}}^{-\text{4}}\text{A}; grading index value p_{\mathit{\text{z}}}=\text{0.1, 0.5}. The numerical results show that the vertical displacement w_1 decreases or the first-two natural frequencies increases when the voltage value V_0 increases in magnitude in the negative direction or the magnetic field value {\mathrm{\Psi }}_0 increases in magnitude in the positive direction. It shows that the structure’s stiffness can be controlled simply when the voltage and magnetic field values are increased or decreased; this is also an advantage only found in smart materials and structures. In addition, the advance of this study is that it uses various types of elastic BCs to overcome the disadvantages of traditional BCs. In this example, the BCs used for the displacement amplitudes and the first and second natural frequencies are large, which is rarely mentioned in previous works. In addition, for FGM, the p_{\mathit{\text{z}}} value increases making the structure become stiffer, so it is easy to see that the natural frequency of the beam increases.

Simultaneous effects of four factors such as different BCs, electric, magnetic load and nonlocal parameters on displacement w_1 and first-two natural frequencies {\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2 of MEE-FG nanobeam as presented in Tab.6. From this table, it can be seen that the change in the nonlocal coefficient {\eta }_m tends to make the texture softer, as shown by increasing vertical displacement and decreasing natural frequency. Especially in this work, we assume that the nonlocal coefficient varies along the beam thickness. It is necessary to have experimental studies or calculate the molecular dynamics to prove the correctness of this theory.

Next, the effect of temperature and humidity difference along with electric and magnetic load on first-three vibration mode values of MEE-FG nanobeam is described by Tab.7 and Tab.8. Evidently, the values of the natural frequencies all decrease as the temperature and humidity difference increases, which indicates that the high temperature and humidity environment will make the structure soften, it is the cause of damage to the structure, so for the structures when working in such environment, it is necessary to provide heat or moisture protection plans to maintain the working state and prolong the service life for structures.

The first mode shape of the MEE-FG nanobeams with various boundary restraint parameters (CC and E2E2) is described as shown in Fig.4. The difference between the traditional and elastic boundaries can be clearly seen. For a traditional boundary CC, at the two ends of the boundary, the value of w\left(x\right) is always zero. For elastic edges E2E2, it will be shifted down a bit, due to the deformation of the elastic springs. In addition, it can be seen that the mode shapes are symmetric because the two ends of the beam are symmetric. The effect of voltage and magnetic field increases or decreases the structure’s stiffness, so the beam thickness significantly influences the structure’s workability.

Fig.5 shows the first natural frequency of the MEE-FG nanobeams. In Fig.5, the beam length is fixed as 10 nm, the length-to-thickness ratio L/h varies, and beam thickness varies from 0.2 to 2 nm, while the length-to-thickness ratio L/h is fixed at 20. From these figures, it can be seen that when voltage and magnetic fields act on the beam structure, the larger the length-to-thickness L/h, the greater the impacts of the electric and magnetic fields on the structure (the natural frequency increases or decreases up to 75%). This is also a remarkable point for this type of structure; it helps engineers and designers make appropriate adjustments in practice.

In engineering practice, most of the constructions or structures are supported by elastic foundations, so determining the impacts of the elastic foundation stiffness coefficient on the structure is very important in calculating and designing structures. Accordingly, Fig.6 describes the impacts of the elastic medium stiffness parameters K_{\mathrm{w}} and K_{\mathrm{s}} on the natural frequency of MEE-FG nanobeam. The foundation stiffness coefficient K_{\mathrm{w}} varies from 0 to 100 and K_{\mathrm{s}} varies from 0 to 10. As expected, these two parameters increase, increasing the natural frequency of the nanobeam. Besides, the shift in frequency with respect to K_{\mathrm{w}} has an almost linear law, and while a change of K_{\mathrm{s}} follows a near nonlinear law.

