A layer-wise theory for the dynamic analysis of functionally graded composite nanobeams under thermal environment using nonlocal boundary conditions and Chebyshev collocation technique

Abstract

In this study, a layer-wise theory satisfying displacement continuity at each layer’s interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen’s nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method.

Keywords

Composite nanobeams functionally graded layers layer-wise theory Eringen’s nonlocal theory nonlocal boundary conditions thermal stresses Chebyshev collocation technique

1. Introduction

In recent years, consistent efforts are going on to reduce the weight of the structures. The popularity of lightweight structures has necessitated the development of high-strength composite materials with low density because of their adaptability to different situations to serve specific purposes and exhibit desirable properties. Keeping the above in view, fiber-reinforced materials gained wide popularity in the design of structural components which are used in various technological situations such as diaphragms used in pressure capsules, aircraft fuselages (Jones, 1975), and beams fabricated out of modern composites, to mention a few. Composite materials such as boron-epoxy, glass-epoxy, Kevlar, graphite, etc. are fabricated to meet the desirability of lightweight, high strength, corrosion resistance, and high-temperature performance. In this regard, several studies dealing with the static and dynamic behavior of composite/laminated beams with various constraints have appeared in the literature, and the imported ones are reported here. Ahmed (1972) developed a finite element technique to study the effect of transverse shear deformation on the natural frequencies of honeycomb sandwich beams. However, the higher-order shear beam theory (HOT) was employed by Rao and Ganesan (1995) to examine the responses of tapered composite beams using finite element formulation. A state-space based differential quadrature method (DQM) was introduced by Chen et al. (2004) to illustrate the vibration behavior of generally composite beams. Lezgy-Nazargah et al. (2011) developed a refined HOT for the vibration analyses of composite/laminated beams using a C¹-continuous shear lock-free finite element model. Recently, the dynamic stiffness method was employed by Damanpack and Khalili (2012) to present the free vibration analysis of a three-layered sandwich symmetric beam based on HOT. Giunta et al. (2013) presented the Navier type closed-form solution for the stability and free vibration analyses of sandwich beams using classical and first-order shear beam theories. The free vibration behavior of soft-core sandwich beams was discussed by Wang and Wang (2016) employing the quadrature element method. The natural frequencies and mode shapes of sandwich beams with honeycomb-corrugation hybrid cores were predicted by Zhang et al. (2017) using the finite element method. Yan et al. (2017) carried out the closed-form solutions for the free vibration analysis of laminated beam on the basis of Carrera Unified Formulation and layer-wise theory. Hui et al. (2017) adopted the hierarchical one-dimensional finite elements and analyzed the vibration behavior of a three-dimensional sandwich beam. A C⁰ finite element beam model was proposed by Chalak et al. (2012) to illustrate the responses of a sandwich beam using higher-order zigzag theory. Employing finite element formulation, Chanthanumataporn and Watanabe (2018) investigated the effect of damping due to ambient air on the vibration characteristics of light sandwich beam considering shear deformation for the core. Wang and Zhao (2019) adopted the Timoshenko beam theory and Chebyshev collocation technique to present the free vibration analysis of a sandwich beam with a metal foam core and resting on the Winkler-Pasternak foundation. Thermally-induced transverse vibrations of sandwich beams were studied by Chen et al. (2019) considering non-uniform cross-section properties. Shu et al. (2021) analyzed the vibration characteristics of elastic metamaterial sandwich beam using multi-physics software COMSOL.

Despite the above merits, a mismatch of mechanical/thermal properties exists at layer interfaces in laminated structures, often making them sustain delamination, debonding, and cracks, especially under a high operating environment. The aforementioned drawbacks experienced by conventional composite material layers can be avoided by replacing them with a new kind of material, that is, functionally graded materials (FGMs), where these problems can be addressed in an optimum/efficient manner. These advanced materials are manufactured with the composition of two different conventional materials at the microscopic level, which leads to continuous change in their properties from one surface to the other ones. The recent contributions made on FG sandwich beams are reported here, to mention a few. Yang et al. (2014) presented the free vibration analysis of FG sandwich beams using a two-dimensional elasticity theory and meshfree boundary-domain integral equation method. The implicit time integration Newmark’s method was used by Şimşek and Al-Shujairi (2017) to study the static, free, and forced vibrations of FG sandwich beam under double moving harmonic loads on the basis of Timoshenko beam theory. Using Euler-Bernoulli beam theory for bending and Vlasov theory for torsion, Kim and Lee (2017) investigated the spatially coupled vibration behavior of thin-walled FG sandwich beams employing the finite element method. Yildirim (2020) presented the free vibration analysis of FG sandwich beams using the complementary functions method and two-dimensional elasticity theory. The static and free vibration behavior of the FG sandwich beam was analyzed by Koutoati et al. (2021) using the finite element method on the basis of Timoshenko first-order theory and higher-order theories. In this view, a critical review of the work on FG sandwich structures is presented by Garg et al. (2021).

