1 Introduction

Recently, nanotechnology is concerned with fabrication of functionally graded materials (FGMs) and engineering structures at a nanoscale, which enables new generation of materials and devices with innovative properties [21]. FGMs are suitable for large structures (i.e., aircraft, space vehicles, automotive industries, optics, barrier coating, nuclear reactors and propulsion systems) and nanostructures (i.e., nanoelectromechanical systems, thin films, shape memory alloys and atomic force microscopes). A FGM is depicted by a continuous graduation of material composition in one or more dimensions from one material to another that provides an elegant solution to the problem of high transverse shear stresses, interface cracking, delamination and residual stresses. Power-law distribution (P-FGM) [3, 21–23, 39, 53, 54, 60] and exponential distribution (E-FGM) [4, 33, 43, 44] are frequently used to depict the variations of material properties distribution of FGMs.

It is observed that the stress concentrations appear in the interface layer when the material is continuously changing rapidly for both power and exponential functions distribution [9]. Therefore, Chi and Chung [15] suggested a sigmoid function, a combination of two types of P-FGM functions, to reduce the stress intensity factors in cracked structure. In addition, structures, which have ceramic constituent at top and bottom surfaces and have metallic core, can sustain a higher temperature more than power and exponential FGM.

By using powder metallurgy and thermal spraying techniques, Kapuria et al. [38] fabricated multilayered FG beam and then experimentally validated the results on static and free vibration. For sigmoid function, Ben-Oumrane et al. [9] analyzed theoretically a flexional bending of Al/Al2O3 S-FGM thick beams according to different beam theories. Mahi et al. [45] presented exact solutions to study the free vibration of a unified higher-order shear deformation theory. Material properties are proposed to be temperature dependent and vary continuously through the thickness according to E-FGM and S-FGM. Fereidoon and Mohyeddin [32] exploited differential quadrature method to analyze bending of thin functionally graded plates. Duc and Cong [17] investigated the nonlinear dynamic response of imperfect symmetric thin sigmoid functionally graded material (S-FGM) subjected to mechanical loads. Lee and Kim [41] studied thermal post-buckling and snap-through instabilities of P-FGM, E-FGM and S-FGM panels in hypersonic flows. Jung and Han [37] illustrated bending behavior of nonlocal S-FGM nanoplates with first-order shear deformation.

Tiny beam is the basic structure used in several applications such as nanoelectromechanical systems (NEMS), nanowires, nanoprobes, atomic force microscope (AFM), nanoactuators and nanosensors. For convincing designing of nanostructure, the size and length-scale effects and the atomic forces should be included in the mathematical formulation. The nonlocal elasticity theory developed by Eringen [29–31] is a promising theory which contains information about the forces between atoms and the internal length scale. The use of nonlocal continuum mechanics has found successful applications in several areas which include fracture mechanics, lattice dispersion of elastic waves, mechanics of dislocations and wave propagation in mechanics [52]. Reddy [51] developed analytical solutions for bending, buckling and vibration of isotropic beams using Euler–Bernoulli, Timoshenko, Reddy and Levinson beam theories.

A lot of researchers have motivated on FG of nanobeams using nonlocal Eringen’s elasticity in differential model. According to power-law distribution of functionally graded nanobeams, Eltaher et al. [21, 22] presented a finite element model to study a static, buckling and dynamic behavior of Euler nanobeams. Eltaher et al. [23] illustrated the effect of neutral plane on natural frequencies and noted that the calculated frequency at mid-plane is overestimated than that at neutral axis. Tounsi et al. [59], Benguediab et al. [7] and Besseghier et al. [10] studied nonlocal chirality and thermal effects on buckling properties of double-walled carbon nanotubes (DWCNTs). Simsek and Yurtcu [55] examined analytically static bending and buckling of Timoshenko and Euler–Bernoulli nanobeams. Eltaher et al. [25, 26] adopted previous model to consider the shear effect by using Timoshenko nanobeams. Reddy et al. [53] developed a nonlinear finite element models for a static bending of FG nanobeams with moderate displacements and rotations. Uymaz [60], Rahmani and Pedram [50] used Navier’s solution to study the free and forced vibration problem of a FG nanobeam. Kiani [39] proposed a mathematical model to explore vibrations and instabilities of moving FG nanobeams. The longitudinal and lateral equations of motion of the moving nanostructure were extracted by employing the nonlocal Rayleigh beam model.

