1 Introduction

Hygro-thermal stresses arising from temperature and moisture variations can affect the mechanical performance of engineering structures even at nanoscale. Also, advanced structural components composed of functionally graded materials (FGMs) may experience intense hygro-thermal environments which show an adverse influence on the stiffness and safety of such structures. Therefore, an accurate evaluation of environmental exposure is necessitated to find the essence of their detrimental influence on the FG structures. Thus, it is of utmost significance to investigate hygro-thermally induced mechanical behavior of these structural elements. To this end, hygro-thermal stress analysis of one-dimensional functionally graded piezoelectric media via analytical solutions is carried out by Akbarzadeh and Chen [1]. Postbuckling of functionally graded (FG) plates under hygro-thermal environments is studied by Lee and Kim [30]. The static behaviors of exponentially inhomogeneous plates subjected to a transverse uniform loading and hygro-thermal conditions are studied by Zenkour [40]. Al Khateeb and Zenkour [4] presented a refined four-variable plate model for bending analysis of advanced plates embedded on elastic foundations in hygro-thermal environments. In another study, Zenkour et al. [41] researched the influence of temperature and moisture on the mechanical behavior of shear-deformable composite FGM plates resting on elastic foundations. Tounsi et al. [36] introduced a refined trigonometric shear deformation theory for thermoelastic bending of FG sandwich plates. Thermomechanical bending response of FGM plates was explored by Hamidi et al. [24] and Bouderba et al. [10]. Bending analysis of FGM plates under hygro-thermomechanical loading using a four-variable refined plate theory was presented by Zidi et al. [42].

Recently, due to the impotency of classical beam theory or Euler–Bernoulli beam theory (EBT) in adequately modeling of the transverse shear deformations as well as shear correction factor dependency of Timoshenko beam theory (TBT), a number of higher-order theories are proposed and applied in analysis of FG structures. EBT neglects the effects of thickness stretching and shear deformation. This simple theory can be successfully used in analysis of slender beams with a large aspect ratio. However, influences of shear and normal deformations may become more considerable for moderately thick beams and plates [6], Helabi et al. [25], and Ait Amar Meziane et al. [3, 7]. Hence, several novel plate theories are suggested accounting for both transverse shear and normal deformations and satisfy the zero traction boundary conditions on the surfaces of the plate without using shear correction factor [9, 11, 31].

Vo et al. [38] provided finite element vibration and buckling analysis of FG sandwich beams through a quasi-3D theory in which both shear deformation and thickness stretching effects are included. Ebrahimi and Salari [12] presented a semi-analytical method for vibrational and buckling analysis of FG nanobeams considering the physical neutral axis position. Recently, Ebrahimi and Barati [16–19] presented static and dynamic modeling of a thermopiezoelectrically actuated nanosize beam subjected to a magnetic field. Most recently small-scale effects on hygro-thermomechanical vibration of temperature-dependent nonhomogeneous nanoscale beams are investigated by Ebrahimi and Barati [20]. Ebrahimi and Barati [21] also proposed a nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams.

Due to the characteristic size of beams applied in nanoelectromechanical systems (NEMs), the small-scale impact on their behavior is prominent. It is clear that above-mentioned works cannot predict the small-size impacts on the nanostructures. Thus, the nonlocal elasticity theory suggested by Eringen [22] is developed. He extended the classical continuum mechanics to describe small-scale influences. Based on the nonlocal constitutive relations of Eringen, several studies have been carried out to investigate the mechanical responses of nanostructures. Peddieson et al. [39] indicated that nonlocal continuum mechanics could be applicable to analysis of nanoscale structures. To extend nonlocal elasticity theory for analysis of FG structures, vibrational behavior of nanosize FG beams via finite element method is explored by Eltaher et al. [23]. Tounsi et al. [35, 36] investigated the nonlocal effects on thermal buckling properties of double-walled carbon nanotubes. Size-dependent mechanical behavior of FG trigonometric shear-deformable nanobeams including neutral surface position concept was introduced by Ahouel et al. [2], while Larbi Chaht et al. [29] explored the bending and buckling of FG nanoscale beams including the thickness stretching effect. Also the chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes were investigated by Benguediab et al. [8].

The influences of various thermal environments on buckling and vibration of nonlocal temperature-dependent FG beams are analyzed by Ebrahimi and Salari [14] using Navier analytical solution. In another work, Ebrahimi and Salari [13] investigated thermomechanical vibration of FG nanobeams with arbitrary boundary conditions applying differential transform method (DTM). Also, Ebrahimi et al. [15] explored the effects of linear and nonlinear temperature distributions on vibration of FG nanobeams.

Hygro-thermal analysis of nanostructures is reported in a few works. Small-scale bending analysis of nanoplates resting on elastic foundation under hygro-thermomechanical environments is performed by Alzahrani et al. [5]. Also, using trigonometric shear deformation plate theory, Sobhy [34] investigated vibrational behavior of orthotropic double-layered graphene sheets subjected to hygro-thermal loading. Therefore, it is evident that there is no published work studying hygro-thermomechanical vibration analysis of FG nanobeams with or without elastic foundation using higher-order shear deformation theories.