The rapid development of the micro-electro-mechanical system (MEMS) and nano-electro-mechanical system (NEMS) led* to the extensive use of beam-type nano components in electrical, civil, mechanical, aerospace industries, etc. (George, 2002; Ho et al., 1997; Lyshevski, 2002). Due to the advantage of lightweight, high stiffness, improved functionality, and durability, researchers throughout the world have been attracted to study their static/dynamic behavior with a fair amount of accuracy using analytical/numerical methods and different theories. Among the recent ones, Nazemnezhad et al. (2014) employed harmonic DQM to present the free vibration analysis of bilayer graphene nanoribbons with interlayer shear effect. The free vibration characteristics of the FG nano sandwich beam embedded on the Pasternak foundation were examined by Asemi et al. (2014) using Timoshenko beam theory and generalized DQM. A finite element model was developed by Subramani et al. (2017) to analyze the free vibration behavior of carbon fiber reinforced laminated sandwich beam on the basis of HOT. Kamali and Nazemnezhad (2018) employed harmonic DQM and examined the nonlocal free vibrations of bilayer graphene nanoribbons. They investigated the interlayer shear and tensile-compressive van der Waals effect on the first four natural frequencies for various boundary conditions. Kim (2019) developed an evaluation technique to predict the first and second order-vibration modes of a sandwich microcantilever beam on the basis of Euler-Bernoulli beam theory.

In recent years, several studies have been conducted to show the effect of thermal stresses on the static and dynamic behavior of sandwich beams. Pradhan and Murmu (2009) presented the thermomechanical vibration analysis of the FG sandwich beam resting on variable Winkler and two-parameter foundation employing modified DQM. Mirzaei and Kiani (2016) presented the nonlinear vibration analysis of temperature-dependent FG sandwich beam with carbon nanotubes (CNTs) as face sheets using equivalent single-layer Timoshenko theory along with von Karman type non-linearity and Ritz formulation. A higher-order shear deformation theory was proposed by Ebrahimi and Farazmandnia (2017) for the thermomechanical vibration analysis of FG CNTs sandwich beams employing Navier’s solution procedure. Li et al. (2019) examined the nonlinear flexural vibrations of FG sandwich beams with negative Poisson’s ratio Honeycomb core using the finite element method. A refined 4-unknown quasi-3D zigzag beam theory was developed by Han et al. (2018) for the stability and free vibration analysis of composite beams under axial mechanical and transverse thermal loading using the Ritz method.

The previous researches on nanostructures are widely focused on the clamped (C) and simply-supported (S) boundaries due to contradiction in results for cantilever nanobeams (Ansari et al., 2018; Lal and Dangi, 2019; Lu et al., 2006; Norouzzadeh et al., 2018; Norouzzadeh and Ansari, 2017; Wang et al., 2007). These studies found that with the increase in nonlocal parameter, the stiffness and frequencies of the nanobeams decreases for all boundary conditions except for cantilever nanobeams. This is due to the use of local boundary conditions rather than nonlocal ones which is further discussed and modified by a few researchers for isotropic and homogeneous nanobeams (Li et al., 2020; Mirzaei and Kiani, 2016; Norouzzadeh and Ansari, 2017; Panc, 1975; Paradoen, 1977).

Several numerical techniques were adopted and developed by researchers based on the characteristics of static and dynamic problems such as linear/nonlinear equations, regular/irregular geometries, implementation processes, etc. Accordingly, these techniques have their advantages and disadvantages. For linear differential equations with regular domains, spectral methods (Chai and Wang, 2022; Wang et al., 2018, 2019; Ye and Wang, 2021) are easy to implement and provide better accuracy with less computational cost while these can’t be used for nonlinear differential equations. So, the mixed type of numerical techniques are useful but for irregular domains, different versions of the finite element method (Wang et al., 2021) are the best choice to study the characteristics of the structural element. Spectral method, being the linear type of differential equations, is the best choice and hence Chebyshev collocation technique is adopted by the authors’ due to its advantage of better accuracy and choice of grid distribution which is always fixed. To the best of authors’ knowledge, Chebyshev collocation method is first reported by Gupta and Lal (1979) to study the axisymmetric vibrations of linearly tapered annular plates under in-plane force.

