Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro–thermo–magnetic environment

Abstract

Herein, a two-node beam element enriched based on the Lagrange and Hermite interpolation function is proposed to solve the governing equation of a functionally graded porous (FGP) curved nanobeam on an elastic foundation in a hygro–thermo–magnetic environment. The material properties of curved nanobeams change continuously along the thickness via a power-law distribution, and the porosity distributions are described by an uneven porosity distribution. The effects of magnetic fields, temperature, and moisture on the curved nanobeam are assumed to result in axial loads and not affect the mechanical properties of the material. The equilibrium equations of the curved nanobeam are derived using Hamilton’s principle based on various beam theories, including the classical theory, first-order shear deformation theory, and higher-order shear deformation theory, and the nonlocal elasticity theory. The accuracy of the proposed method is verified by comparing the results obtained with those of previous reliable studies. Additionally, the effects of different parameters on the free vibration behavior of the FGP curved nanobeams are investigated comprehensively.

Graphical abstract

Keywords

functionally graded porous material / curved nanobeam / hygro–thermo–magnetic / enriched finite element method

1 Introduction

Owing to the rapid development of the material industry in recent decades, micro- and nanosized structures are increasingly and widely used in various fields, including medicine, electronics, biotechnology, and micro-electromechanics. Consequently, scientists have become more interested in analyzing the mechanical behavior of these micro- and nanosized structures. However, the classical continuum theories applied to conventional structures are no longer exact when they are used to analyze micro- and nanoscale structural elements, such as nanobeams, nanoplates, and nanoshells. To resolve this problem and obtain the most accurate results, scientists have proposed modified elasticity theories, such as the strain gradient elasticity, couple stress, Eringen’s nonlocal elasticity, and general nonlocal elasticity [1–7]. Among the above theories, Eringen’s nonlocal elasticity theory has been the most developed for analyzing nanoscale structures [8–22].

Functionally graded materials (FGMs) are primarily fabricated using metals and ceramics, in which the material properties continuously change in one or two directions, to achieve the desired mechanical features. Over recent decades, the utilization of FGMs has increased significantly; therefore, understanding the mechanical behavior of FGM micro/nanostructures is essential [23–30]. Meanwhile, the static bending, free vibration, and buckling behaviors of functionally graded (FG) nanobeams have been investigated extensively based on various theories by many scientists. Assadi and Farshi [31] investigated the size-dependent vibrations of curved nanobeams and rings by considering the surface energy effect. Ansari et al. [32] investigated the free vibration behavior of curved FG microbeams with varying material properties in the thickness direction using the modified strain gradient elasticity theory coupled with the first-order shear deformation beam theory. Medina et al. [33] investigated the snap-through buckling of electrostatically driven initially curved prestressed microbeams. Ebrahimi and Barati [34] proposed a nonlocal strain-gradient theory for the dynamic simulation of inhomogeneous curved nanobeams embedded in an elastic medium. Hosseini and Rahmani [35] performed a free vibration analysis of shallow and deep curved FG nanobeams using the nonlocal elasticity theory established by Eringen. Arefi and Zenkour [36] investigated the transient response of a sandwich-curved nanobeam on a Pasternak-type elastic foundation in a magneto–electro–elastic environment. In their study, the governing equations of motion were derived using nonlocal magneto–electro–elastic theory and Hamilton’s principle. She et al. [37] investigated the free vibration behavior of porous nanotubes for the first time. In their study, the nonlocal strain gradient theory was combined with a refined beam model to establish a size-dependent model of nanotubes. Ebrahimi and Barati [38] used the nonlocal strain gradient elasticity theory and a higher-order refined beam theory to analyze the damping vibration characteristics of FG nanobeams on a viscoelastic foundation under hygro–thermo conditions. Ganapathi et al. [39] presented a finite element approach for the free vibration analysis of curved nanobeams based on nonlocal higher-order shear deformation theory. Rezaiee-Pajand [40] developed an efficient and high-performance four-node isoparametric beam element for the thermo–mechanical nonlinear analysis of functionally graded porous (FGP) beams.

Most of the studies pertaining to curved FG nanobeams above involve primarily analytical and semi-analytical methods, which are limited to some complex geometric and boundary conditions. Hence, numerical methods such as isogeometric analysis [41–44] and finite element methods [45–50] which are more advantageous compared with analytical and semi-analytic methods, have been proposed. However, using the standard finite element method is ineffective for solving problems that require substantial computations. Therefore, enriched finite element methods must be devised to reduce the computational cost while ensuring convergence and accuracy. A few enriched finite element methods have been published [51–56], which are superior to the standard finite element method and thus can be used for solving problems requiring substantial computations.

A review of the literature shows that the mechanical behavior of curved FG nanobeams has been investigated extensively and remarkable achievements have been realized. However, the mechanical behavior of curved nanobeams composed of FGP materials in hygro–thermo–magnetic environments has not been analyzed hitherto. In addition, investigations into the behavior of FG-curved nanobeams using the enriched finite element method are few. Therefore, this study is conducted to fill the abovementioned research gaps; in particular, an enriched finite element method is proposed to perform a free vibration analysis of FGP-curved nanobeams on an elastic foundation in a hygro–thermo–magnetic environment based on various beam theories. The proposed model considers a two-node beam element based on hierarchical functions to enrich the Lagrange and Hermite interpolation functions and provide a good convergent beam element. The material properties vary continuously with the thickness via a power-law distribution, and the porosity for manufacturing curved beams are defined by an uneven distribution. The effects of the magnetic fields, temperature, and moisture on the curved nanobeam are assumed to result in initial axial stresses and not affect the mechanical properties of the material. The elastic foundation is assumed to be of the Winkler–Pasternak type. The numerical results are compared with results reported in the literature to validate the model. Additionally, the effects of the material properties, power-law index, porosity volume fraction, temperature, and moisture on the free vibration behavior of the FGP curved nanobeams are analyzed comprehensively.

The remainder of this paper is organized as follows: Section 2 presents the theoretical formulations, mechanical model, and materials. Section 3 provides the verification examples and numerical results. Section 4 presents the highlights and conclusions of the study.

2 Theoretical formulation

2.1 FGP nanobeams

Consider a curved rectangular cross-section FGP nanobeam with mean radius R_x, opening angle \phi, length a=R_x\phi, width b, and thickness h, as shown in Fig.1 The material properties are assumed to vary continuously via the power-law distribution from the top surface (\mathit{\text{z}}=h/2) to the bottom surface (\mathit{\text{z}}=-h/2). The top surface is composed of ceramic, whereas the bottom surface is composed of metal. The elastic foundation is of the Winkler–Pasternak type, which comprises two parameters, i.e., the Winkler stiffness coefficient (k_{\text{w}}) and shearing stiffness coefficient (k_{\text{s}}). The curved nanobeam is immersed in a hygro–thermo–magnetic environment, which is assumed to result in only initial axial stresses.