Chaht et al. [14] presented bending and buckling behavior of FGM size-dependent nanobeams including the thickness stretching effect. Ebrahimi and Salari [18] used both Navier method and a semi-analytical differential transform method to investigate a free flexural vibrational of FG Euler nanobeams. Ebrahimi and Boreiry [19] investigated surface effects on nonlocal vibrational behavior of nanobeams. Salehipour et al. [54] presented a modified nonlocal elasticity theory for functionally graded materials by using an imaginary nonlocal strain tensor to directly obtain the nonlocal stress tensor. Rahmani and Jandaghian [49] developed an analytical solution of a buckling of third-order nanofunctionally graded beam by using Rayleigh–Ritz technique. Filiz and Aydogdu [33] studied a wave propagation in embedded functionally graded nanotubes conveying fluid. The material properties are changing exponentially in the thickness direction. Eltaher et al. [27] presented a review on nonlocal elastic models for bending, buckling, vibrations and wave propagation of nanoscale beams. Agwa and Eltaher [1] presented a vibration behavior of carbyne nanomechanical mass sensors with surface effect. Ebrahimi and Barati [20] investigated vibration behavior of magneto-electro-thermo-elastic functionally graded nanobeams based on a higher-order shear deformation beam theory. Hosseini and Rahmani [36] presented free vibration analysis of shallow and deep curved functionally graded (FG) nonlocal nanobeam. Sourki and Hoseini [56] investigated a vibration of a cracked microbeam based on the modified couple stress theory within the framework of Euler–Bernoulli beam theory. Eltaher et al. [28] presented nonlinear analysis of size-dependent and material-dependent nonlocal carbon nanotubes.

To satisfy the zero traction boundary conditions on the surfaces, Eltaher et al. [12] and Zidi et al. [62] studied thermo- and hygro-thermo-mechanical bending response of FG plates using refined shear deformation theory. Meziane et al. [47], Hebali et al. [35], Belabed et al. [5], Mahi and Tounsi [46] and Bennoun et al. [8] developed a simple and accurate shear deformation theory for bending and free vibration of FG plates without requiring any shear correction factor. Bourada et al. [13] presented a simple shear and normal deformations theory for FG beams. Hamidi et al. [34] presented an accurate sinusoidal plate theory for the thermomechanical bending analysis of FG sandwich plates. Yahia et al. [61] studied wave propagation in FG plates with porosities using various higher-order shear deformation plate theories. Bellifa et al. [6] and Ahouel et al. [2] investigated size-dependent mechanical behavior of FG trigonometric shear deformable nanobeams including neutral surface position concept. Bounouara et al. [11] proposed zeroth-order shear deformation theory to study vibration of FG plate with parabolic variation within the plate thickness and vanish on the plate surfaces.

From the literature review and to the best of the authors’ knowledge, it can be concluded that no researchers have attempted to use a sigmoidal function and symmetric power distributions for FG nanobeams. In fact, all of the related studies use the power and exponential distribution for FG nanobeams. The present study is intended to fill this gap in the literature by considering a sigmoidal distribution through thickness of nanobeams. So, this paper presented a free vibration of a new FG nonlocal nanobeam. The manuscript is organized as follows. Section 2 describes the mathematical formulation and governing equations of sigmoidal FG nanobeams with Eringen’s nonlocal elasticity and Euler–Bernoulli kinematics assumptions. Section 3 summarizes the displacement finite element model to study a buckling stability and free vibration of nanobeam. In Sect. 4, a code validation and numerical results are discussed. Section 5 summarizes concluding remarks.

2 Problem formulation

2.1 Spatial material graduation functions

Functionally graded materials (FGMs) are produced by combining various materials continuously though a specific spatial direction. The simplest and accepted homogenization methods to estimate the effective properties at micromechanics level are Voigt rule [48] and Mori–Tanaka mode [58]. The mechanical properties are graded across the thickness according to the Voigt model [42, 40].