During the above survey, it is observed that most of the researchers used conventional beam theories to analyze the vibrational behavior of laminated/composite/sandwich beams. However, two-dimensional elasticity theory (Setoodeh et al., 2012; Yildirim, 2020), layer-wise Euler-Bernoulli theory (Kamali and Nazemnezhad, 2018; Nazemnezhad et al., 2014), and refined four-unknown quasi-3D zigzag beam theory (Han et al., 2018) obtained the displacements of each layer independently. Setoodeh et al. (2012) investigated the free vibration behavior of elastically supported sandwich beams under thermal environment by adopting a two-dimensional elasticity theory together with layer-wise theory and DQM. According to this theory, the displacement field of each layer is assumed independently by avoiding their mutual impact. Similarly, Xiang and Yang (2008) presented the free and forced vibration analysis of thermally-induced non-uniform laminated FG Timoshenko beam by employing DQM. Considering these, the present work includes the effect of the displacement field of every layer on the others and the layer-wise theory developed by Ferreira et al. (2008) is adopted to study the effect of thermal environment on the vibration characteristics of the temperature-dependent FG composite nanobeams. The material properties of the nanobeams are varied as a power-law function across the thickness. The size-dependency of Type 1 and Type 2 composite nanobeams is analyzed by developing nonlocal boundary conditions for Eringen’s nonlocal elasticity theory. Hamilton’s principle is used to derive the governing equation of motion for such a model. The resulting equations are discretized by the Chebyshev method to obtain the frequency equations for clamped: clamped, clamped: simply-supported, simply-supported: simply-supported, and clamped: free composite FG nanobeams, and solved by MATLAB. The effects of various parameters such as volume fraction index, layer thickness ratio, nonlocal parameter, and temperature difference on the fundamental frequencies of the beam are analyzed. Comparison of results with published results obtained from other theories/approximate methods is presented for some special cases.

2. Theoretical formulation

Consider a functionally graded sandwich nanobeam in a Cartesian coordinate system with length $L$ , and $h_{1}$ , $h_{2}$ , and $h_{3}$ , as thicknesses for layers one, two, and three, respectively. $z = 0$ be the middle plane of the nanobeam. Both types of configurations for composite FG nanobeams are shown in Figure 1. For Type 1, the facings of the nanobeam are made of functionally graded materials while the core is metallic. However, for Type 2, the core is made of functionally graded material, the upper and lower facings are ceramic and metal, respectively.

Figure 1.

Composite beam: type 1: homogeneous core and functionally graded faces; type 2: functionally graded core and homogeneous faces.

The FG material is a classic ceramic-metal composition in the thickness direction of the beam. Following the rule of mixture, the temperature-dependent mechanical properties of FG material are defined as (Lal and Saini, 2019):

P (z, T) = {P_{c} (T) - P_{m} (T)} V_{c} + P_{m} (T)

(1)

where

P_{c} (T)

and

P_{m} (T)

denotes the mechanical properties for ceramic and metal, respectively;

T

represents the temperature and;

V_{c}

being the volume fraction of ceramic constituent at any point

z

in the structure which has different representation for both type of composite construction (Pandey and Pradyumna, 2015), that is,

(i) Type 1:

V_{c} (z) = {\begin{cases} {(\frac{z - a_{1}}{a_{0} - a_{1}})}^{n} z \in [\begin{array}{l} a_{0}, & a_{1} \end{array}] \\ 0 z \in [\begin{array}{l} a_{1}, & a_{2} \end{array}] \\ {(\frac{z - a_{2}}{a_{3} - a_{2}})}^{n} z \in [\begin{array}{l} a_{2}, & a_{3} \end{array}] \end{cases}

(2a)

(ii) Type 2:

V_{c} (z) = {\begin{cases} 0 z \in [\begin{array}{l} a_{0}, & a_{1} \end{array}] \\ {(\frac{z - a_{1}}{a_{2} - a_{1}})}^{n} z \in [\begin{array}{l} a_{1}, & a_{2} \end{array}] \\ 1 z \in [\begin{array}{l} a_{2}, & a_{3} \end{array}] \end{cases}

(2b)

Here, the parameter

n

(

\geq

0) controls the material distribution along the thickness of the beam and

a_{0} = - (h_{1} + h_{2} / 2)

a_{1} = - h_{2} / 2

a_{2} = h_{2} / 2

, and

a_{3} = (h_{3} + h_{2} / 2)

The temperature-dependent mechanical properties (Young’s modulus

E

, mass density

ρ

, thermal conductivity

k

, thermal expansion coefficient

α

) of the constituents are given as:

P_{i} (T) = P_{0} (P_{- 1} T^{- 1} + 1 + P_{1} T^{1} + P_{2} T^{2} + P_{3} T^{3}), i = m, c

(3)

where the experimental values of

P_{j}

(j = −1, 0, 1, 2, 3) for ceramic and metal are listed in Table 1.

Table 1.

Mechanical properties for ceramic and metal.