In this study, a unified higher-order beam theory containing parabolic, sinusoidal, hyperbolic, exponential, and inverse cotangential functions is developed to investigate the influences of moisture and temperature rise due to various hygro-thermal loads on small-amplitude vibration of nanosize FGM beams resting on elastic foundation. These theories provide a constant transverse displacement and higher-order variation of axial displacement through the depth of the nanobeam so that there is no need for any shear correction factors. Three types of environmental condition namely uniform, linear, and sinusoidal hygro-thermal loading are studied. The material properties of FGM nanobeam are regarded to be temperature dependent and are graded in the thickness direction via power-law model. An analytical solution is applied to solve the governing equations derived from Hamilton’s principle. Several numerical and illustrative results are presented to indicate the effects of the shear deformation, various hygro-thermal environments, gradient index, nonlocality, and elastic foundation parameters on the hygro-thermomechanical vibration of FG nanobeams.

2 Theory and formulation

2.1 Power-law FG nanobeam model

A FG nanobeam system of length a, width b, and thickness h is shown in Fig. 1. The effective material properties of the nonlocal FG beam including Young’s modulus \( E_{f} \), moisture expansion coefficient \( \beta_{f} \), thermal expansion \( \alpha_{f} \), and mass density \( \rho_{f} \) change continuously in the z-axis direction (thickness direction) according to power-law model. Hence, the effective material properties, \( P_{f} \), can be stated as:

$$ P_{f} = P_{\text{c}} V_{\text{c}} + P_{\text{m}} V_{\text{m}} $$
(1)

In which \( P_{\text{m}} \), \( P_{\text{c}} \), \( V_{\text{m}} \), and \( V_{\text{c}} \) denote the metal and ceramic material properties and its volume fractions which have a relation with the following form:

$$ V_{\text{c}} + V_{\text{m}} = 1 $$
(2)

The ceramic phase volume fraction of the nanobeam is described as:

$$ V_{\text{c}} = \left( {\frac{z}{h} + \frac{1}{2}} \right)^{P} $$
(3)

Here p is the power-law index which is a nonnegative variable and estimates the material distribution through the nanobeam thickness, and z is the distance from the mid-surface of the nanobeam. It is clear that when p is equal to zero, the FG nanobeam is fully ceramic. So, according to Eqs. (1)–(2), the effective material properties of the nonlocal FGM beam including Young’s modulus (\( E \)), mass density (\( \rho \)), thermal expansion (\( \alpha \)), and moisture expansion coefficient (\( \beta \)) can be expressed in the following form:

$$ \begin{aligned} & E(z) = \left( {E_{\text{c}} - E_{\text{m}} } \right)\left( {\frac{z}{h} + \frac{1}{2}} \right)^{p} + E_{\text{m}} \\ & \rho (z) = \left( {\rho_{\text{c}} - \rho_{\text{m}} } \right)\left( {\frac{z}{h} + \frac{1}{2}} \right)^{p} + \rho_{\text{m}} \\ & \alpha (z) = \left( {\alpha_{\text{c}} - \alpha_{\text{m}} } \right)\left( {\frac{z}{h} + \frac{1}{2}} \right)^{p} + \alpha_{\text{m}} \\ & \beta (z) = \left( {\beta_{\text{c}} - \beta_{\text{m}} } \right)\left( {\frac{z}{h} + \frac{1}{2}} \right)^{p} + \beta_{\text{m}} \\ \end{aligned} $$
(4)

For more precise anticipation of FGMs behavior under extreme temperature fields, material properties must be dependent on temperature. Therefore, temperature-dependent coefficients of material phases can be expressed according to the following nonlinear equation:

$$ P = P_{0} \left( {P_{ - 1} \;T^{ - 1} + 1 + P_{1} \;T + P_{2} \;T^{2} + P_{3} \;T^{3} } \right) $$
(5)

where \( P_{0} ,P_{ - 1} ,P_{1} ,P_{2} \) and \( P_{3} \) are the temperature-dependent coefficients which are tabulated in Table 2 for \( {\text{Si}}_{ 3} {\text{N}}_{ 4} \) and \( {\text{SUS}}\, 3 0 4 \). The bottom and top surfaces of FG nanobeam are supposed to be fully metal (\( {\text{SUS}}\, 3 0 4 \)) and fully ceramic (\( {\text{Si}}_{ 3} {\text{N}}_{ 4} \)), respectively.

Fig. 1
figure 1

Geometry of functionally graded nanobeam resting on elastic foundation

Full size image

2.2 Kinematic relations

Based on the refined shear deformation beam theories, the displacement field at any point of the beam can be written as:

$$ u_{x} \left( {x,z} \right) = u\left( x \right) - z\frac{{\partial w_{\text{b}} }}{\partial x} - f(z)\frac{{\partial w_{\text{s}} }}{\partial x} $$
(6)
$$ u_{z} (x,z) = w_{\text{b}} (x) + w_{\text{s}} (x) $$
(7)

where \( u \) is longitudinal displacement and \( w_{\text{b}} ,\,w_{\text{s}} \) are the bending and shear components of transverse displacement of a point on the midplane of the beam. \( f(z) \) is the shape function determining the distribution of the transverse shear strain and shear stress through the thickness of the beam for which various beam models (parabolic, sinusoidal, hyperbolic, exponential and inverse cotangential) are presented in Table 1. Nonzero strains of the present beam model are expressed as follows:

$$ \varepsilon_{xx} = \frac{\partial u}{\partial x} - z\frac{{\partial^{2} w_{\text{b}} }}{{\partial x^{2} }} - f(z)\frac{{\partial^{2} w_{\text{s}} }}{{\partial x^{2} }} $$
(8)
$$ \gamma_{xz} = g\frac{{\partial w_{\text{s}} }}{\partial x} $$
(9)

where \( g(z) = 1 - {\text{d}}f/{\text{d}}z \). By using the Hamilton’s principle, in which the motion of an elastic structure in the time interval t1 < t < t2 is so that the integral with respect to time of the total potential energy is extremum:

$$ \int_{0}^{t} {\delta (U + V - K)\,{\text{d}}t = 0} $$
(10)