Owing to the porosity of the material during the fabrication of FGP nanobeams, the following variation law pertaining to the mechanical characteristics can be considered [57]

(1)\{\begin{array}{c}S\left(\mathit{\text{z}}\right)=S_{\text{m}}+\left(S_{\text{c}}-S_{\text{m}}\right)V_{\text{c}}-\vartheta \left(S_{\text{c}}+S_{\text{m}}\right),\\ V_{\text{c}}={\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{p_{\text{z}}},\end{array}

where S represents the effective material properties, such as Young’s modulus E, mass density \rho, and Poisson’s ratio \nu, thermal expansion coefficient \alpha, and moisture expansion coefficient \beta ; V_{\text{c}} is the volume fraction of the ceramic; p_{\text{z}} is the power-law index, which is a positive real number and determines the material distribution in the thickness direction of the curved nanobeam. The subscripts ‘m’ and ‘c’ denote the metallic and ceramic constituents, respectively. \vartheta is the control coefficient for the porosity of the nanobeam and defined as follows [57]:

(2)\vartheta =\frac{\chi }{2}\left(1-\frac{2\left|\mathit{\text{z}}\right|}{h}\right),

where \chi is the coefficient of evaluation of the porosity volume of the curved nanobeam.

2.2 Kinematic relations

The variations in the thickness direction of the nonzero displacement components, i.e., the axial and transverse components (U_x and U_{\mathit{\text{z}}}, respectively), are approximated as follows [58]:

(3)\{\begin{array}{c}{U}_x\left(x,\mathit{\text{z}},t\right)=\left(1+\frac{\mathit{\text{z}}}{R_x}\right)u_0\left(x,t\right)-\mathit{\text{z}}\frac{\mathrm{\partial }{w}^{\text{b}}\left(x,t\right)}{\mathrm{\partial }x}-f\left(\mathit{\text{z}}\right)\frac{\mathrm{\partial }{w}^{\text{s}}\left(x,t\right)}{\mathrm{\partial }x},\\ U_{\mathit{\text{z}}}\left(x,\mathit{\text{z}},t\right)=w^{\text{b}}\left(x,t\right)+w^{\text{s}}\left(x,t\right),\end{array}

where u_0, w^{\text{b}}, and w^{\text{s}} are the three unknowns corresponding to the mid-plane axial displacement, transverse bending, and transverse shear displacement components, respectively.

The definition of the function f\left(\mathit{\text{z}}\right) in Eq. (3) allows us to recover some theories in the literature as follows.

• Classical beam theory (CLT): f\left(\mathit{\text{z}}\right)=0, w^{\text{s}}=0.

• First-order shear deformation beam theory (FSDT) [9]: f\left(\mathit{\text{z}}\right)=0.

• Higher-order trigonometric shear beam theory (HSDT) [62]: f\left(\mathit{\text{z}}\right)=\mathit{\text{z}}-\text{sin}{\displaystyle \frac{\text{π}\mathit{\text{z}}}{h}}.

Based on the displacement field in Eq. (3), the nonzero linear strain components of the FGP curved nanobeam are expressed as follows.

(4)\begin{array}{rl}{\epsilon }_{xx}=& \frac{\mathrm{\partial }{U}_x}{\mathrm{\partial }x}+\frac{U_{\mathit{\text{z}}}}{R_x}=\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}+\frac{w^{\text{b}}+w^{\text{s}}}{R_x}\\ & -\mathit{\text{z}}\left(\frac{{\mathrm{\partial }}^2{w}^{\text{b}}}{\mathrm{\partial }{x}^2}-\frac{1}{R_x}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}\right)-f\left(\mathit{\text{z}}\right)\frac{{\mathrm{\partial }}^2{w}^{\text{s}}}{\mathrm{\partial }{x}^2}\\ =& {\epsilon }^1+\mathit{\text{z}}{\epsilon }^2+f\left(\mathit{\text{z}}\right){\epsilon }^3,\end{array}

(5){\gamma }_{x\mathit{\text{z}}}=\frac{\mathrm{\partial }{U}_x}{\mathrm{\partial }\mathit{\text{z}}}+\frac{\mathrm{\partial }{U}_{\mathit{\text{z}}}}{\mathrm{\partial }x}-\frac{u_0}{R_x}=g\left(\mathit{\text{z}}\right)\frac{\mathrm{\partial }{w}^{\text{s}}}{\mathrm{\partial }x}=g\left(\mathit{\text{z}}\right){\gamma }^1,

where

\begin{array}{}\text{(6a)}& \mathrm{F}\mathrm{S}\mathrm{D}\mathrm{T}[59]:g\left(\mathit{\text{z}}\right)=\frac{5}{4}\cos \frac{\text{π}\mathit{\text{z}}}{h},\end{array}

\begin{array}{}\text{(6b)}& \mathrm{H}\mathrm{S}\mathrm{D}\mathrm{T}:g\left(\mathit{\text{z}}\right)=1-\frac{\mathrm{\partial }f\left(\mathit{\text{z}}\right)}{\mathrm{\partial }\mathit{\text{z}}}=\cos \frac{\text{π}\mathit{\text{z}}}{h}.\end{array}

Based on Eq. (6), the components of the transverse shear strain {\gamma }_{x\mathit{\text{z}}} satisfy the zero shear stress conditions at the top and bottom surfaces of the curved nanobeam of the FSDT. This problem is novel, as the shear correction factor k=5/6 and reduction integral are not necessary for addressing shear locking.

In fact, the equations above can be summarized in the form of vectors as follows.

(7){\epsilon }_{xx}={\epsilon }^1+\mathit{\text{z}}{\epsilon }^2+f\left(\mathit{\text{z}}\right){\epsilon }^3,{\gamma }_{x\mathit{\text{z}}}=g\left(\mathit{\text{z}}\right){\gamma }^1,

where

(8)\begin{array}{rl}& {\epsilon }^1=\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}+\frac{w^{\text{b}}+w^{\text{s}}}{R_x},{\epsilon }^2=-\frac{{\mathrm{\partial }}^2{w}^{\text{b}}}{\mathrm{\partial }{x}^2}+\frac{\mathrm{\partial }{u}_0}{R_x\mathrm{\partial }x},\\ & {\epsilon }^3=-\frac{{\mathrm{\partial }}^2{w}^{\text{s}}}{\mathrm{\partial }{x}^2},{\gamma }^1=\frac{\mathrm{\partial }{w}^{\text{s}}}{\mathrm{\partial }x}.\end{array}

2.3 Equations of motion

The general equations of motion of the FGP curved nanobeam are derived by applying Hamilton’s principle, as follows [9]:

(9){\int }_{t_1}^{t_2}\left(\delta U^{\text{b}}+\delta V-\delta K\right)\text{d}t=0,

where \delta U^{\text{b}} denote the variation in the strain energy, \delta V is the variation in the potential energy of the external loads, and \delta K is the variation in the kinetic energy.