	Material	$P_{0}$	$P_{1}$	$P_{2}$	$P_{3}$
E (Pa)	SUS304	201.04 × 10⁹	3.079 × 10⁻⁴	−6.534 × 10⁻⁷	0
E (Pa)	Si₃N₄	348.43 × 10⁹	−3.07 × 10⁻⁴	2.160 × 10⁻⁷	−8.946 × 10⁻¹¹
ρ (kg/m³)	SUS304	8166	0	0	0
ρ (kg/m³)	Si₃N₄	2370	0	0	0
α (1/K)	SUS304	12.33 × 10⁻⁶	8.086 × 10⁻⁴	0	0
α (1/K)	Si₃N₄	5.8723 × 10⁻⁶	9.095 × 10⁻⁴	0	0
k (W/mK)	SUS304	12.04	0	0	0
k (W/mK)	Si₃N₄	9.19	0	0	0

As the material is distributed asymmetrically about the mid-plane of nanobeam, the physical neutral plane would be different from the mid-plane of the beam. According to the elastic mechanics theory for vibration analysis of FG structures, the mathematical condition for the physical neutral plane is defined as,

\int_{a_{0}}^{a_{1}} (z - z_{0}) E^{(1)} (z, T) d z + \int_{a_{1}}^{a_{2}} (z - z_{0}) E^{(2)} (z, T) d z + \int_{a_{2}}^{a_{3}} (z - z_{0}) E^{(1)} (z, T) d z = 0

(4)

where

z_{0}

is the distance of the physical neutral plane from the mid-plane of the nanobeam.

The value of $z_{0}$ is obtained by solving equation (4), gives

z_{0} = \frac{\int_{a_{0}}^{a_{1}} z E^{(1)} (z, T) d z + \int_{a_{1}}^{a_{2}} z E^{(2)} (z, T) d z + \int_{a_{2}}^{a_{3}} z E^{(1)} (z, T) d z}{\int_{a_{0}}^{a_{1}} E^{(1)} (z, T) d z + \int_{a_{1}}^{a_{2}} E^{(2)} (z, T) d z + \int_{a_{2}}^{a_{3}} E^{(1)} (z, T) d z}

2.1. Thermal analysis

The temperature profile for the present FG composite nanobeams is obtained by solving the following one-dimensional steady state heat conduction equation:

\frac{d}{d z} (k (z) \frac{d T}{d z}) = 0

(5)

with thermal boundary conditions

T = T_{u}

and

T = T_{l}

z = a_{3}

and

z = a_{0}

, respectively,

k (z) = {\begin{cases} k^{(1)} z \in [\begin{array}{l} a_{0}, & a_{1} \end{array}] \\ k^{(2)} z \in [\begin{array}{l} a_{1}, & a_{2} \end{array}] \\ k^{(3)} z \in [\begin{array}{l} a_{2}, & a_{3} \end{array}] \end{cases} and T (z) = {\begin{cases} T^{(1)} z \in [\begin{array}{l} a_{0}, & a_{1} \end{array}] \\ T^{(2)} z \in [\begin{array}{l} a_{1}, & a_{2} \end{array}] \\ T^{(3)} z \in [\begin{array}{l} a_{2}, & a_{3} \end{array}] \end{cases}

(6)

where

k^{(i)}

and

T^{(i)}

are the thermal conductivity and temperature variations for

i^{t h}

layer (i = 1, 2, 3), respectively. However, the continuity conditions on temperature variations at interface layers are defined as:

T^{(1)} (a_{1}) = T^{(2)} (a_{1}), T^{(2)} (a_{2}) = T^{(3)} (a_{2})

(7a)

{k^{(1)} \frac{d T^{(1)}}{d z}}_{z = a_{1}} = {k^{(2)} \frac{d T^{(2)}}{d z}}_{z = a_{1}}, {k^{(2)} \frac{d T^{(2)}}{d z}}_{z = a_{2}} = {k^{(3)} \frac{d T^{(3)}}{d z}}_{z = a_{2}}

(7b)

Using the above conditions, the expressions for temperature variation of each layer are obtained for Type 1 and Type 2 FG composite nanobeams, given as:

Type 1:

T^{(1)} (z) = T_{m 1} + (T_{l} - T_{m 1}) A (z), T^{(3)} (z) = T_{m 2} + (T_{u} - T_{m 2}) \hat{A} (z),

T^{(2)} (z) = [(T_{m 2} - T_{m 1}) z - a_{1} T_{m 2} + a_{2} T_{m 1}] / (a_{2} - a_{1})

where

A (z) = \frac{1}{C} [\frac{z - a_{1}}{a_{0} - a_{1}} - \frac{k_{c m}}{(n + 1)} {(\frac{z - a_{1}}{a_{0} - a_{1}})}^{n + 1} + \frac{k_{c m}^{2}}{(2 n + 1)} {(\frac{z - a_{1}}{a_{0} - a_{1}})}^{2 n + 1} \dots + \frac{k_{c m}^{4}}{(4 n + 1)} {(\frac{z - a_{1}}{a_{0} - a_{1}})}^{4 n + 1}]