Here \( U \) is strain energy, \( V \) is work done by external forces, and K is kinetic energy. The virtual strain energy can be written as:

$$ \delta \,U = \int_{v} {\sigma_{ij} \delta \,\varepsilon_{ij} \,{\text{d}}V = } \int_{v} {(\sigma_{xx} \delta \,\varepsilon_{xx} + \sigma_{xz} \delta \,\gamma_{xz} )\,{\text{d}}V} $$
(11)

Substituting Eqs. (8)–(9) into Eq. (11) yields:

$$ \delta \,U = \int_{0}^{L} {\left( {N\frac{{{\text{d}}\delta u}}{{{\text{d}}x}} - M_{\text{b}} \frac{{{\text{d}}^{2} \delta w_{\text{b}} }}{{{\text{d}}x^{2} }} - M_{\text{s}} \frac{{{\text{d}}^{2} \delta w_{\text{s}} }}{{{\text{d}}x^{2} }} + Q\frac{{{\text{d}}\delta w_{\text{s}} }}{{{\text{d}}x}}} \right)} \,{\text{d}}x $$
(12)

In which the variables introduced in arriving at the last expression are defined as follows:

$$ N = \int_{A} {\sigma_{xx} } \,{\text{d}}A,\;M_{\text{b}} = \int_{A} {z\sigma_{xx} } \,{\text{d}}A,\;M_{\text{s}} = \int_{A} {f\sigma_{xx} } \,{\text{d}}A,\;Q = \int_{A} {g\sigma_{xz} } \,{\text{d}}A $$
(13)

The first variation of the work done by applied forces can be written in the form:

$$ \begin{aligned} \delta V & = \int_{0}^{L} \left( {(N^{T} + N^{H} )\left( {\frac{{{\text{d}}(w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x}}\frac{{{\text{d}}\delta (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x}}} \right) - k_{w} \delta \,(w_{\text{b}} + w_{\text{s}} )} \right. \\ &\left. \quad + k_{p} \left( {\frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} + \frac{{{\text{d}}^{2} \delta (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }}} \right)\right){\text{d}}x \\ \end{aligned} $$
(14)

where \( k_{w} \) and \( k_{p} \) are linear and shear coefficient of elastic foundation, respectively, and \( N^{T} \) and \( N^{H} \) are applied forces due to temperature and moisture change as:

$$ \begin{aligned} & N^{T} = \int_{ - h/2}^{h/2} {E(z,T)\,\alpha (z,T)\,(T - T_{0} )\,{\text{d}}z} , \\ & N^{H} = \int_{ - h/2}^{h/2} {E(z,T)\,\beta (z,T)\,(C - C_{0} )\,{\text{d}}z} \\ \end{aligned} $$
(15)

where T 0 and C 0 are the reference temperature and moisture concentrations, respectively. The variation of the kinetic energy can be expressed as:

$$ \begin{aligned} \delta \,K & = \int_{0}^{L} \left( {I_{0} \left[{\frac{{{\text{d}}u}}{{{\text{d}}t}}\frac{{{\text{d}}\delta\,u}}{{{\text{d}}t}} + \left( {\frac{{{\text{d}}w_{b}}}{{{\text{d}}t}} + \frac{{{\text{d}}w_{s} }}{{{\text{d}}t}}}\right)\left( {\frac{{{\text{d}}\delta \,w_{b} }}{{{\text{d}}t}} +\frac{{{\text{d}}\delta \,w_{s} }}{{{\text{d}}t}}} \right)} \right]- I_{1} \left({\frac{{{\text{d}}u}}{{{\text{d}}t}}\frac{{{\text{d}}^{2} \delta w_{b} }}{{{\text{d}}x{\text{d}}t}} + \frac{{{\text{d}}^{2} w_{b}}}{{{\text{d}}x{\text{d}}t}}\frac{{{\text{d}}\delta u}}{{{\text{d}}t}}} \right)} \right. \\ & \quad + I_{2} \left({\frac{{{\text{d}}^{2} w_{b}}}{{{\text{d}}x{\text{d}}t}}\frac{{{\text{d}}^{2} \delta w_{b}}}{{{\text{d}}x{\text{d}}t}}} \right) - J_{1} \left({\frac{{{\text{d}}u}}{{{\text{d}}t}}\frac{{{\text{d}}^{2} \delta w_{s} }}{{{\text{d}}x{\text{d}}t}} + \frac{{{\text{d}}^{2} w_{s}}}{{{\text{d}}x{\text{d}}t}}\frac{{{\text{d}}\delta u}}{{{\text{d}}t}}} \right) + K_{2} \left( {\frac{{{\text{d}}^{2}w_{s} }}{{{\text{d}}x{\text{d}}t}}\frac{{{\text{d}}^{2} \delta w_{s}}}{{{\text{d}}x{\text{d}}t}}} \right) \\ &\left. \quad + J_{2}\left( {\frac{{{\text{d}}^{2} w_{b}}}{{{\text{d}}x{\text{d}}t}}\frac{{{\text{d}}^{2} \delta w_{s}}}{{{\text{d}}x{\text{d}}t}} + \frac{{{\text{d}}^{2} w_{s}}}{{{\text{d}}x{\text{d}}t}}\frac{{{\text{d}}^{2} \delta w_{b}}}{{{\text{d}}x{\text{d}}t}}} \right)\right){\text{d}}x \\\end{aligned} $$
(16)

where

$$ (I_{0} ,I_{1} ,J_{1} ,I_{2} ,J_{2} ,K_{2} ) = \int_{A} {\rho (z)(1,z,f,z^{2} } ,zf,f^{2} ){\text{d}}A $$
(17)