The volume fraction of materials can be expressed as:

$$ V_{\text{c}} = \left( {\frac{1}{2} + \frac{z}{h}} \right)^{k} \quad \left( {0 \le k < \infty } \right) $$
(1a)
$$ V_{\text{c}} + V_{\text{m}} = 1 $$
(1b)

where V is the volume fraction, k is the nonnegative power exponent, and subscripts c and m represent the ceramic and metal, respectively. In the current analysis, symmetric power function (SP-FGM) and sigmoid function S-FGM are proposed.

The symmetric power function can be depicted by:

$$ P_{1} \left( z \right) = \left( {P_{\text{surf}} - P_{\text{core}} } \right)\left( {\frac{ - 2z}{h}} \right)^{k} + \,P_{\text{core}} \quad \left( {\frac{ - h}{2} \le z \le 0} \right) $$
(2a)
$$ P_{2} \left( z \right) = \left( {P_{\text{surf}} - P_{\text{core}} } \right)\left( {\frac{2z}{h}} \right)^{k} +\, P_{\text{core}} \quad \left( {0 \le z \le \frac{h}{2}} \right) $$
(2b)

However, the sigmoid functional distribution can be described by:

$$ P_{1} \left( z \right) = P_{\text{m}} V_{\text{m}} + P_{\text{C}} V_{\text{c}} = P_{\text{m}} + \frac{1}{2}\left( {P_{\text{c}} - P_{\text{m}} } \right)\left( {1 + \frac{2z}{h}} \right)^{k} \quad \left( {\frac{ - h}{2} \le z \le 0} \right) $$
(3a)
$$ P_{2} \left( z \right) = P_{\text{m}} V_{\text{m}} + P_{\text{C}} V_{\text{c}} = P_{\text{c}} - \frac{1}{2}\left( {P_{\text{c}} - P_{\text{m}} } \right)\left( {1 - \frac{2z}{h}} \right)^{k} \quad \left( {0 \le z \le \frac{h}{2}} \right) $$
(3b)

where P is material properties [Young’s modulus (E), density (ρ) or Poisson’s ratio (υ)] and subscripts surf, core, m and c are surface, core, metal and ceramics, respectively. The FG beam in the current manuscript is composed of aluminum metal [E m = 70 GPa, ρ m = 2.7 g/cm3 and υ m is 0.3] and ceramics of alumina [E c = 380 GPa, ρ c = 3.96 g/cm3 and υ c = 0.3]. Delale and Erdogan [16] proved that the effect of Poisson’s ratio on the deformation is much less than Young’s modulus. So, the Poisson’s ratio is assumed to be constant in this analysis. The distribution of Young’s modulus and mass density through the beam thickness for ceramic–metal–ceramic (CMC), metal–ceramic–metal (MCM) and sigmoidal distribution is presented in Figs. 1, 2 and 3, respectively.

Fig. 1
figure 1

Variation of Young’s modulus and mass density through the beam thickness according to SP-FGM (CMC)

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Fig. 2
figure 2

Variation of Young’s modulus and mass density through the beam thickness according to SP-FGM (MCM)

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Fig. 3
figure 3

Variation of Young’s modulus and mass density through the beam thickness according to sigmoidal function

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2.2 Geometrical fit conditions

Based on the Euler–Bernoulli theory, plane sections perpendicular to the axis of the beam before deformation remain plane, rigid and rotate such that they remain perpendicular to the (deformed) axis after deformation. The assumptions amount to neglecting the Poisson effect and transverse strains, Reddy (2014). The displacement field can be assumed as:

$$ u\left( {x,z} \right) = u_{0} \left( x \right) - z\frac{{{\text{d}}w_{0} \left( x \right)}}{{{\text{d}}x}} $$
(4a)
$$ w\left( {x,z} \right) = w_{0} \left( x \right) $$
(4b)