\hat{A} (z) = \frac{1}{C} [\frac{z - a_{2}}{a_{3} - a_{2}} - \frac{k_{c m}}{(n + 1)} {(\frac{z - a_{2}}{a_{3} - a_{2}})}^{n + 1} + \frac{k_{c m}^{2}}{(2 n + 1)} {(\frac{z - a_{2}}{a_{3} - a_{2}})}^{2 n + 1} \dots + \frac{k_{c m}^{4}}{(4 n + 1)} {(\frac{z - a_{2}}{a_{3} - a_{2}})}^{4 n + 1}]

C = 1 - \frac{k_{c m}}{(n + 1)} + \frac{k_{c m}^{2}}{(2 n + 1)} - \frac{k_{c m}^{3}}{(3 n + 1)} + \frac{k_{c m}^{4}}{(4 n + 1)},

T_{m 1} = \frac{[T_{u} h_{1} C + T_{l} (h_{3} C + h_{2})]}{[C (h_{1} + h_{3}) + h_{2}]}, T_{m 2} = \frac{[T_{l} h_{3} C + T_{u} (h_{1} C + h_{2})]}{[C (h_{1} + h_{3}) + h_{2}]}, k_{c m} = \frac{(k_{c} - k_{m})}{k_{m}}

Type 2:

T^{(1)} = \frac{[(T_{m 1} - T_{l}) z + a_{1} T_{l} - a_{0} T_{m 1}]}{h_{1}}, T^{(3)} = \frac{[(T_{u} - T_{m 2}) z + a_{3} T_{m 2} - a_{2} T_{u}]}{h_{3}},

T^{(2)} = T_{m 1} + (T_{m 2} - T_{m 1}) A (z)

A (z) = \frac{1}{C} [\frac{z - a_{1}}{a_{2} - a_{1}} - \frac{k_{c m}}{(n + 1)} {(\frac{z - a_{1}}{a_{2} - a_{1}})}^{n + 1} + \frac{k_{c m}^{2}}{(2 n + 1)} {(\frac{z - a_{1}}{a_{2} - a_{1}})}^{2 n + 1} \dots + \frac{k_{c m}^{4}}{(4 n + 1)} {(\frac{z - a_{1}}{a_{2} - a_{1}})}^{4 n + 1}]

T_{m 1} = \frac{[T_{u} k_{m} k^{(3)} h_{1} + T_{l} k^{(1)} (k_{c} h_{3} G + k^{(3)} h_{2} C)]}{[k^{(3)} (k_{m} h_{1} + k^{(1)} h_{2} C) + G k^{(1)} k_{c} h_{3}]}, T_{m 2} = \frac{[T_{l} k_{c} k^{(1)} h_{3} G + T_{u} k^{(3)} (k_{m} h_{1} + k^{(1)} h_{2} C)]}{[k^{(3)} (k_{m} h_{1} + k^{(1)} h_{2} C) + G k^{(1)} k_{c} h_{3}]},

C = 1 - k_{c m} + k_{c m}^{2} - k_{c m}^{3} + k_{c m}^{4}

2.2. Euler-Lagrange equations of FG composite nanobeam

Following the kinematics of a three-layered FG composite nanobeam with first-order shear deformation theory for each layer, the in-plane displacements $u^{(i)}$ and transverse displacements $w^{(i)}$ for center and top layer are given by (Pandey and Pradyumna, 2015):

u^{(2)} (x, z, t) = u_{0} (x, t) + (z^{(2)} - z_{0}) φ^{(2)} (x, t), w^{(2)} (x, z, t) = w_{0} (x, t),

u^{(3)} (x, z, t) = u_{0} (x, t) + (z^{(3)} - z_{0}) φ^{(3)} (x, t) + (h_{2} / 2) φ^{(2)} (x, t) + (h_{3} / 2) φ^{(3)} (x, t), w^{(3)} (x, z, t) = w_{0} (x, t)

where

φ^{(i)}

is the rotation of mid-plane’s normal about the

x

-axis;

u_{0}

and

w_{0}

are the displacements of the mid-plane.