The variation in the strain energy of the FGP curved nanobeam \delta U^{\text{b}} for the eth element with length l_e can be expressed as

(10)\begin{array}{rl}\delta U^{\text{b}}=& \underset{l_e}{\overset{}{\int }}\underset{-h/2}{\overset{h/2}{\int }}\left({\sigma }_{xx}\delta {\epsilon }_{xx}+{\sigma }_{x\mathit{\text{z}}}\delta {\gamma }_{x\mathit{\text{z}}}\right)\text{d}\mathit{\text{z}}\text{d}x\\ =& {\int }_{l_{\text{e}}}^{}({\mathit{N}}^x\left(\frac{\mathrm{\partial }\delta u_0}{\mathrm{\partial }x}+\frac{\delta w^{\text{b}}+\delta w^{\text{s}}}{R_x}\right)\\ & -{\mathit{M}}^x\left(\frac{{\mathrm{\partial }}^2\delta w^{\text{b}}}{\mathrm{\partial }{x}^2}-\frac{\mathrm{\partial }\delta u_0}{R_x\mathrm{\partial }x}\right)+{\mathit{L}}^x\frac{{\mathrm{\partial }}^2\delta w^{\text{s}}}{\mathrm{\partial }{x}^2}-{\mathit{Q}}^x\frac{\mathrm{\partial }\delta w^{\text{s}}}{\mathrm{\partial }x})\text{d}x,\end{array}

where {\mathit{N}}^x,{\mathit{Q}}^x,{\mathit{M}}^x, and {\mathit{L}}^x are the axial compression force, shear force, bending moment, and higher-order bending moment in the x-direction, respectively.

(11)\begin{array}{rl}& {\mathit{N}}^x={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\sigma }_{xx}\text{d}\mathit{\text{z}},{\mathit{M}}^x={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\overline{\mathit{\text{z}}}{\sigma }_{xx}\text{d}\mathit{\text{z}},\\ & {\mathit{L}}^x={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\overline{f}{\sigma }_{xx}\text{d}\mathit{\text{z}},{\mathit{Q}}^x={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\sigma }_{x\mathit{\text{z}}}\overline{g}\text{d}\mathit{\text{z}},\end{array}

where \overline{\mathit{\text{z}}}=\mathit{\text{z}}-t_0,\overline{f}=f\left(\mathit{\text{z}}-t_0\right),\overline{g}=g\left(\mathit{\text{z}}-t_0\right); t_0 is the distance from the geometrical middle surface to the physically neutral surface of the FGP curved nanobeam, which is defined as follows:

(12)t_0=\frac{{\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\mathit{\text{z}}\text{d}\mathit{\text{z}}}}{{\displaystyle {\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\text{d}\mathit{\text{z}}}}.

The variation in the potential energy \delta V of the applied forces is expressed as

(13)\delta V=\underset{l_e}{\overset{}{\int }}\left(-q_{\mathit{\text{z}}}+k_{\text{w}}{U}_{\mathit{\text{z}}}-k_{\text{s}}\frac{{\mathrm{\partial }}^2{U}_{\mathit{\text{z}}}}{\mathrm{\partial }{x}^2}-f^{\text{HT}}\frac{{\mathrm{\partial }}^2{U}_{\mathit{\text{z}}}}{\mathrm{\partial }{x}^2}-\eta h{H}_x\frac{{\mathrm{\partial }}^2{U}_{\mathit{\text{z}}}}{\mathrm{\partial }{x}^2}\right)\delta U_{\mathit{\text{z}}}\text{d}x,

where q_{\mathit{\text{z}}} is the mechanical force exerting in the direction perpendicular to the tangent of the curved beam, f^{\text{HT}} is the hygro–thermo load [60] due to the temperature and moisture of the environment, \eta is the magnetic permeability, and H_x is the magnetic field. Mathematically, the effect of a hygro–thermo environment can be determined based on the forces due to temperature increase and moisture absorption, as follows:

(14)\begin{array}{rl}& f^{\text{HT}}=f^{\text{H}}+f^{\text{T}},f^{\text{H}}=-{\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\left(\mathit{\text{z}}\right)\beta \left(\mathit{\text{z}}\right)\mathrm{\Delta }C\text{d}\mathit{\text{z}},\\ & f^{\text{T}}=-{\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\left(\mathit{\text{z}}\right)\alpha \left(\mathit{\text{z}}\right)\mathrm{\Delta }T\text{d}\mathit{\text{z}},\end{array}

where \mathrm{\Delta }T and \mathrm{\Delta }C denote the changes in temperature and moisture, respectively, which conform to the law of uniform distribution [60].

The variation in kinetic energy \delta K is expressed as

(15)\begin{array}{rl}\delta K=& {\int }_{l_{\text{e}}}^{}{\int }_{-\frac{h}{2}}^{\frac{h}{2}}\rho \left(\mathit{\text{z}}\right)\left(\frac{\mathrm{\partial }{U}_x}{\mathrm{\partial }t}\frac{\mathrm{\partial }\delta U_x}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{U}_{\mathit{\text{z}}}}{\mathrm{\partial }t}\frac{\mathrm{\partial }\delta U_{\mathit{\text{z}}}}{\mathrm{\partial }t}\right)\text{d}\mathit{\text{z}}\text{d}x\\ =& \underset{l_e}{\overset{}{\int }}(I_1\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }t}\frac{\mathrm{\partial }\delta u_0}{\mathrm{\partial }t}+I_2\frac{\mathrm{\partial }\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }t}\frac{\mathrm{\partial }\left(\delta w^{\text{b}}+\delta w^{\text{s}}\right)}{\mathrm{\partial }t}\\ & +I_3\left(\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2\delta w^{\text{b}}}{\mathrm{\partial }x\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^2{w}^{\text{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{\mathrm{\partial }\delta u_0}{\mathrm{\partial }t}\right)\\ & +I_6\left(\frac{{\mathrm{\partial }}^2{w}^{\text{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2\delta w^{\text{s}}}{\mathrm{\partial }x\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^2{w}^{\text{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2\delta w^{\text{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\right)+I_7\frac{{\mathrm{\partial }}^2{w}^{\text{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2\delta w^{\text{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\\ & +I_4\left(\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2\delta w^{\text{s}}}{\mathrm{\partial }x\mathrm{\partial }t}+\frac{{\mathrm{\partial }}^2{w}^{\text{s}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{\mathrm{\partial }\delta u_0}{\mathrm{\partial }t}\right)+I_5\frac{{\mathrm{\partial }}^2{w}^{\text{b}}}{\mathrm{\partial }x\mathrm{\partial }t}\frac{{\mathrm{\partial }}^2\delta w^{\text{b}}}{\mathrm{\partial }x\mathrm{\partial }t})\text{d}x,\end{array}

where

\left(I_1,I_3,I_4\right)={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\rho \left(\mathit{\text{z}}\right)\left(1+\frac{\overline{\mathit{\text{z}}}}{R_x}\right)\left(1+\frac{\overline{\mathit{\text{z}}}}{R_x},-\overline{\mathit{\text{z}}},-\overline{f}\right)\text{d}\mathit{\text{z}},

(16)\left(I_2,I_5,I_6,I_7\right)={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\rho \left(\mathit{\text{z}}\right)\left(1,{\overline{\mathit{\text{z}}}}^2,\overline{\mathit{\text{z}}}\overline{f},{\overline{f}}^2\right)\text{d}\mathit{\text{z}}.