where u and w are the total displacements along the coordinate (x), and u 0 and w 0 denote the axial and transverse displacements of a point on the neutral axis. According to Euler hypothesis, the only nonzero strain is

$$ \varepsilon_{xx} \left( {x,z} \right) = \frac{\text{d}}{{{\text{d}}x}}\left[ {u_{0} \left( x \right) - z\frac{{{\text{d}}w_{0} \left( x \right)}}{{{\text{d}}x}}} \right] = \frac{{{\text{d}}u_{0} \left( x \right)}}{{{\text{d}}x}} - z\frac{{{\text{d}}^{2} w_{0} \left( x \right)}}{{{\text{d}}x^{2} }} = \varepsilon_{xx}^{0} + z\varepsilon_{xx}^{1} $$
(5)

and nonzero classical stress can be presented by:

$$ \sigma_{xx} \left( {x,z} \right) = E\left( z \right)\varepsilon_{xx} \left( {x,z} \right) = E\left( z \right)\left[ {\varepsilon_{xx}^{0} + z\varepsilon_{xx}^{1} } \right] $$
(6)

Axial and bending moment can be written as:

$$ N_{xx} = \mathop \int \limits_{A} \sigma_{xx} dA = A_{11} \varepsilon_{xx}^{0} + B_{11} \varepsilon_{xx}^{1} $$
(7a)
$$ M_{xx} = \mathop \int \limits_{A} z\sigma_{xx} dA = B_{11} \varepsilon_{xx}^{0} + D_{11} \varepsilon_{xx}^{1} $$
(7b)

where

$$ \left[ {A_{11} , B_{11} , D_{11} } \right] = b\mathop \int \limits_{h} E\left( z \right) \left[ {1, z, z^{2} } \right]{\text{d}}z = b \left[ {\mathop \int \limits_{{ - \frac{h}{2}}}^{0} E_{1} \left( z \right) \left[ {1, z, z^{2} } \right]{\text{d}}z + \mathop \int \limits_{0}^{{\frac{h}{2}}} E_{2} \left( z \right) \left[ {1, z, z^{2} } \right]{\text{d}}z} \right] $$
(8)

2.3 Nonlocal stress–strain relations

Based on the nonlocal differential Eringen elasticity theory, the nonlocal constitutive relation can be described by [31]:

$$ \left( {1 - \left( {e_{0} a} \right)^{2 } \nabla^{2} } \right)\sigma_{ij} = t_{ij} $$
(9)

where ∇2 is the Laplacian operator, t ij is the classical macroscopic stress tensor, e 0 is a material constant, and a is the internal characteristic length. For Euler–Bernoulli nonlocal FG beam, Eq. (9) can be simplified as

$$ \sigma_{xx} - \mu \frac{{\partial^{2} \sigma_{xx} }}{{\partial x^{2} }} = E\left( z \right)\varepsilon_{xx} ,\quad \left( {\mu = e_{0}^{2} a^{2} } \right) $$
(10)

and nonlocal axial and bending moment can be described by:

$$ N - \mu \frac{{\partial^{2} N}}{{\partial x^{2} }} = A_{11} \varepsilon_{xx}^{0} + B_{11} \varepsilon_{xx}^{1} $$
(11a)
$$ M - \mu \frac{{\partial^{2} M}}{{\partial x^{2} }} = B_{11} \varepsilon_{xx}^{0} + D_{11} \varepsilon_{xx}^{1} $$
(11b)

2.4 Nonlocal equations of motion

According to Hamilton’s principle, the equation of motion of functionally graded beam can be derived to the following

$$ A_{11} \frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + B_{11} \frac{{\partial^{3} w_{o} }}{{\partial x^{3} }} + \left( {1 - \mu \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)f = I_{0} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} - I_{1} \frac{{\partial^{3} w_{0} }}{{\partial t^{2} \partial x}} - \mu \left[ { I_{0} \frac{{\partial^{4} u_{0} }}{{\partial t^{2} \partial x^{2} }} - I_{1} \frac{{\partial^{5} w_{0} }}{{\partial t^{2} \partial x^{3} }}} \right] $$
(12a)
$$ B_{11} \left( {\frac{{{\text{d}}^{3} u_{0} }}{{{\text{d}}x^{3} }}} \right) + D_{11} \frac{{{\text{d}}^{4} w_{0} }}{{{\text{d}}x^{4} }} + \left( {1 - \mu \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)q + \left( {1 - \mu \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)\left( {\bar{N}\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right) = \left( {1 - \mu \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)\left[ {I_{0} \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{3} u_{0} }}{{\partial t^{2} \partial x}} - I_{2} \frac{{\partial^{4} w_{0} }}{{\partial t^{2} \partial x^{2} }}} \right] $$
(12b)