The strains for the center and top layers of the composite nanobeam are given by,

ε_{x x}^{(2)} = \frac{\partial u_{0}}{\partial x} + (z^{(2)} - z_{0}) \frac{\partial φ^{(2)}}{\partial x}, γ_{x z}^{(2)} = \frac{\partial w_{0}}{\partial x} + φ^{(2)},

ε_{x x}^{(3)} = \frac{\partial u_{0}}{\partial x} + (z^{(3)} - z_{0}) \frac{\partial φ^{(3)}}{\partial x} + \frac{h_{2}}{2} \frac{\partial φ^{(2)}}{\partial x} + \frac{h_{3}}{2} \frac{\partial φ^{(3)}}{\partial x}, γ_{x z}^{(3)} = \frac{\partial w_{0}}{\partial x} + φ^{(3)},

Following the stress-strain relationship for a beam, the stresses of the center and top layers can be obtained in terms of displacements as:

σ_{x x}^{(2)} = E^{(2)} (z, T) {\frac{\partial u_{0}}{\partial x} + (z^{(2)} - z_{0}) \frac{\partial φ^{(2)}}{\partial x}}, σ_{x z}^{(2)} = G^{(2)} (z, T) (\frac{\partial w_{0}}{\partial x} + φ^{(2)}),

σ_{x x}^{(3)} = E^{(3)} (z, T) {\frac{\partial u_{0}}{\partial x} + (z^{(3)} - z_{0}) \frac{\partial φ^{(3)}}{\partial x} + \frac{h_{2}}{2} \frac{\partial φ^{(2)}}{\partial x} + \frac{h_{3}}{2} \frac{\partial φ^{(3)}}{\partial x}}, σ_{x z}^{(3)} = G^{(3)} (z, T) (\frac{\partial w_{0}}{\partial x} + φ^{(3)})

For the sake of brevity, the displacements, strains, and stresses of the bottom layer have not been mentioned as they can be obtained from displacements, strains, and stresses of top layer by replacing $φ^{(3)}$ , $z^{(3)}$ , $h_{2}$ , and $h_{3}$ by $φ^{(1)}$ , $z^{(1)}$ , $- h_{2}$ , and $- h_{1}$ , respectively. The normal stress, shear stress, and moment resultants of each layer are given by:

[\begin{array}{l} N_{x x}^{(1)} \\ M_{x x}^{(1)} \end{array}] = \int_{a_{0}}^{a_{1}} σ_{x x}^{(1)} [\begin{array}{l} 1 \\ (z^{(1)} - z_{0}) \end{array}] d z, [\begin{array}{l} N_{x x}^{(2)} \\ M_{x x}^{(2)} \end{array}] = \int_{a_{1}}^{a_{2}} σ_{x x}^{(2)} [\begin{array}{l} 1 \\ (z^{(2)} - z_{0}) \end{array}] d z, [\begin{array}{l} N_{x x}^{(3)} \\ M_{x x}^{(3)} \end{array}] = \int_{a_{2}}^{a_{3}} σ_{x x}^{(1)} [\begin{array}{l} 1 \\ (z^{(3)} - z_{0}) \end{array}] d z,

Q_{x z}^{(1)} = \int_{a_{0}}^{a_{1}} σ_{x z}^{(1)} d z, Q_{x z}^{(2)} = \int_{a_{1}}^{a_{2}} σ_{x z}^{(2)} d z, Q_{x z}^{(3)} = \int_{a_{2}}^{a_{3}} σ_{x z}^{(3)} d z

Following Reddy (2003, 2006), the variations in the total strain and kinetic energy of the composite nanobeam are:

δ U = \int_{0}^{L} [\int_{a_{0}}^{a_{1}} (σ_{x x}^{(1)} δ ε_{x x}^{(1)} + σ_{x z}^{(1)} δ γ_{x z}^{(1)}) d z + \int_{a_{1}}^{a_{2}} (σ_{x x}^{(2)} δ ε_{x x}^{(2)} + σ_{x z}^{(2)} δ γ_{x z}^{(2)}) d z + \int_{a_{2}}^{a_{3}} (σ_{x x}^{(3)} δ ε_{x x}^{(3)} + σ_{x z}^{(3)} δ γ_{x z}^{(3)}) d z] d x

= \int_{0}^{L} [\begin{array}{l} (N_{x x}^{(1)} + N_{x x}^{(2)} + N_{x x}^{(3)}) \frac{\partial δ u_{0}}{\partial x} + (M_{x x}^{(1)} - \frac{h_{1}}{2} N_{x x}^{(1)}) \frac{\partial δ φ^{(1)}}{\partial x} + (M_{x x}^{(2)} + \frac{h_{2}}{2} (N_{x x}^{(3)} - N_{x x}^{(1)})) \frac{\partial δ φ^{(2)}}{\partial x} + \\ (M_{x x}^{(3)} + \frac{h_{3}}{2} N_{x x}^{(3)}) \frac{\partial δ φ^{(3)}}{\partial x} + (Q_{x z}^{(1)} + Q_{x z}^{(2)} + Q_{x z}^{(3)}) \frac{\partial δ w_{0}}{\partial x} + Q_{x z}^{(1)} δ φ^{(1)} + Q_{x z}^{(2)} δ φ^{(2)} + Q_{x z}^{(3)} δ φ^{(3)} \end{array}] d x

(8)