The following Euler–Lagrange equations are obtained by inserting Eqs. (12)–(16) in Eq. (10) when the coefficients of \( \delta u,\delta \,w_{\text{b}} \) and \( \delta \,w_{\text{s}} \) are equal to zero:

$$ \frac{\partial N}{\partial x} = I_{0} \frac{{{\text{d}}^{2} u}}{{{\text{d}}t^{2} }} - I_{1} \frac{{{\text{d}}^{3} w_{\text{b}} }}{{{\text{d}}x{\text{d}}t^{2} }} - J_{1} \frac{{{\text{d}}^{3} w_{\text{s}} }}{{{\text{d}}x{\text{d}}t^{2} }} $$
(18)
$$ \begin{aligned} \frac{{{\text{d}}^{2} M_{\text{b}} }}{{{\text{d}}x^{2} }} & = (N^{T} + N^{H} )\frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} + I_{0} \left( {\frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}t^{2} }} + \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}t^{2} }}} \right) \\ & \quad + I_{1} \frac{{{\text{d}}^{3} u}}{{{\text{d}}x{\text{d}}t^{2} }} - I_{2} \frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} - J_{2} \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} \\ & \quad + k_{w} (w_{\text{b}} + w_{\text{s}} ) - k_{p} \frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} \\ \end{aligned} $$
(19)
$$ \begin{aligned} \frac{{{\text{d}}^{2} M_{\text{s}} }}{{{\text{d}}x^{2} }} + \frac{{{\text{d}}Q}}{{{\text{d}}x}} & = (N^{T} + N^{H} )\frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} + I_{0} \left( {\frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}t^{2} }} + \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}t^{2} }}} \right) \\ & \quad + J_{1} \frac{{{\text{d}}^{3} u}}{{{\text{d}}x{\text{d}}t^{2} }} - J_{2} \frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} - K_{2} \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} \\ & \quad + k_{w} (w_{\text{b}} + w_{\text{s}} ) - k_{p} \frac{{{\text{d}}^{2} (w_{\text{b}} + w_{s} )}}{{{\text{d}}x^{2} }} \\ \end{aligned} $$
(20)
Table 1 Shape functions
Full size table

2.3 The nonlocal elasticity model for FG nanobeam

According to Eringen nonlocal elasticity model [22] which contains wide-range interactions between points in a continuum solid, the stress state at a point inside a body is regarded to be function of all neighbor points’ strains. Hence, in the present work in order to capture the small-size impacts, nonlocal elasticity theory is implemented in which a linear differential framework of constitutive equations is expressed as:

$$ (1 - (e_{0} a)\nabla^{2} )\sigma_{kl} = t_{kl} $$
(21)

In which \( \nabla^{2} \) denotes the Laplacian operator. Therefore, the scale length \( e_{0} \,a \) considers the influences of small size on the response of nanoscale structures. Thus, the constitutive relations of nonlocal theory for a FG nanobeam can be stated as:

$$ \sigma_{xx} - \mu \frac{{\partial^{2} \sigma_{xx} }}{{\partial x^{2} }} = E(z)\varepsilon_{xx} $$
(22)
$$ \sigma_{xz} - \mu \frac{{\partial^{2} \sigma_{xz} }}{{\partial x^{2} }} = G(z)\gamma_{xz} $$
(23)

where \( (\mu = (e_{0} a)^{2} ) \). By integrating Eqs. (22) and (23) over the area of nanobeam’s cross section, the following relations for a nonlocal refined FG beam model can be obtained:

$$ N - \mu \frac{{{\text{d}}^{2} N}}{{{\text{d}}x^{2} }} = A\frac{{{\text{d}}u}}{{{\text{d}}x}} - B\frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}x^{2} }} - B_{\text{s}} \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}x^{2} }} $$
(24)
$$ M_{\text{b}} - \mu \frac{{{\text{d}}^{2} M_{\text{b}} }}{{{\text{d}}x^{2} }} = B\frac{{{\text{d}}u}}{{{\text{d}}x}} - D\frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}x^{2} }} - D_{\text{s}} \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}x^{2} }} $$
(25)
$$ M_{\text{s}} - \mu \frac{{{\text{d}}^{2} M_{\text{s}} }}{{{\text{d}}x^{2} }} = B_{\text{s}} \frac{{{\text{d}}u}}{{{\text{d}}x}} - D_{\text{s}} \frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}x^{2} }} - H_{\text{s}} \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}x^{2} }} $$
(26)
$$ Q - \mu \frac{{{\text{d}}^{2} Q}}{{{\text{d}}x^{2} }} = A_{\text{s}} \frac{{{\text{d}}w_{\text{s}} }}{{{\text{d}}x}} $$
(27)

where the cross-sectional rigidities are calculated as follows:

$$ (A,B,B_{\text{s}} ,D,D_{\text{s}} ,H_{\text{s}} ) = \int_{A} {E(z)\,(1\,,z,f,z^{2} ,zf,f^{2} )\,} {\text{d}}A $$
(28)
$$ A_{\text{s}} = \int_{A} {g^{2} G(z)} \,{\text{d}}A $$
(29)