where f is the axial distributed force in x-direction, q is the transverse distributed force in z-direction, and \( \bar{N} \) is the axial compressive load normal to the cross section and applied at the neutral axis. Inertia terms I 0, I 1 and I 2 can be described by:

$$ \left[ {I_{0} , I_{1} , I_{2} } \right] = b\mathop \int \limits_{h} \rho \left( z \right) \left[ {1, z, z^{2} } \right]{\text{d}}z = b \left[ {\mathop \int \limits_{{ - \frac{h}{2}}}^{0} \rho_{1} \left( z \right) \left[ {1, z, z^{2} } \right]{\text{d}}z + \mathop \int \limits_{0}^{{\frac{h}{2}}} \rho_{2} \left( z \right) \left[ {1, z, z^{2} } \right]{\text{d}}z} \right] $$
(13)

3 Numerical formulation

The displacement components at the mid-plane (that is coincident with neutral plane in the current material distributions) of a beam element can be described as [24]:

  • In-plane displacement u 0

    $$ u_{0}^{\left( e \right)} \left( {x, t} \right) = \sum\limits_{i = 1}^{2} {N_{i} U_{i} \left( t \right) = N_{1} U_{1} \left( t \right) + N_{2} U_{2} \left( t \right)} \quad {\text{where}}\;i\,{ = 1,2 } $$
    (14a)

  • Transverse displacement w o

    $$ w_{0}^{\left( e \right)} \left( {x,t} \right) = \mathop \sum \limits_{k = 1}^{4} \tilde{N}_{k} \tilde{W}_{k} = \tilde{N}_{1} W_{1} + \tilde{N}_{2} \theta_{1} + \tilde{N}_{3} W_{2} + \tilde{N}_{4} \theta_{2} $$
    (14b)

where U, W and θ are the nodal displacements and slope, respectively. N i is the Lagrangian interpolation function for in-plane displacement, and \( \tilde{N}_{\text{k}} \) is the Hermitian interpolation shape function for transverse displacements. The variational statement of nonlocal Euler–Bernoulli beam has the following form [57]:

$$ \begin{aligned} & \int_{0}^{T} {\int_{0}^{L} {\left\{ {\left( {\left[ { - \int_{{ - \frac{h}{2}}}^{0} {E_{1} (z){\text{d}}z - } \int_{0}^{{\frac{h}{2}}} {E_{2} (z ) {\text{d}}z} } \right]\frac{{\partial u_{0} }}{\partial x}\frac{{\partial \delta u_{0} }}{\partial x} + \left[ {\int_{{ - \frac{h}{2}}}^{0} {zE_{1} (z){\text{d}}z - } \int_{0}^{{\frac{h}{2}}} {zE_{2} (z ) {\text{d}}z} } \right]\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial \delta u_{0} }}{\partial x}} \right.} \right.} } \\ & \quad + \left[ {\int_{{ - \frac{h}{2}}}^{0} {zE_{1} (z){\text{d}}z + } \int_{0}^{{\frac{h}{2}}} {zE_{2} (z ) {\text{d}}z} } \right]\frac{{\partial u_{0} }}{\partial x}\frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }} + \left[ { - \int_{{ - \frac{h}{2}}}^{0} {z^{2} E_{1} (z){\text{d}}z - } \int_{0}^{{\frac{h}{2}}} {z^{2} E_{2} (z ) {\text{d}}z} } \right]\left. {\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial \delta w_{0} }}{{\partial x^{2} }}} \right) \\ & \quad + \left( {f\delta u_{0} + \mu \frac{\partial f}{\partial x}\frac{{\partial \delta u_{0} }}{\partial x}} \right) + \left( {q\delta w_{0} - \mu q\frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }}} \right) + \left( {\bar{N}\frac{{\partial w_{0} }}{\partial x}\frac{{\partial \delta w_{0} }}{\partial x} - \mu \bar{N}\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }}} \right) \\ & \quad + \left( {I_{0} \frac{{\partial u_{0} }}{\partial t}\frac{{\partial \delta u_{0} }}{\partial t} - \mu I_{0} \frac{{\partial^{3} u_{0} }}{{\partial t^{2} \partial x}}\frac{{\partial \delta u_{0} }}{\partial x}} \right) + \left( {I_{0} \frac{{\partial w_{0} }}{\partial t}\frac{{\partial \delta w_{0} }}{\partial t} + \mu I_{0} \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}\frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }} + I_{2} \frac{{\partial^{2} w_{0} }}{\partial t\partial x}\frac{{\partial^{2} \delta w_{0} }}{\partial t\partial x} - \mu I_{2} \frac{{\partial^{4} w_{0} }}{{\partial t^{2} \partial x^{2} }}\frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }}} \right) \\ & \quad + \left. {\left( {I_{1} \frac{{\partial^{2} u_{0} }}{\partial t\partial x}\frac{{\partial \delta w_{0} }}{\partial t} + \mu I_{1} \frac{{\partial^{3} u_{0} }}{{\partial t^{2} \partial x}}\frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }} - I_{1} \frac{{\partial^{2} w_{0} }}{\partial t\partial x}\frac{{\partial \delta u_{0} }}{\partial t} + I_{1} \mu \frac{{\partial^{4} w_{0} }}{{\partial t^{2} \partial x^{2} }}\frac{{\partial \delta u_{0} }}{\partial x}} \right)} \right\}{\text{d}}x{\text{d}}t + \int_{0}^{t} {\left[ {\bar{N}_{B} \delta u_{0} + \bar{V}_{B} \delta w_{0} + \bar{M}_{B} \frac{{\partial \delta w_{0} }}{\partial x}} \right]_{0}^{L} {\text{d}}t = 0} \\ \end{aligned} $$
(15)

By substituting Eq. (14) into Eq. (15) and integrating over the domain, the following equation of motion is derived

$$ \left( {M_{\text{l}} + \mu M_{\text{nl}} } \right) \ddot{\textit{{Y}}} + K_{\text{s}} Y + K_{\text{G}} Y = F + Q $$
(16)

where M l and Mnl are local and nonlocal mass matrices, respectively. K s is the stiffness matrix of FG beam, K G is the geometrical stiffness matrix, Y is the generalized displacement vector, and F and Q are the distributed force vector and concentrated force vector, respectively. The element matrices and force vectors can be represented by:

  • The mass matrices can be represented by

    $$ M_{\text{l}} = \mathop \int \limits_{0}^{l} I_{0} N_{i} N_{j} {\text{d}}x + \mathop \int \limits_{0}^{l} \left( {I_{0} \tilde{N}_{k} \tilde{N}_{\text{l}} + I_{2} \frac{{\partial \tilde{N}_{k} }}{\partial x}\frac{{\partial \tilde{N}_{1} }}{\partial x}} \right) {\text{d}}x + \mathop \int \limits_{0}^{l} \left( {I_{1} \frac{{\partial N_{i} }}{\partial x}\tilde{N}_{1} + I_{1} \frac{{\partial^{2} \tilde{N}_{1} }}{{\partial x^{2} }}N_{i} } \right) {\text{d}}x $$
    (17a)
    $$ M_{\text{nl}} = - \mathop \int \limits_{0}^{l} I_{0} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} {\text{d}}x + \mathop \int \limits_{0}^{l} \left( {I_{0} \tilde{N}_{k} \frac{{\partial^{2} \tilde{N}_{1} }}{{\partial x^{2} }} - I_{2} \frac{{\partial^{2} \tilde{N}_{k} }}{{\partial x^{2} }}\frac{{\partial^{2} \tilde{N}_{1} }}{{\partial x^{2} }}} \right) {\text{d}}x + \mathop \int \limits_{0}^{l} \left( {I_{1} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial^{2} \tilde{N}_{1} }}{{\partial x^{2} }} + I_{1} \frac{{\partial^{2} \tilde{N}_{1} }}{{\partial x^{2} }}\frac{{\partial N_{i} }}{\partial x}} \right) {\text{d}}x $$
    (17b)