δ V = \int_{0}^{L} {\begin{cases} \int_{a_{0}}^{a_{1}} ρ^{(1)} (\frac{\partial u^{(1)}}{\partial t} \frac{\partial δ u^{(1)}}{\partial t} + \frac{\partial w^{(1)}}{\partial t} \frac{\partial δ w^{(1)}}{\partial t}) d z + \int_{a_{1}}^{a_{2}} ρ^{(2)} (\frac{\partial u^{(2)}}{\partial t} \frac{\partial δ u^{(2)}}{\partial t} + \frac{\partial w^{(2)}}{\partial t} \frac{\partial δ w^{(2)}}{\partial t}) d z \\ + \int_{a_{2}}^{a_{3}} ρ^{(3)} (\frac{\partial u^{(3)}}{\partial t} \frac{\partial δ u^{(3)}}{\partial t} + \frac{\partial w^{(3)}}{\partial t} \frac{\partial δ w^{(3)}}{\partial t}) d z \end{cases}} d x

= \int_{0}^{L} {\begin{cases} \begin{array}{l} [\begin{array}{l} (I_{0}^{(1)} + I_{0}^{(2)} + I_{0}^{(3)}) \frac{\partial u_{0}}{\partial t} + (I_{1}^{(1)} - \frac{h_{1}}{2} I_{0}^{(1)}) \frac{\partial φ^{(1)}}{\partial t} + \\ {I_{1}^{(2)} - \frac{h_{2}}{2} (I_{0}^{(1)} - I_{0}^{(3)}) \frac{\partial φ^{(2)}}{\partial t}} \frac{\partial φ^{(2)}}{\partial t} + (I_{1}^{(3)} + \frac{h_{3}}{2} I_{0}^{(3)}) \frac{\partial φ^{(3)}}{\partial t} \end{array}] \frac{\partial δ u_{0}}{\partial t} + \\ [\begin{array}{l} {I_{1}^{(2)} + \frac{h_{2}}{2} (I_{0}^{(3)} - I_{0}^{(1)})} \frac{\partial u_{0}}{\partial t} + \frac{h_{2}}{2} (\frac{I_{0}^{(1)} h_{1}}{2} - I_{1}^{(1)}) \frac{\partial φ^{(1)}}{\partial t} + \\ {I_{2}^{(2)} + \frac{h_{2}^{2}}{4} (I_{0}^{(1)} + I_{0}^{(3)})} \frac{\partial φ^{(2)}}{\partial t} + \frac{h_{2}}{2} (I_{1}^{(3)} + \frac{I_{0}^{(3)} h_{3}}{2}) \frac{\partial φ^{(3)}}{\partial t} \end{array}] \frac{\partial δ φ^{(2)}}{\partial t} + \\ [(I_{1}^{(1)} - \frac{I_{0}^{(1)} h_{1}}{2}) \frac{\partial u_{0}}{\partial t} + (I_{2}^{(1)} - I_{1}^{(1)} h_{1} + \frac{I_{0}^{(1)} h_{1}^{2}}{4}) \frac{\partial φ^{(1)}}{\partial t} + \frac{h_{2}}{2} (\frac{I_{0}^{(1)} h_{1}}{2} - I_{1}^{(1)}) \frac{\partial φ^{(2)}}{\partial t}] \frac{\partial δ φ^{(1)}}{\partial t} \\ + [(I_{1}^{(3)} + \frac{I_{0}^{(3)} h_{3}}{2}) \frac{\partial u_{0}}{\partial t} + \frac{h_{2}}{2} (\frac{I_{0}^{(3)} h_{3}}{2} + I_{1}^{(3)}) \frac{\partial φ^{(2)}}{\partial t} + (I_{2}^{(3)} + I_{1}^{(3)} h_{3} + \frac{I_{0}^{(3)} h_{3}^{2}}{4}) \frac{\partial φ^{(3)}}{\partial t}] \frac{\partial δ φ^{(3)}}{\partial t} \end{array} \\ + [(I_{0}^{(1)} + I_{0}^{(2)} + I_{0}^{(3)}) \frac{\partial w_{0}}{\partial t}] \frac{\partial δ w_{0}}{\partial t} \end{cases}} d x

(9)

where

[\begin{array}{l} I_{0}^{(2)} \\ I_{1}^{(2)} \\ I_{2}^{(2)} \end{array}] = \int_{a_{1}}^{a_{2}} ρ^{(2)} [\begin{array}{l} 1 \\ (z^{(2)} - z_{0}) \\ {(z^{(2)} - z_{0})}^{2} \end{array}] d z, [\begin{array}{l} I_{0}^{(1)} \\ I_{1}^{(1)} \\ I_{2}^{(1)} \end{array}] = \int_{a_{0}}^{a_{1}} ρ^{(1)} [\begin{array}{l} 1 \\ (z^{(1)} - z_{0}) \\ {(z^{(1)} - z_{0})}^{2} \end{array}] d z, [\begin{array}{l} I_{0}^{(3)} \\ I_{1}^{(3)} \\ I_{2}^{(3)} \end{array}] = \int_{a_{2}}^{a_{3}} ρ^{(3)} [\begin{array}{l} 1 \\ (z^{(1)} - z_{0}) \\ {(z^{(1)} - z_{0})}^{2} \end{array}] d z