The nonlocal governing equations of refined shear-deformable FG nanobeams in terms of the displacement can be obtained by inserting \( N \), \( M_{\text{b}} \), \( M_{\text{s}} \), and \( Q \) from Eqs. (24)–(27), respectively, into Eqs. (18)–(20) as follows:

$$ \begin{aligned} & A\frac{{{\text{d}}^{2} u}}{{{\text{d}}x^{2} }} - B\frac{{\partial^{3} w_{\text{b}} }}{{\partial x^{3} }} - B_{\text{s}} \frac{{\partial^{3} w_{\text{s}} }}{{\partial x^{3} }} - I_{0} \frac{{{\text{d}}^{2} u}}{{{\text{d}}t^{2} }} + I_{1} \frac{{{\text{d}}^{3} w_{\text{b}} }}{{{\text{d}}x{\text{d}}t^{2} }} + J_{1} \frac{{{\text{d}}^{3} w_{\text{s}} }}{{{\text{d}}x{\text{d}}t^{2} }} \\ & \quad + \mu \,\left( {I_{0} \frac{{{\text{d}}^{4} u}}{{{\text{d}}x^{2} {\text{d}}t^{2} }} - I_{1} \frac{{{\text{d}}^{5} w_{\text{b}} }}{{{\text{d}}x^{3} {\text{d}}t^{2} }} - J_{1} \frac{{{\text{d}}^{5} w_{\text{s}} }}{{{\text{d}}x^{3} {\text{d}}t^{2} }}} \right) = 0 \\ \end{aligned} $$
(30)
$$ \begin{aligned} & B\frac{{{\text{d}}^{3} u}}{{{\text{d}}x^{3} }} - D\frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{4} }} - D_{\text{s}} \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{4} }} - (N^{T} + N^{H} )\frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} \\ & \quad - I_{0} \left( {\frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}t^{2} }} + \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}t^{2} }}} \right) - I_{1} \frac{{{\text{d}}^{3} u}}{{{\text{d}}x{\text{d}}t^{2} }} + I_{2} \frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} \\ & \quad + J_{2} \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} - k_{\text{w}} (w_{\text{b}} + w_{\text{s}} ) + k_{\text{p}} \frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} \\ & \quad + \mu \left( ( + (N^{T} + N^{H} )\frac{{{\text{d}}^{4} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{4} }} + I_{0} \left( {\frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} + \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }}} \right) + I_{1} \frac{{{\text{d}}^{5} u}}{{{\text{d}}x^{3} {\text{d}}t^{2} }} \right. \\ &\quad \left.- I_{2} \frac{{{\text{d}}^{6} w_{\text{b}} }}{{{\text{d}}x^{4} {\text{d}}t^{2} }} - J_{2} \frac{{{\text{d}}^{6} w_{\text{s}} }}{{{\text{d}}x^{4} {\text{d}}t^{2} }} + k_{\text{w}} \frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} - k_{\text{p}} \frac{{{\text{d}}^{4} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{4} }} \right) = 0 \\ \end{aligned} $$
(31)
$$ \begin{aligned} & B_{s} \frac{{{\text{d}}^{3} u}}{{{\text{d}}x^{3} }} - D_{\text{s}} \frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{4} }} - H_{\text{s}} \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{4} }} + A_{\text{s}} \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}x^{2} }} \\ & \quad - (N^{T} + N^{H} )\frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} - I_{0} \left( {\frac{{{\text{d}}^{2} w_{\text{b}} }}{{{\text{d}}t^{2} }} + \frac{{{\text{d}}^{2} w_{\text{s}} }}{{{\text{d}}t^{2} }}} \right) - J_{1} \frac{{{\text{d}}^{3} u}}{{{\text{d}}x{\text{d}}t^{2} }} \\ & \quad + J_{2} \frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} + K_{2} \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} - k_{\text{w}} (w_{\text{b}} + w_{\text{s}} ) + k_{\text{p}} \frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} \\ & \quad + \mu \left((N^{T} + N^{H} )\frac{{{\text{d}}^{4} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{4} }} + I_{0} \left( {\frac{{{\text{d}}^{4} w_{\text{b}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }} + \frac{{{\text{d}}^{4} w_{\text{s}} }}{{{\text{d}}x^{2} {\text{d}}t^{2} }}} \right) + J_{1} \frac{{{\text{d}}^{5} u}}{{{\text{d}}x^{3} {\text{d}}t^{2} }} \right.\\ & \left.\quad - J_{2} \frac{{{\text{d}}^{6} w_{\text{b}} }}{{{\text{d}}x^{4} {\text{d}}t^{2} }} - K_{2} \frac{{{\text{d}}^{6} w_{\text{s}} }}{{{\text{d}}x^{4} {\text{d}}t^{2} }} + k_{\text{w}} \frac{{{\text{d}}^{2} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{2} }} - k_{\text{p}} \frac{{{\text{d}}^{4} (w_{\text{b}} + w_{\text{s}} )}}{{{\text{d}}x^{4} }}\right) = 0 \\ \end{aligned} $$
(32)