Substituting Eqs. (10), (13), and (15) into Eq. (9) and performing mathematical manipulations yields the following element motion equations:

(17)\delta u_0:\frac{\mathrm{\partial }{\mathit{N}}^x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\mathit{M}}^x}{R_x\mathrm{\partial }x}=I_1\frac{{\mathrm{\partial }}^2{u}_0}{\mathrm{\partial }{t}^2}+I_3\frac{{\mathrm{\partial }}^3{w}^{\text{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }x}+I_4\frac{{\mathrm{\partial }}^3{w}^{\text{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }x},

(18)\begin{array}{rl}\delta w^{\text{b}}:& \frac{{\mathrm{\partial }}^2{\mathit{M}}^x}{\mathrm{\partial }{x}^2}-\frac{{\mathit{N}}^x}{R_x}=-q_{\mathit{\text{z}}}+I_2\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{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}^{\text{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-I_6\frac{{\mathrm{\partial }}^4{w}^{\text{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}\\ & -k_{\text{w}}\left(w^{\text{b}}+w^{\text{s}}\right)+\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{x}^2},\end{array}

(19)\begin{array}{rl}\delta w^{\text{s}}:& \frac{{\mathrm{\partial }}^2{\mathit{L}}^x}{\mathrm{\partial }{x}^2}+\frac{\mathrm{\partial }{\mathit{Q}}^x}{\mathrm{\partial }x}-\frac{{\mathit{N}}^x}{R_x}=-q_{\mathit{\text{z}}}+I_2\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{t}^2}\\ & -I_4\frac{{\mathrm{\partial }}^3{u}_0}{\mathrm{\partial }x\mathrm{\partial }{t}^2}-I_6\frac{{\mathrm{\partial }}^4{w}^{\text{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-I_7\frac{{\mathrm{\partial }}^4{w}^{\text{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}\\ & -k_{\text{w}}\left(w^{\text{b}}+w^{\text{s}}\right)+\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{x}^2}.\end{array}

2.4 Nonlocal elasticity theory

The equations of nonlocal elastic solids are considered herein and are formulated as follows:

(20){\sigma }_{ij,j}+q_i=\rho {\ddot{u}}_i,

where i,j represent the symbols \mathit{\text{z}},x, respectively; q_i is the body load; \rho is the mass density.

According to classical local elasticity theories, the stress at a point depends only on the strain components at that point, whereas nonlocal elasticity theories assume that the stress at a point depends on the strain components at all points of the continuum.

The basic constitutive equations for a nonlocal homogeneous material without a body force can be written as [1,2,13]

(21){\sigma }_{ij}={\int }_{\psi }^{}\alpha \left(\left[x^{\prime }-x\right],\tau \right){\sigma }_{ij}\left(x^{\prime }\right)\text{d}{x}^{\prime },

where \alpha \left(\left[x^{\prime }-x\right],\tau \right) is the nonlocal kernel function, {\sigma }_{ij} is the local stress tensor at point x, and \tau is a length-scale parameter.

According to Eringen’s theory along with certain assumptions, integro-partial differential equations of nonlocal elasticity can be simplified to partial differential equations [1,2,13]. Thus, the nonlocal constitutive behavior of a Hookean solid can be represented by the following differential constitutive relation:

(22){\sigma }_{ij}-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}{\sigma }_{ij}=C_{ijkl}{\epsilon }_{ij},

where \mu =\left(e_{0}a\right)\text{nm} represents the small-scale effect in nanostructures.

According to the nonlocal theory, Eq. (22) can be adapted for an elastic curved nanobeam as follows:

(23){\sigma }_{xx}-{\mu }^2\frac{{\mathrm{\partial }}^2{\sigma }_{xx}}{\mathrm{\partial }{x}^2}=E\left(\mathit{\text{z}}\right){\epsilon }_{xx},{\sigma }_{x\mathit{\text{z}}}-{\mu }^2\frac{{\mathrm{\partial }}^2{\sigma }_{x\mathit{\text{z}}}}{\mathrm{\partial }{x}^2}={\displaystyle \frac{E\left(\mathit{\text{z}}\right)}{2\left(1+v\left(\mathit{\text{z}}\right)\right)}}{\gamma }_{x\mathit{\text{z}}}.

The following constitutive relations are obtained using Eqs. (11) and (23):

(24)\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right){\mathit{N}}^x=\left(\mathbb{A}{\epsilon }^1+\mathbb{B}{\epsilon }^2+{\mathbb{B}}^{\text{b}}{\epsilon }^3\right),

(25)\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right){\mathit{M}}^x=\left(\mathbb{B}{\epsilon }^1+\mathbb{F}{\epsilon }^2+{\mathbb{F}}^{\text{b}}{\epsilon }^3\right),

(26)\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right){\mathit{L}}^x=\left({\mathbb{B}}^{\text{b}}{\epsilon }^1+{\mathbb{F}}^{\text{b}}{\epsilon }^2+\mathbb{H}{\epsilon }^3\right),

(27)\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right){\mathit{Q}}^x={\mathbb{A}}^{\text{b}}{\gamma }^1,

where

(28)\begin{array}{rl}& \left(\mathbb{A},\mathbb{B},{\mathbb{B}}^{\text{b}},\mathbb{F},{\mathbb{F}}^{\text{b}},\mathbb{H}\right)={\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\left(\mathit{\text{z}}\right)\left(1,\overline{\mathit{\text{z}}},\overline{f},{\overline{\mathit{\text{z}}}}^2,\overline{\mathit{\text{z}}}\overline{f},{\overline{f}}^2\right)\text{d}\mathit{\text{z}},\\ & {\mathbb{A}}^{\text{b}}={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\displaystyle \frac{E\left(\mathit{\text{z}}\right)}{2\left(1+v\left(\mathit{\text{z}}\right)\right)}}{\overline{g}}^2\text{d}\mathit{\text{z}}.\end{array}

Substituting Eqs. (24)–(27) into Eqs. (17)–(19) yields the following equations for the motion of an FGP curved nanobeam:

(29)\frac{\mathrm{\partial }{\mathit{N}}^x}{\mathrm{\partial }x}+\frac{1}{R_x}\frac{\mathrm{\partial }{\mathit{M}}^x}{\mathrm{\partial }x}=\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)\left(I_1\frac{{\mathrm{\partial }}^2{u}_0}{\mathrm{\partial }{t}^2}+I_3\frac{{\mathrm{\partial }}^3{w}^{\text{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }x}+I_4\frac{{\mathrm{\partial }}^3{w}^{\text{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }x}\right),

(30)\begin{array}{rl}\frac{{\mathrm{\partial }}^2{\mathit{M}}^x}{\mathrm{\partial }{x}^2}& -\frac{{\mathit{N}}^x}{R_x}=\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)(-q_{\mathit{\text{z}}}+I_2\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{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}^{\text{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-I_6\frac{{\mathrm{\partial }}^4{w}^{\text{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}\\ & -k_{\text{w}}\left(w^{\text{b}}+w^{\text{s}}\right)+\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{x}^2}),\end{array}