  • Element stiffness matrix can be calculated by

    $$ K_{\text{u}} = \mathop \int \limits_{0}^{l} \left[ { - \mathop \int \limits_{{ - \frac{h}{2}}}^{0} E_{1} \left( z \right){\text{d}}z - \mathop \int \limits_{0}^{{\frac{h}{2}}} E_{2} \left( z \right){\text{d}}z} \right]\frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x}{\text{d}}x\quad {\text{where}}\;i\;{\text{and}}\;j = 1,2 $$
    (17c)
    $$ K_{\text{w}} = \mathop \int \limits_{0}^{l} \left[ { - \mathop \int \limits_{{ - \frac{h}{2}}}^{0} z^{2} E_{1} \left( z \right){\text{d}}z - \mathop \int \limits_{0}^{{\frac{h}{2}}} z^{2} E_{2} \left( z \right){\text{d}}z} \right]\frac{{\partial \tilde{N}_{k} }}{\partial x}\frac{{\partial \tilde{N}_{1} }}{\partial x}{\text{d}}x\quad {\text{where}}\;k\;{\text{and}}\;l = 1,2,3,4 $$
    (17d)
    $$ K_{\text{uw}} = \mathop \int \limits_{0}^{l} \left[ {\mathop \int \limits_{{ - \frac{h}{2}}}^{0} zE_{1} \left( z \right){\text{d}}z + \mathop \int \limits_{0}^{{\frac{h}{2}}} zE_{2} \left( z \right){\text{d}}z} \right]\frac{{\partial^{2} \tilde{N}_{k} }}{{\partial x^{2} }}\frac{{\partial N_{i} }}{\partial x}{\text{d}}x + \mathop \int \limits_{0}^{l} \left[ {\mathop \int \limits_{{ - \frac{h}{2}}}^{0} zE_{1} \left( z \right){\text{d}}z + \mathop \int \limits_{0}^{{\frac{h}{2}}} zE_{2} \left( z \right){\text{d}}z} \right]\frac{{\partial N_{i} }}{\partial x}\frac{{\partial^{2} \tilde{N}_{k} }}{{\partial x^{2} }}{\text{d}}x $$
    (17e)
    $$ K_{\text{s}} = K_{\text{u}} + K_{\text{w}} + K_{\text{uw}} $$
    (17f)

  • Element geometrical stiffness matrix can be represented by

    $$ K_{\text{G}} = \mathop \int \limits_{0}^{L} \left[ { - \bar{N}\frac{{\partial \tilde{N}_{k} }}{\partial x}\frac{{\partial \tilde{N}_{1} }}{\partial x} + \mu \bar{N}\frac{{\partial^{2} \tilde{N}_{k} }}{{\partial x^{2} }}\frac{{\partial^{2} \tilde{N}_{1} }}{{\partial x^{2} }}} \right]{\text{d}}x $$
    (17g)

  • The force vector can be represented by

    $$ F = q\mathop \int \limits_{0}^{L} \left[ {\tilde{N}_{k} - \mu \frac{{\partial^{2} \tilde{N}_{k} }}{{\partial x^{2} }}} \right] {\text{d}}x + \mathop \int \limits_{0}^{L} \left[ {fN_{i} + \mu \frac{\partial f}{\partial x}\frac{{\partial N_{i} }}{\partial x}} \right] {\text{d}}x $$
    (17h)

4 Numerical results

Here numerical examples are considered. In all cases, the dimensions of beam geometry are described as Eltaher et al. [21]. These values are used only for the purpose of numerically evaluating the parametric effects of nonlocal parameter (μ), material distribution (k) and functional distribution (P). The beam was assumed to be simply supported at both ends: a mesh of 50 elements (with linear approximation of u and hermite cubic approximation of w).