3 Solution procedures

Here, to satisfy simply supported boundary condition for hygro-thermomechanical free vibration of FG nanobeams, analytical solution of the coupled governing equations is proposed. To this purpose, the displacement variables adopted are of the form:

$$ u(x,t) = \sum\limits_{n = 1}^{\infty } {U_{n} \cos \,(\alpha x)} \,{\text{e}}^{{i\omega_{n} t}} $$
(33)
$$ w_{b} (x,t) = \sum\limits_{n = 1}^{\infty } {W_{bn} \sin \,(\alpha x)} \,{\text{e}}^{{i\omega_{n} t}} $$
(34)
$$ w_{s} (x,t) = \sum\limits_{n = 1}^{\infty } {W_{sn} \sin \,(\alpha x)} \,{\text{e}}^{{i\omega_{n} t}} $$
(35)

where \( \alpha = \frac{n\pi }{L} \) and (\( U_{n} \),\( W_{bn} \),\( W_{sn} \)) are the unknown Fourier coefficients. Inserting Eqs. (33)–(35) into Eqs. (30)–(32), respectively, leads to:

$$ \left\{ {\,\left( {\begin{array}{*{20}c} {k_{11} } & {k_{12} } & {k_{13} } \\ {k_{21} } & {k_{22} } & {k_{23} } \\ {k_{31} } & {k_{32} } & {k_{33} } \\ \end{array} } \right) - \bar{\omega }_{n}^{2} \left( {\begin{array}{*{20}c} {m_{11} } & {m_{12} } & {m_{13} } \\ {m_{21} } & {m_{22} } & {m_{23} } \\ {m_{31} } & {m_{32} } & {m_{33} } \\ \end{array} } \right)} \right\}\left\{ \begin{aligned} U_{n} \hfill \\ W_{bn} \hfill \\ W_{sn} \hfill \\ \end{aligned} \right\} = 0 $$
(36)

where

$$ \begin{aligned} & k_{1,1} = - A\alpha^{2} ,\quad k_{1,2} = B\alpha^{3} ,\quad k_{1,3} = B_{s} \alpha^{3} ,\quad k_{2,3} = - D_{s} \alpha^{4} , \\ & k_{2,2} = (N^{T} + N^{H} )\alpha^{2} (1 + \mu \alpha^{2} ) - k_{p} \alpha^{2} (1 + \mu \alpha^{2} ) - k_{w} (1 + \mu \alpha^{2} ) - D\alpha^{4} , \\ & k_{3,3} = (N^{T} + N^{H} )\alpha^{2} (1 + \mu \alpha^{2} ) - k_{p} \alpha^{2} (1 + \mu \alpha^{2} ) - k_{w} (1 + \mu \alpha^{2} ) - A_{\text{s}} \alpha^{2} - H_{\text{s}} \alpha^{4} , \\ & m_{1,1} = I_{0} (1 + \mu \alpha^{2} ),\quad m_{1,2} = - I_{1} \alpha - \mu I_{1} \alpha^{3} ,\quad m_{1,3} = - J_{1} \alpha - \mu J_{1} \alpha^{3} , \\ & m_{2,2} = I_{0} (1 + \mu \alpha^{2} ) + I_{2} \alpha^{2} + \mu I_{2} \alpha^{4} ,\quad m_{2,3} = I_{0} (1 + \mu \alpha^{2} ) + J_{2} \alpha^{2} + \mu J_{2} \alpha^{4} ; \\ & m_{3,3} = I_{0} (1 + \mu \alpha^{2} ) + K_{2} \alpha^{2} + \mu K_{2} \alpha^{4} , \\ \end{aligned} $$
(37)

4 Various hygro-thermal environments

4.1 Uniform moisture and temperature rise

For a FG nanobeam at reference moisture concentration C0 and reference temperature \( T_{0} \), the moisture and temperature are uniformly raised to a final value C and \( T \), respectively, in which the moisture and temperature change are ∆T = T − T 0 and ∆C = C − C 0.

4.2 Linear moisture and temperature rise

For a FG nanobeam for which the plate thickness is thin enough, the moisture and temperature distributions are linearly variable through the thickness as follows:

$$ T = T_{m} + (\Delta T)\left( {\frac{1}{2} + \frac{z}{h}} \right),\quad C = C_{m} + (\Delta C)\left( {\frac{1}{2} + \frac{z}{h}} \right) $$
(38)

The \( (\Delta T,\Delta C) \) in Eq. (38) could be defined \( \Delta T = T_{c} - T_{m} ,\quad \Delta C = C_{c} - C_{m} \).

4.3 Sinusoidal moisture and temperature rise

The moisture and temperature fields when FG nanobeam is exposed to sinusoidal moisture/temperature rise across the thickness can be defined as [32]:

$$ T = T_{m} + \Delta T\,\left( {1 - \text{Cos} \frac{\pi }{2}\left( {\frac{1}{2} + \frac{z}{h}} \right)} \right),\quad C = C_{m} + \Delta C\left( {1 - \text{Cos} \frac{\pi }{2}\left( {\frac{1}{2} + \frac{z}{h}} \right)} \right) $$
(39)

where ∆T = T c  − T m and ∆C = C c  − C m are temperature and moisture change.