(31)\begin{array}{rl}\frac{{\mathrm{\partial }}^2{\mathit{L}}^x}{\mathrm{\partial }{x}^2}& +\frac{\mathrm{\partial }{\mathit{Q}}^x}{\mathrm{\partial }x}-\frac{{\mathit{N}}^x}{R_x}=\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)(-q_{\mathit{\text{z}}}+I_2\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{t}^2}\\ & -I_4\frac{{\mathrm{\partial }}^3{u}_0}{\mathrm{\partial }x\mathrm{\partial }{t}^2}-I_6\frac{{\mathrm{\partial }}^4{w}^{\text{s}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}-I_7\frac{{\mathrm{\partial }}^4{w}^{\text{b}}}{\mathrm{\partial }{t}^2\mathrm{\partial }{x}^2}\\ & -k_{\text{w}}\left(w^{\text{b}}+w^{\text{s}}\right)+\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2\left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{x}^2}).\end{array}

Using Galerkin’s method and weight functions \delta u_0,\delta w^{\text{b}}, and \delta w^{\text{s}} in Eqs. (29)–(31), respectively, yields the following equation:

(32)\begin{array}{rl}& {\int }_{\psi }^{}{\sigma }_{ij,j}\delta {\epsilon }_{ij}\text{d}\mathit{\text{{textit{z}}}}\text{d}x+\underset{S}{\stackrel{}{{\displaystyle \int }}}\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)\\ & \cdot \left(k_{\text{w}}{U}_{\mathit{\text{z}}}-\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}{U}_{\mathit{\text{z}}}\right)\delta U_{\mathit{\text{z}}}\text{d}x\\ & +{\int }_{\psi }^{}\left(1-{\mu }^2{\mathrm{\nabla }}^2\right)\rho {\ddot{U}}_i\delta U_i\text{d}\mathit{\text{z}}\text{d}x\\ =& {\int }_{\psi }^{}\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)q_{\mathit{\text{z}}}\delta U_{\mathit{\text{z}}}\text{d}x+{{\displaystyle \int }}_{{\text{Λ}}_{\text{g}}}^{}{\sigma }_{ij}{n}_j\delta U_i\mathrm{d}{\text{Λ}}_{\text{g}},\end{array}

where {\mathrm{\Lambda }}_{\text{g}} denotes the Neumann boundary. In this study, traction on the Neumann boundary is disregarded [61]. Hence, Eq. (32) is expressed as follows:

(33)\begin{array}{rl}& {\int }_{\psi }^{}{\sigma }_{ij,j}\delta {\epsilon }_{ij}\text{d}\mathit{\text{z}}\text{d}x+\underset{S}{\stackrel{}{{\displaystyle \int }}}\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)\\ & \cdot \left(k_{\text{w}}{U}_i-\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}{U}_{\mathit{\text{z}}}\right)\delta U_{\mathit{\text{z}}}\text{d}x\\ & +{\int }_{\psi }^{}\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)\rho {\ddot{U}}_i\delta U_i\text{d}\mathit{\text{z}}\text{d}x\\ =& {\int }_{\psi }^{}\left(1-{\mu }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\right)q_{\mathit{\text{z}}}\delta U_{\mathit{\text{z}}}\text{d}x.\end{array}

Subsequently, Eq. (33) can be rewritten as

(34)\begin{array}{rl}& {\int }_{l_e}^{}({\epsilon }^{\text{T}}Q\delta \epsilon +k_{\text{w}}{\left(w^{\text{b}}+w^{\text{s}}\right)}^{\text{T}}\delta \left(w^{\text{b}}+w^{\text{s}}\right)\\ & -\left({\mathit{P}}_{\text{ma}}^{\text{T}}{\mathit{H}}_{\text{ma}}\delta {\mathit{P}}_{\text{ma}}+{\mu }^2\frac{\mathrm{\partial }{\mathit{P}}_{\text{ma}}^{\text{T}}}{\mathrm{\partial }x}{\mathit{H}}_{\text{ma}}\frac{\mathrm{\partial }\delta {\mathit{P}}_{\text{ma}}^{\text{T}}}{\mathrm{\partial }x}\right)\\ & -q_{\mathit{\text{z}}}\left(\delta {\left(w^{\text{b}}+w^{\text{s}}\right)}^{\text{T}}-{\mu }^2\frac{{\mathrm{\partial }}^2{\left(w^{\text{b}}+w^{\text{s}}\right)}^{\text{T}}}{\mathrm{\partial }{x}^2}\right)\\ & +\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x+{\mu }^2{k}_{\text{w}}\right)\frac{\mathrm{\partial }{\left(w^{\text{b}}+w^{\text{s}}\right)}^{\text{T}}}{\mathrm{\partial }x}\frac{\mathrm{\partial }\delta \left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }x}\\ & +{\mu }^2\left(k_{\text{s}}+f^{\text{HT}}+\eta h{H}_x\right)\frac{{\mathrm{\partial }}^2{\left(w^{\text{b}}+w^{\text{s}}\right)}^{\text{T}}}{\mathrm{\partial }{x}^2}\frac{{\mathrm{\partial }}^2\delta \left(w^{\text{b}}+w^{\text{s}}\right)}{\mathrm{\partial }{x}^2})\text{d}x=0,\end{array}

where

(35)\begin{array}{rl}& \mathit{\epsilon }=\left\{\begin{array}{c}{\epsilon }^1\\ {\epsilon }^2\\ {\epsilon }^3\\ {\gamma }^1\end{array}\right\},\mathit{Q}=\left[\begin{array}{cccc}\mathbb{A}& \mathbb{B}& {\mathbb{B}}^{\text{b}}& 0\\ \mathbb{B}& \mathbb{F}& {\mathbb{F}}^{\text{b}}& 0\\ {\mathbb{B}}^{\text{b}}& {\mathbb{F}}^{\text{b}}& \mathbb{H}& 0\\ 0& 0& 0& {\mathbb{A}}^{\text{b}}\end{array}\right],\\ & {\mathit{H}}_{\text{ma}}=\left[\begin{array}{ccccc}{I}_1& 0& 0& I_3& I_4\\ 0& I_2& I_2& 0& 0\\ 0& I_2& I_2& 0& 0\\ I_3& 0& 0& I_5& I_6\\ I_4& 0& 0& I_6& I_7\end{array}\right],{\mathit{P}}_{\text{ma}}=\left\{\begin{array}{c}{u}_0\\ w^{\text{b}}\\ w^{\text{s}}\\ {\displaystyle \frac{\mathrm{\partial }{w}^{\text{b}}}{\mathrm{\partial }x}}\\ {\displaystyle \frac{\mathrm{\partial }{w}^{\text{s}}}{\mathrm{\partial }x}}\end{array}\right\}.\end{array}

2.5 Navier’s method

Prior to using the enriched finite element method to compute the FGP curved nanobeam, we devised an analytical method based on Navier’s theory to confirm the accuracy and reliability of the proposed enriched finite element method. The displacements of the simply supported curved nanobeam based on Navier’s solution are as follows [62]:

(36)u_0=\sum _{m=1}^{\mathrm{\infty }}{U}_{0m}\sin \left({\lambda }_{m}x\right)\text{sin}\left({\omega }_{m}t+{\phi }_{0m}\right),