4.3 Stress, electric and magnetic field

Similar to Subsection 4.2, the stresses {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right),{\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right), the electric displacement D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) and magnetic induction B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) varies along the beam thickness due to the influence of the factors presented in graph form. First, the impacts of voltage difference V_0 on the values of normal stress {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right), shear stress {\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right), electric displacement D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) and magnetic induction B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) of SS MEE-FG nanobeam is shown in Fig.6. It is easy to see that the value of voltage V_0 increases in the positive direction, causing the stress value to increase, besides, it also increases the magnitude value of D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right), B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right). The variation of {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right),B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) along the beam thickness are almost nonlinear curves, and the lines intersect at the same point (these curves occur due to the shift in mechanical properties along the beam thickness). Whereas the shear stress is a deviated parabola and D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) are non-intersecting nonlinear curves. It is important to note that the shear stress {\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) will be zero at the top \left(\mathit{\text{z}}=h/2\right) and bottom \left(\mathit{\text{z}}=-h/2\right) faces of the nanobeam, which the classical first-order shear strain hypothesis with a shear correction factor of 5/6 cannot be realized.

In contrast to the increase in voltage value V_0 (Fig.7), the increase in the value of {\Psi }_0 (Fig.8) in the positive direction causes the normal stress value {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right) to decrease and increase in value. The maximum value of the shear stress {\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right). Besides, increasing the value of {\Psi }_0 leads to a negligible change in the value of D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) along the girder thickness while the value of B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) changes rapidly along the thickness and is larger toward the underside of the beam \left(\mathit{\text{z}}=-h/2\right).

Next, Fig.9 describes the impacts of the nonlocal parameter {\eta }_m on the static displacement of the MEE-FG nanobeam. The parameter {\eta }_m changes at five values including 0, 0.5, 1.0, 1.5, 2.0 nm. It is easy to see that the stress values all increase as the nonlocal parameter increases; this can be explained as the coefficient. The nonlocal increase causes the nanobeam structure to soften, thus increasing the displacement and stress values. In addition, the valuesD_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right),B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) also tend to increase when the nonlocal coefficient changes, these lines intersect at \mathit{\text{z}}=0.

Fig.10 shows the change of {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right),{\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right),D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right),B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) along the thickness of MEE-FG nanobeam with various values p_{\mathit{\text{z}}}. It can be seen that the increase in the p_{\mathit{\text{z}}} parameter causes most of the mechanical properties of the beam to increase. At position \mathit{\text{z}}=-h/2 the value of D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) is the same in the cases where p_{\mathit{\text{z}}} changes and is significantly different when \mathit{\text{z}}=h/2; while according to the beam thickness, the lines B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) are close together at the point \mathit{\text{z}}=-0.3h.

Fig.11 and Fig.12 show the effect of temperature and humidity differences on stresses {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right),{\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right), the electric displacement D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) and magnetic induction B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) along the girder thickness. It can be observed that increasing the temperature and humidity differences leads to an increase in the magnitude of the stresses, the electric displacement D_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right) and magnetic induction B_{\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right). However, they do not change much between the maximum and minimum values, which is consistent with reality because FG materials can work in high-temperature and humid environments.

5 Conclusions

Based on IGA and refined nonlocal higher-order shear hypothesis, the free oscillation and bending behavior of MEE-FG nanobeam subject to elastic BCs in a hygro-temperature environment on Pasternak medium are presented. The magneto-electric boundary condition and Maxwell equation are used to calculate the variation of electric and magnetic potentials along the thickness direction of the nanobeam. Elastic BCs help to overcome the limitations of traditional boundary conditions that have been studied previously. The accuracy and convergence of the proposed method are carefully verified by comparing the numerical results with reputable publications. From an in-depth assessment of the impacts of factors on bending and free oscillation characteristics, the paper has the following conclusions.

1) The elastic BCs suggested in this study bring the structural analysis closer to engineering reality compared to the utilization of conventional boundary conditions.

2) The voltage applied to the beam increases in the positive direction, causing the structure’s stiffness to decrease, while the value of the magnetic field increases in the positive direction, making the structure stiffer. Therefore, the change of the two values of voltage and magnetic field can control the vibration of the structure easily

3) The temperature and humidity environment causes the structure’s stiffness to decrease, while the elastic medium stiffness makes the structure stiffer.

4) The thinner the beam thickness, the more obvious the electromagnetic effect is through increased and decreased structural stiffness.

The new points of the present work:

The IGA combined refined higher-order shear theory method was first used in the MEE-FG nanobeam’s static bending and vibration analysis with elastic boundary conditions.

The results of this paper can be used as a benchmark in comparing MEE-FG nanobeam structural calculations. In the future, the algorithm and calculation program of the article can be developed to calculate the nonlinear dynamics, linear and nonlinear dynamic stability problems of MEE-FG nanobeam and for the MEE-FG plate and shell structures.

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