5 Numerical results and discussion

The results of various numerical analyses are provided in this section for hygro-thermomechanical vibration analysis of a simply supported nanoscale FGM beam modeled via various higher-order shear deformation theories (PSDT, TSDT, HSDT, ESDT, and ITSDT). Material properties of the nonlocal FG beam such as Young’s modulus, Poisson ratio, thermal, and moisture expansion coefficients vary through the thickness direction according to power-law model. Temperature-dependent material properties of nonlocal P-FGM beam which is made from steel (\( {\text{SUS}}\, 3 0 4 \)) with \( \beta_{\text{m}} = 0.0005 \) and silicon nitride (\( {\text{Si}}_{ 3} {\text{N}}_{ 4} \)) with \( \beta_{\text{c}} = 0 \) are given in Table 2. The geometry dimensions of the nanobeam are: L (length) = 10 nm, b (width) = 1 nm, and h (thickness) = varied. It is supposed that the temperature rise in fully metal surface of FG nanobeam to reference temperature \( T_{0} \) is \( T_{m} - T_{0} = 5\,{\text{K}} \). For comparison study, numerical results are provided to show the validity of present model in the analysis of nanostructures.

Table 2 Temperature-dependent material properties of FGM constituents [13]
Full size table

Therefore, natural frequency of presented third-order FG nanobeam under linear temperature rise is compared with those obtained by Ebrahimi and Salari [13] for Timoshenko beam model when nonlocal parameter changes from 0 to 4 nm2 in Table 3. It is proved that the present model can evaluate the vibrational behavior of FG nanobeams with excellent agreement. Also, for better presentation of the results, the following dimensionless quantities are adopted:

$$ \hat{\omega } = \omega L^{2} \sqrt {\frac{{\rho_{c} A}}{{E_{c} I}}} ,\quad K_{w} = k_{w} \frac{{L^{4} }}{{E_{c} I}},\quad K_{p} = k_{p} \frac{{L^{2} }}{{E_{c} I}} $$
(40)

The variations of the dimensionless frequencies of FG nanobeam resting on elastic foundation for various beam theories and three environmental conditions called uniform, linear, and sinusoidal hygro-thermal loadings at L/h = 20 are presented in Tables 4, 5, and 6. It is observed that all of the proposed higher-order beam theories provide approximately same results for free vibration of FG nanobeams, and only some negligible differences exist. Also, it is obvious that frequency results of higher-order theories are lower than classical beam theory due to the reason that CBT is impotent to capture shear deformation effect. For any type of hygro-thermomechanical loading, as nonlocal parameter grows, the dimensionless frequency diminishes. The reason is the lower rigidity of the nanobeam when its size reduces.

Table 3 Comparison of the nondimensional frequency of a FG nanobeam under linear temperature rise without elastic foundation (L/h = 20)
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Table 4 Variation of the fundamental nondimensional frequencies of S–S FG nanobeam under uniform hygro-thermal loading for various beam theories (\( K_{w} = K_{p} = 0 \), L/h = 20)
Full size table
Table 5 Variation of the fundamental nondimensional frequencies of S–S FG nanobeam under linear hygro-thermal loading for various beam theories (\( K_{w} = K_{p} = 0 \), L/h = 20)
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Table 6 Variation of the fundamental nondimensional frequencies of S–S FG nanobeam under sinusoidal hygro-thermal loading for various beam theories (\( K_{w} = K_{p} = 0 \), L/h = 20)
Full size table

Also, the rise of moisture and temperature degrades plate stiffness and natural frequencies regardless of hygro-thermal loading type. Therefore, humidity or moisture changes have a notable effect on the mechanical responses of size-dependent FG nanobeams. Moreover, it is found that sinusoidal distribution of temperature and moisture (sinusoidal hygro-thermal loading) provides higher natural frequency than other hygro-thermal loads, while uniform hygro-thermal loading has the lowest one.

The variation of dimensionless natural frequency of simply supported third-order FG nanobeam versus uniform, linear, and sinusoidal moisture concentration rise in prebuckling domain for different nonlocal parameters at p = 1, L/h = 20, K w  = K p  = 0 and ΔT = 40 (K) is plotted in Fig. 2. At a prescribed humidity condition, the nonlocal beam model estimates lower natural frequency than local beam model. Also, it can be seen that for any type of hygro-thermal loading, the dimensionless frequency degrades with the increase in moisture.

Fig. 2
figure 2

Influence of moisture and nonlocal parameter on the dimensionless frequency of the S–S FG beam for various hygro-thermal loadings (\( p = 1,\;L/h = 20 \), K w  = K p  = 0, ΔT = 40 (K))

Full size image

Therefore, moisture concentration and nonlocality have a softening impact on the beam structure and should be considered in the analysis of size-dependent FG nanobeams. Figure 3 shows the dimensionless frequency of S–S higher-order FG nanobeam as function of various temperature rises for different values of moisture concentration at p = 1, L/h = 20, K w  = K p  = 0 and µ = 2 (nm)2. It is known that beam structures may buckle with the rise of temperature which creates compressive axial loads. Therefore, for all hygro-thermal loadings with the temperature increment, the natural frequency of FG nanobeam reaches to zero nearby the critical temperature point. This feature refers to stiffness degradation of nanobeam when the temperature grows. After the branching point, the increase in temperature yields larger values of natural frequency. Moreover, it is seen that moisture concentration has a significant impact on the pre-/postbuckling configuration of FG nanobeam under hygro-thermal loads.