(37)w^{\text{b}}=\sum _{m=1}^{\mathrm{\infty }}{W}_m^{\text{b}}\sin \left({\lambda }_{m}x\right)\text{sin}\left({\omega }_{m}t+{\phi }_{0m}\right),

(38)w^{\text{s}}=\sum _{m=1}^{\mathrm{\infty }}{W}_m^{\text{s}}\sin \left({\lambda }_{m}x\right)\text{sin}\left({\omega }_{m}t+{\phi }_{0m}\right),

where, {\lambda }_m=m\text{π}/a, m is the mode number; \overline{\mathit{d}}={\left[\begin{array}{cc}{U}_{0m}& \begin{array}{cc}{W}_m^{\text{b}}& W_m^{\text{s}}\end{array}\end{array}\right]}^{\text{T}} represents the amplitudes; {\omega }_m is the natural frequency; {\phi }_{0m} is the initial angle of the curved nanobeam. Substituting Eqs. (36)–(38) into Eqs. (29)–(31), respectively, yields

(39)\left(\mathit{K}-{\omega }_m^2\mathit{M}\right)\overline{\mathit{d}}=0,

where K and M represent the stiffness and mass matrices, respectively. These matrices are defined in Electronic Supplementary Material. Using the system of algebraic eigenvalues expressed in Eq. (39), the natural frequency and mode shapes of the FGP curved nanobeams with simply supported boundary conditions are obtained.

2.6 Enriched beam element

A conventional two-node beam element with a length of l_e can be obtained via the energy equations in the preceding section by interpolating the axial and transverse displacements using the Lagrange and Hermite interpolation functions [63], respectively.

(40)u_0={\mathit{N}}^{\text{T}}{\mathit{d}}_{\text{eu}},w^{\text{b}}={\mathit{H}}^{\text{T}}{\mathit{d}}_{\text{eb}},w^{\text{s}}={\mathit{H}}^{\text{T}}{\mathit{d}}_{\text{es}},

where the Lagrange interpolation function \mathit{N}={\left[\begin{array}{cc}{N}_0& N_1\end{array}\right]}^{\text{T}} and Hermite interpolation function \mathit{H}={\left[\begin{array}{cccc}{H}_0& H_1& H_2& H_3\end{array}\right]}^{\text{T}} are detailed in Electronic Supplementary Material. Additionally, the nodal displacement vectors are defined as follows:

(41)\begin{array}{rl}& {\mathit{d}}_{\text{eu}}={\left[\begin{array}{cc}{u}_{01}& u_{02}\end{array}\right]}^{\text{T}},{\mathit{d}}_{\text{eb}}={\left[\begin{array}{cccc}{w}_1^{\text{b}}& {\theta }_1^{\text{b}}& w_2^{\text{b}}& {\theta }_2^{\text{b}}\end{array}\right]}^{\text{T}},\\ & {\mathit{d}}_{\text{es}}={\left[\begin{array}{cccc}{w}_1^{\text{s}}& {\theta }_1^{\text{s}}& w_2^{\text{s}}& {\theta }_2^{\text{s}}\end{array}\right]}^{\text{T}},\end{array}

where {\theta }^{\alpha }={\displaystyle \frac{\mathrm{\partial }{w}^{\alpha }}{\mathrm{\partial }x}}\left(\alpha =\mathrm{b},\mathrm{s}\right).

To increase the convergence rate of the solution method, the preceding interpolation functions are enhanced by adding higher-order functions to the original Lagrange and Hermite interpolation functions. In this study, four higher-order hierarchical functions were considered. In this instance, the interpolation from Eq. (40) is superimposed using the following equation:

(42)\begin{array}{rl}& u_0={\mathit{N}}^{\text{T}}{\mathit{d}}_{\text{eu}}+\stackrel{⌣}{{\mathit{N}}^{\text{T}}}{\stackrel{⌣}{\mathit{d}}}_{\text{eu}},w^{\text{b}}={\mathit{H}}^{\text{T}}{\mathit{d}}_{\text{eb}}++\stackrel{⌣}{{\mathit{N}}^{\text{T}}}{\stackrel{⌣}{\mathit{d}}}_{\text{eb}},\\ & w^{\text{s}}={\mathit{H}}^{\text{T}}{\mathit{d}}_{\text{es}}++\stackrel{⌣}{{\mathit{N}}^{\text{T}}}{\stackrel{⌣}{\mathit{d}}}_{\text{es}},\end{array}

where \stackrel{⌣}{\mathit{N}}={\left[\begin{array}{cccc}{N}_2& N_3& N_4& N_5\end{array}\right]}^{\text{T}} and \stackrel{⌣}{\mathit{H}}={\left[\begin{array}{cccc}{H}_4& H_5& H_6& H_7\end{array}\right]}^{\text{T}} are the added matrices of the fifth- and seventh-order interpolation functions, respectively. Meanwhile, {\stackrel{⌣}{\mathit{d}}}_{\text{eu}},{\stackrel{⌣}{\mathit{d}}}_{\text{eb}}, and {\stackrel{⌣}{\mathit{d}}}_{\text{es}} are the added vectors of unknowns in the following form:

(43)\begin{array}{rl}& {\stackrel{⌣}{\mathit{d}}}_{\text{eu}}={\left[a_{\text{u1}}{a}_{\text{u2}}{a}_{\text{u3}}{a}_{\text{u4}}\right]}^{\text{T}},{\stackrel{⌣}{\mathit{d}}}_{\text{eb}}={\left[a_{\text{b1}}{a}_{\text{b2}}{a}_{\text{b3}}{a}_{\text{b4}}\right]}^{\text{T}},\\ & {\stackrel{⌣}{\mathit{d}}}_{\text{es}}={\left[a_{\text{s1}}{a}_{\text{s2}}{a}_{\text{s3}}{a}_{\text{s4}}\right]}^{\text{T}},\end{array}

where a_{\text{u}\alpha }, a_{\text{b}\alpha }, and a_{\text{s}\alpha } (\alpha =1,2,\dots ,4) are the additional degrees of freedom.

The hierarchical functions N_m\left(m=2,3,...,5\right) and H_n\left(n=4,5,...,7\right) in Eq. (42) increases the degrees of freedom of the element without affecting those at the element nodes [64]. N_{m+1} and H_{n+1} are defined as follows [65]:

(44)N_m=\sqrt{\frac{2m-1}{2}}{\int }_{-1}^{\xi }{L}_{m-1}\left(\xi \right)\text{d}\xi ,m⩾2,

(45)H_{n+1}={\int }_{-1}^{\xi }{N}_n\left(\xi \right)\text{d}\xi ,

where \xi ={\displaystyle \frac{2x}{l_{\text{e}}}}-1 is the natural coordinate and L_m the Legendre polynomial of order m. The explicit forms of interpolation functions N_i and H_{i+2} (i=2,3,...,5) are provided in Electronic Supplementary Material.