Table 1 presents the first five nondimensional frequencies of S-FGM with varying nonlocal parameter (μ) and material distribution (k). Fixing material distribution parameter and varying the nonlocal parameter results in a significant change in the natural frequencies. During this study, it is also found, as others have, that for simply supported nanobeams, natural frequencies decrease as the nonlocal parameter increases. For a case in hand, as the nonlocal parameter changes from 0 to 4 × 10−12, the first and fifth frequencies reduce by about 15.5 and 60 %, respectively, at a constant material distribution k = 0.5. This emphasizes the significance of the nonlocal effect on the natural frequency of beam. It is also noted that for a sigmoidal distribution, the natural frequencies decreased by increasing the material parameter distribution (k), as shown in Table 1. The first fundamental frequency is reduced by 5 % as k changing from 0 to 10. Qualitative behavior of Table 1 for the first fundamental frequency λ 1 is illustrated in Fig. 4. From Fig. 4, it is observed that the fundamental frequency is reduced smoothly by changing the material distribution from 0 to 10 for a certain value of nonlocal parameter.

Table 1 Dimensionless frequencies for different material distribution and nonlocal parameters for S-FGM
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Fig. 4
figure 4

Variation of the fundamental frequency for varying material distribution and nonlocal parameter of S-FGM

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The effects of both size effect and material distribution on the first five fundamental frequencies of ceramics–metal–ceramics functionally graded nanobeam are presented in Table 2 and Fig. 5. As concluded form Table 2 and Fig. 5, the frequencies are assumed to be constant as the material distribution changes from 0 to 1; however, they are reduced significantly as the material parameter changes from 1 to 10. In addition, the nonlocal parameter tends to reduce the fundamental frequency of FG beam at a specific material distribution.

Table 2 Dimensionless frequencies for different material distribution and nonlocal parameters for SP-FGM (CMC)
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Fig. 5
figure 5

Variation of the fundamental frequency for varying material distribution and nonlocal parameter of SP-FGM (CMC)

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The variation of frequencies of metal–ceramics–metal functional graded nanobeams with a nonlocal parameter and material distribution is illustrated in Table 3. It is noted that the nonlocal parameter has the same effect on the natural frequencies for SP-FGM (MCM) as SP-FGM (CMC) and S-FGM. It is observed that the increase in material distribution tends to increase the frequency at a constant nonlocal parameter. For example, at a zero nonlocal parameter, the first frequency increased from 9.9109 to 17.4289 as material distribution increased from 0 to 10. Graphical illustration of nonlocal parameter and material distribution effects on the fundamental frequency of MCM functionally graded nanobeam is shown in Fig. 6. From this figure, it can be concluded that the first frequency increased linearly with a material graduation and decreased with nonlocal parameter.

Table 3 Dimensionless frequencies for different material distribution and nonlocal parameters for SP-FGM (MCM)
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Fig. 6
figure 6

Variation of the fundamental frequency for varying material distribution and nonlocal parameter of SP-FGM (MCM)

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5 Conclusions

A numerical finite element model was developed to investigate the dynamic behavior of both nonlinear symmetric power and sigmoid functionally graded nonlocal nanobeams. The assumed material distributions are symmetrical with mid-plane. Nonlocal differential Eringen’s elasticity is proposed to consider the size dependency of nanobeam.

The most findings of this study may be summarized as:

  • The nonlocal size parameter tends to decrease the frequencies of nanobeams.

  • The material parameter k has different effects on the frequencies of nanobeams. By increasing k, the frequencies decrease in case of S-FGM distribution and increase in case of SP-FGM (MCM). However, in case of SP-FGM (CMC), the frequencies are constant in the range of 0 ≤ k ≤ 1 and decreased in the range 1 ≤ k ≤ 10.

  • The proposed model can give designers and engineers a scope for proper selection of material distribution, especially in manufacturing of nanosensors and nanoactuators.