Fig. 3
figure 3

Influence of moisture concentration on the dimensionless frequency of the S–S FG nanobeam with respect to various temperature rises (\( p = 1,\;L/h = 20 \), K w  = K p  = 0)

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Therefore, increasing the value of moisture concentration leads to lower values of the critical buckling temperature, especially in the case of uniform temperature rise. So, imposing a linear or sinusoidal hygro-thermal loading can delay the critical buckling temperature.

Figure 4 indicates the influence of elastic foundation on pre-/post-buckling vibrational behavior of FG nanobeam versus various temperature rises under thermal ΔC = 0 and hygro-thermal ΔC = 2 loadings when p = 1, L/h = 20 and µ = 2 (nm)2. It is found that an increase in Winkler and Pasternak parameters leads to a remarkable postponement in critical buckling temperature. Also, with the presence of moisture or humidity, the critical temperature has shifted to the left which highlights destructive influence of hygroscopic condition on the beam structure.

Fig. 4
figure 4

Influence of elastic foundation on the dimensionless frequency of the S–S FG nanobeam with respect to temperature change for thermal, ΔC = 0 and hygro-thermal, ΔC = 2 environments (\( p = 1,\;L/h = 20 \), µ = 2 (nm)2)

Full size image

Figure 5 illustrates the variation of natural frequency with respect to gradient index for various thermal and hygro-thermal loadings with and without elastic foundation at L/h = 20 and µ = 2 (nm)2. It is observable that under any type of environmental conditions, the natural frequency reduces with the rise in gradient index, prominently for lower gradient indexes. Moreover, for all values of gradient index, existence of elastic foundation enhances the beam structure and increases the dimensionless frequency. Also, it is found that the effect of moisture concentration on the frequency responses of FG nanobeams is more significant for larger values of gradient index. This is due to the fact that lower values of gradient index are correspond to more portion of the ceramic phase which has a moisture expansion coefficient equal to zero (\( \beta_{c} = 0 \)).

Fig. 5
figure 5

Influence of material graduation on the dimensionless frequency of the S–S FG nanobeam for thermal, ΔC = 0 and hygro-thermal ΔC = 2 environments (\( L/h = 20 \), µ = 2 (nm)2, ΔT = 40 (K))

Full size image

The influence of slenderness ratio on the natural frequencies of FG nanobeams for various types of thermal and hygro-thermal loadings with and without elastic foundation at p = 1 and µ = 2 (nm)2 is demonstrated in Fig. 6. For all environmental conditions, the dimensionless frequency increases for lower slenderness ratios and then reduces for higher slenderness ratios which indicates the significance of shear deformation when the beam thickness is large. Also, the influence of moisture concentration on natural frequencies is negligible at lower slenderness ratios so that it becomes outstanding for larger values of slenderness ratio. So, thinner nanobeams are more affected by the hygro-thermal loadings.

Fig. 6
figure 6

Influence of slenderness ratio on the dimensionless frequency of the FG nanobeam for various moisture rises (\( p = 1 \), µ = 2 (nm)2)

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The variation of the dimensionless frequency of S–S higher-order FG nanobeam with respect to Winkler and Pasternak parameters for different uniform and linear moisture concentrations is presented in Fig. 7, at p = 1, L/h = 20 and µ = 2 (nm)2.

Fig. 7
figure 7

Influence of elastic foundation parameters on the dimensionless frequency of the FG nanobeam for various moisture rises (\( p = 1 \), µ = 2 (nm)2, ΔT = 40 (K))

Full size image

In this figure it is seen that with increase in the Winkler and Pasternak parameters, dimensionless frequency increases for all values of moisture concentration. Also, it is found that the influence of the Pasternak parameter (\( K_{p} \)) on the nondimensional frequency is more prominent than that of the Winkler parameter (\( K_{w} \)). So, it is very important to regard the shear layer of an elastic foundation in the analysis of FG nanostructures.

6 Conclusions

In this paper, hygro-thermomechanical free vibration analysis of embedded functionally graded size-dependent nanobeams exposed to various hygro-thermal loadings is performed by using different higher-order beam theories. Three kinds of environmental conditions namely, uniform, linear, and sinusoidal hygro-thermal loadings are investigated.

Temperature-dependent material properties of nonlocal FG beam change gradually according to the power-law distribution. Size-dependent description of nanobeam is conducted using nonlocal elasticity theory of Eringen. Applying Navier solution the nonlocal coupled governing equations obtained from Hamilton’s principle are solved. Finally the impacts of moisture concentration, temperature rise, shear deformation, nonlocal parameter, elastic foundation, material composition, and slenderness ratio on the vibrational characteristics of nanosize FG beams are explored. As a general consequence, all of the higher-order theories provide accurate and approximately same results for the hygro-thermal vibrational behavior of FG nanobeams. The results of higher-order theories are smaller than of classical beam theory since they consider the shear deformation effect. Also, it is seen that the influence of moisture or humidity is significant for higher values of gradient index and slenderness ratio. Moreover, it is found that at a prescribed environmental condition, nonlocality and gradient index have a notable decreasing effect on the natural frequency of FG nanobeams. Also, various hygro-thermal loadings estimate different values of natural frequency and critical temperature so that uniform and sinusoidal hygro-thermal environments produce smallest and largest amount of dimensionless frequency, respectively. Also, it is indicated that with an increase in Winkler or Pasternak parameter, the beam becomes more rigid, and the dimensionless frequency of FG nanobeams enlarges.