The degree-of-freedom vector of the elements is defined as {\mathit{d}}_{\text{e}}={\left[{\mathit{d}}_{\text{eu}}{\mathit{d}}_{\text{eb}}{\mathit{d}}_{\text{es}}{\stackrel{⌣}{\mathit{d}}}_{\text{eu}}{\stackrel{⌣}{\mathit{d}}}_{\text{eb}}{\stackrel{⌣}{\mathit{d}}}_{\text{es}}\right]}^{\text{T}}. Accordingly, Eq. (42) can be rewritten as

(46)u_0={\left({\mathit{N}}^u\right)}^{\text{T}}{\mathit{d}}_{\text{e}},w^{\text{b}}={\left({\mathit{Q}}^{\text{b}}\right)}^{\text{T}}{\mathit{d}}_{\text{e}},w^{\text{s}}={\left({\mathit{Q}}^{\text{s}}\right)}^{\text{T}}{\mathit{d}}_{\text{e}}.

Substituting Eq. (46) into Eq. (34) and then performing the assembling procedure yields the following equation for a curved nanobeam:

(47)\mathit{M}\ddot{\hat{\mathit{d}}}+\hat{\mathit{K}}\hat{\mathit{d}}=0,

where \hat{\mathit{K}}={{\displaystyle \sum }}_{e=1}^{Ne}{\mathit{K}}_e is the global stiffness matrix, \hat{\mathit{M}}={{\displaystyle \sum }}_{e=1}^{Ne}{\mathit{M}}_e is the global mass matrix, \hat{\mathit{d}} is the global degree-of-freedom vector, and N_e is the total number of elements. The details of all the matrices are provided in Electronic Supplementary Material. Considering the harmonic nature of the free vibration of the structure under investigation, we can substitute \hat{\mathit{d}}={\mathit{D}}_0\text{sin}\left(\omega t+{\phi }_0\right) into Eq. (47). The obtained results can be expressed in the following algebraic eigenvalue form:

(48)\left(\hat{\mathit{K}}-{\omega }^2\hat{\mathit{M}}\right){\mathit{D}}_0=0,

where \omega is the natural frequency of the FGP curved nanobeam and {\phi }_0 is the initial phase angle. By solving Eq. (48), the natural frequency and mode shapes of the FGP curved nanobeams under different boundary conditions can be obtained.

2.7 Boundary conditions of curved beams when applying the enriched FEM

For the finite element analysis, the boundary conditions of the beams were considered based on the constraints of the geometrical factors. Four types of various boundary conditions (BCs) were analyzed in the numerical examples, i.e., simply-supported (SS), clamped-clamped (CC), clamped-supported (CS), and clamped-free (CF) boundaries.

- SS:

- CC:

- CS:

u_0=w^{\text{b}}=w^{\text{s}}=\frac{\mathrm{\partial }{w}^{\text{b}}}{\mathrm{\partial }x}=\frac{\mathrm{\partial }{w}^{\text{s}}}{\mathrm{\partial }x}=0\mathrm{a}\mathrm{t}x=0;w^{\text{b}}=w^{\text{s}}=0\mathrm{a}\mathrm{t}x=a.

- CF:

u_0=w^{\text{b}}=w^{\text{s}}=\frac{\mathrm{\partial }{w}^{\text{b}}}{\mathrm{\partial }x}=\frac{\mathrm{\partial }{w}^{\text{s}}}{\mathrm{\partial }x}=0\mathrm{a}\mathrm{t}x=0.

When using the finite element method or other numerical methods to investigate curved beams, the simply supported boundary condition restricts the displacement in the normal direction and does not restrict the displacement in the tangential direction with the radius of the curved beam. Using the enriched FEM, the degrees of freedom added to the beam element does not affect the elimination of the boundary conditions.

3 Numerical results and discussion

In this section, we present verification examples for validating the efficiency of the proposed enriched finite element formulation for the free vibration analysis of FGP curved nanobeams on an elastic foundation in a hygro–thermo–magnetic environment. Subsequently, the effects of the magnetic field, nonlocal coefficient, porosity coefficient, temperature change, moisture change, opening angle, elastic foundation stiffness, and other parameters on the free vibration behavior of FGP curved nanobeams are investigated. Navier’s solution is only applied to the analysis of FGP curved nanobeam structures with SS boundary conditions, whereas the finite element method can be employed for all types of boundary conditions. The mechanical characteristics of the \text{Al}/\text{A}{\text{l}}_2{\text{O}}_3 porous curved nanobeams investigated are presented in Tab.1.

The normalized parameters are expressed as follows:

(49)\begin{array}{rl}& {\mathrm{\Omega }}_m={\omega }_m{a}^2\sqrt{\frac{{\rho }_{\text{c}}A}{E_{\text{c}}I}},K_{\text{w}}=\frac{k_{\text{w}}{a}^4}{D_{\text{c}}},\\ & K_{\text{s}}=\frac{k_{\text{s}}{a}^2}{D_{\text{c}}},H_x^{\ast }=\frac{\eta h{H}_x{a}^2}{D_{\text{c}}},\end{array}

where

(50)D_{\text{c}}=\frac{E_{\text{c}}{h}^3}{12\left(1-{\nu }_{\text{c}}^2\right)},A=bh,I=\frac{b{h}^3}{12}.

Unless otherwise specified, the initial geometrical dimensions and mechanical parameters of the FGP curved nanobeam are as follows.

For the data in the tables, a=10\text{nm},b=1\text{nm},\mathrm{\Delta }T=100\text{K},\mathrm{\Delta }C=2\mathrm{\%},p_{\mathit{\text{z}}}=1,K_{\text{w}}=50,K_{\text{s}}=5, \chi = 0.2.

For the data in the figures, a/h=5,\mu =2,p_{\mathit{\text{z}}}=0.5, \chi =0.1,K_{\text{w}}=10,K_{\text{s}}=10,\mathrm{\Delta }T=50\text{K},\mathrm{\Delta }C=1\mathrm{\%}, and H_x^{\text{*}}=2.

3.1 Accuracy and convergence studies

To validate the present theoretical formulation, the convergence and accuracy of the first and second natural frequencies of the fully simply supported FG curved nanobeams were determined, and the results are listed in Tab.2. Based on this table, the results obtained using Navier’s solution, the enriched FEM, and the conventional FEM [39] can be compared. The results yielded by the enriched FEM were consistent with those yielded by Navier’s solution and the conventional FEM [39]. In a previous study [39], the results were extracted using a three-node beam element, and a curved beam was discretized using 32 elements. The results shown in the table indicates that using only one enriched element yielded accurate results, which is expected to incur lower computational costs. Tab.3 and Tab.4 list the first and second natural frequencies of the curved nanobeams with CF and CC boundary conditions, respectively. For the case involving the CF boundary condition, the first natural frequency of the proposed method converged with two elements. For the case involving the CC boundary condition, six beam elements were required. In addition, the conventional FEM [39] required at least 32 three-node beam elements to yield converged results. Hence, we confirm that the proposed enriched finite element method guarantees accuracy and reliability.

3.2 Parametric studies

The effects of geometric and material parameters, including the magnetic field H_x^{\ast }, temperature change \mathrm{\Delta }T\left(\text{K}\right), moisture change \mathrm{\Delta }C\left(\mathrm{\%}{\text{H}}_2\text{O}\right), power-law index p_{\mathit{\text{z}}}, nonlocal coefficient \mu \left(\text{nm}\right), porosity coefficient \chi, elastic foundation stiffness coefficients (K_{\text{w}}\text{and}{K}_{\text{s}}), thickness ratio a/h, opening angle \phi, and boundary conditions, on the free vibration response of the FGP curved nanobeam are analyzed and discussed in this subsection. Tab.5–Tab.8 and the Fig.2–Fig.8 present the first and second natural frequencies of the FGP curved nanobeam on an elastic foundation in a hygro–thermo–magnetic environment, respectively. The input parameters included the opening angle \left(\phi ={30}^{\text{o}}\text{and}{120}^{\text{o}}\right), beam thickness ratio (a/h= 5, 10, 20), nonlocal coefficient (\mu =0,1,2\text{nm}), magnetic field (H_x^{\text{*}}=0,1,5), and boundary conditions (CC, SS, CS, and CF). Based on these data, the first nondimensional natural frequency {\mathrm{\Omega }}_1 when \phi =30\text{°} is less than that when \phi =120\text{°} for the CC boundary condition. Furthermore, based on the tables and figures, the nondimensional natural frequency for the case involving the CLT is the highest. However, when the thickness of the curved beam decreases, the effect of the shear strain {\gamma }_{x\mathit{\text{z}}} diminishes, thus resulting in the same results for all three beam theories.

However, in the cases involving the CS, SS, and CF boundary conditions, the opposite results were obtained, i.e., the second natural frequency {\mathrm{\Omega }}_2 decreased as the opening angle \phi increased from 30\text{°} to 120\text{°}. For the FGP curved nanobeam with thickness ratios a/h=10,20 and the CS nanobeam with thickness ratio a/h=5 under the CC condition, the second natural frequency {\mathrm{\Omega }}_2 increased with the opening angle \phi. When the nonlocal coefficient \mu increased, the stiffness of the structure decreased. Meanwhile, when the nonlocal coefficient μ increased, the stiffness of the structure decreased, which decreased the nondimensional natural frequency. In addition, an increase in the magnetic field H_x^{\text{*}} resulted in an increase in the nondimensional natural frequency.

Fig.2 shows the effect of the magnetic field H_x^{\text{*}} on the nondimensional natural frequency of the FGP curved nanobeam for various boundary conditions. An increase in the magnetic field H_x^{\text{*}} resulted in an increase in the first nondimensional natural frequency {\mathrm{\Omega }}_1, where the increase was almost linear. Among the four boundary conditions considered in this subsection, the CC boundary condition yielded the most variable results. When the opening angle \phi increased, frequency {\mathrm{\Omega }}_1 increased, whereas the other boundaries yielded the opposite results. In addition, this phenomenon occurred only in the case of the FGP curved nanobeam without an elastic foundation, i.e., not in the cases of the FGP curved nanobeam on an elastic foundation. Fig.3 and Fig.4 show the free vibration responses of the FGP curved nanobeam for various boundary conditions as functions of temperature \mathrm{\Delta }T\left(\text{K}\right) and moisture \mathrm{\Delta }C\left(\mathrm{\%}{\text{H}}_{\text{2}}\text{O}\right). As the temperature or moisture increased, the stiffness of the beam decreased, which consequently decreased the nondimensional natural frequency. In contrast to the increase in temperature or moisture, which reduced the nondimensional natural frequency, an increase in the porosity coefficient resulted in an increase in the nondimensional natural frequency of the FGP curved nanobeam, as shown in Fig.5. Additionally, the increase in the nondimensional natural frequency {\mathrm{\Omega }}_1 presented a nonlinear curve. This is reasonable because the increase in the porosity coefficient \chi decreases the stiffness of the nanobeam. Fig.6 shows the relationship between the nondimensional natural frequency {\mathrm{\Omega }}_1 and the change in the power-law index p_{\mathit{\text{z}}}. For all the cases of open angles, when the values of the power-law index varied in the range of p_{\mathit{\text{z}}}=0, the nondimensional natural frequency {\mathrm{\Omega }}_1 of the FGP nanobeam with the SS and CF boundary conditions was the lowest. The effect of the nonlocal coefficient \mu on the natural frequency was independent of the boundary conditions and geometrical structure. Meanwhile, changes in the temperature \mathrm{\Delta }T\left(\text{K}\right) and moisture \mathrm{\Delta }C\left(\text{\%}{\text{H}}_{\text{2}}\text{O}\right) were affected significantly by the boundary conditions and geometrical structure.

The elastic foundation stiffness is a parameter that significantly affects the nondimensional natural frequency of the FGP curved nanobeam. As shown in Fig.7, an increase in the elastic foundation stiffness resulted in an increase in the total stiffness of the structure, which caused the nondimensional natural frequency {\mathrm{\Omega }}_1 to increase rapidly. In particular, an increase in the stiffness coefficient k_{\text{w}} resulted in a linear increase in the nondimensional natural frequency {\mathrm{\Omega }}_1, whereas an increase in the slide stiffness coefficient k_{\text{s}} resulted in a nonlinear increase in {\mathrm{\Omega }}_1.

Finally, Fig.8 shows the first two free vibration mode shapes of the CC FGP curved nanobeam for various opening angles \phi. When the open angle \phi or curvature of the beam increased, the displacement u_0 increased as well.

4 Conclusions

An enriched finite element method was proposed herein for the free vibration analysis of an FGP curved nanobeam on an elastic medium in a hygro–thermo–magnetic environment. The proposed elements were based on different beam theories combined with Navier’s analysis solution, nonlocal elasticity theory, and enriched technology using conventional Lagrange and Hermite interpolations with hierarchical functions. The accuracy of the proposed element was confirmed by comparing the obtained results with results presented in the literature. The following conclusions were inferred from the numerical results.

1) To obtain the first nondimensional natural frequency of the FGP curved nanobeam, only one curved beam element was required for the SS boundary condition; however, two elements were required for the CF boundary condition, and six elements for the CC and CS boundary conditions.

2) For the CC boundary condition, opening angle \phi =90\text{°} yielded the highest natural frequency value.

3) Based on the different boundary conditions considered, an increase in the nonlocal coefficient \mu, temperature, and moisture reduced the natural frequency.

4) For an uneven distribution, an increase in the porosity coefficient resulted in an increase in the nondimensional natural frequency. Furthermore, the elastic foundation stiffness increased the total stiffness of the structures. Therefore, an increase in the coefficient of the elastic foundation stiffness increased the natural frequency.

5) For the free vibration analysis of an FGP curved nanobeam on an elastic foundation, the enriched beam element significantly improved the convergence and accuracy of the numerical results.

6) The effects of nonlocalness, porosity, curvature, and elastic foundation stiffness on the free vibration responses of an FGP curved nanobeam on an elastic foundation can be evaluated simultaneously.

7) The results of this study can be used as a benchmark for analyzing curved FG nanobeams. In addition, the proposed method can be applied to analyze nanostructures with mechanical properties that vary in many directions or problems involving dynamic stability and linear/nonlinear dynamics.

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