The total work done by the thermal stresses is expressed as:

δ W_{T} = \int_{0}^{L} {\begin{cases} \int_{a_{0}}^{a_{1}} E^{(1)} (z, T) α^{(1)} (z, T) (T^{(1)} - T_{0}) \frac{\partial w_{0}}{\partial x} \frac{\partial δ w_{0}}{\partial x} d z + \\ \int_{a_{1}}^{a_{2}} E^{(2)} (z, T) α^{(2)} (z, T) (T^{(2)} - T_{0}) \frac{\partial w_{0}}{\partial x} \frac{\partial δ w_{0}}{\partial x} d z + \\ \int_{a_{2}}^{a_{3}} E^{(3)} (z, T) α^{(3)} (z, T) (T^{(3)} - T_{0}) \frac{\partial w_{0}}{\partial x} \frac{\partial δ w_{0}}{\partial x} d z \end{cases}} d x = \int_{0}^{L} (N_{T}^{(1)} + N_{T}^{(2)} + N_{T}^{(3)}) \frac{\partial w_{0}}{\partial x} \frac{\partial δ w_{0}}{\partial x} d x

(10)

where

T_{0}

is the reference temperature, and

N_{T}^{(i)} = \int_{a_{i - 1}}^{a_{i}} E^{(i)} (z, T) α^{(i)} (z, T) (T^{(i)} - T_{0}) d z, i = 1, 2, 3

Using the Hamilton’s energy principle $δ \int_{t_{1}}^{t_{2}} (V - U - W_{T}) d t = 0$ and usual simplification leads to the Euler-Lagrange equilibrium equations

\frac{\partial}{\partial x} (N_{x x}^{(1)} + N_{x x}^{(2)} + N_{x x}^{(3)}) = [\begin{array}{l} (I_{0}^{(1)} + I_{0}^{(2)} + I_{0}^{(3)}) \frac{\partial^{2} u_{0}}{\partial t^{2}} + (I_{1}^{(1)} - \frac{h_{1}}{2} I_{0}^{(1)}) \frac{\partial^{2} φ^{(1)}}{\partial t^{2}} + \\ {I_{1}^{(2)} - \frac{h_{2}}{2} (I_{0}^{(1)} - I_{0}^{(3)})} \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} + (I_{1}^{(3)} + \frac{h_{3}}{2} I_{0}^{(3)}) \frac{\partial^{2} φ^{(3)}}{\partial t^{2}} \end{array}]

(11a)

\frac{\partial}{\partial x} (M_{x x}^{(1)} - \frac{h_{1}}{2} N_{x x}^{(1)}) - Q_{x z}^{(1)} = [\begin{array}{l} (I_{1}^{(1)} - \frac{I_{0}^{(1)} h_{1}}{2}) \frac{\partial^{2} u_{0}}{\partial t^{2}} + \frac{h_{2}}{2} (\frac{I_{0}^{(1)} h_{1}}{2} - I_{1}^{(1)}) \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} \\ + (I_{2}^{(1)} - I_{1}^{(1)} h_{1} + \frac{I_{0}^{(1)} h_{1}^{2}}{4}) \frac{\partial^{2} φ^{(1)}}{\partial t^{2}} \end{array}]

(11b)

\frac{\partial}{\partial x} (M_{x x}^{(2)} + \frac{h_{2}}{2} (N_{x x}^{(3)} - N_{x x}^{(1)})) - Q_{x z}^{(2)} = [\begin{array}{l} {I_{1}^{(2)} + \frac{h_{2}}{2} (I_{0}^{(3)} - I_{0}^{(1)})} \frac{\partial^{2} u_{0}}{\partial t^{2}} + \frac{h_{2}}{2} (\frac{I_{0}^{(1)} h_{1}}{2} - I_{1}^{(1)}) \frac{\partial^{2} φ^{(1)}}{\partial t^{2}} + \\ {I_{2}^{(2)} + \frac{h_{2}^{2}}{4} (I_{0}^{(1)} + I_{0}^{(3)})} \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} + \frac{h_{2}}{2} (I_{1}^{(3)} + \frac{I_{0}^{(3)} h_{3}}{2}) \frac{\partial^{2} φ^{(3)}}{\partial t^{2}} \end{array}]

(11c)

\frac{\partial}{\partial x} (M_{x x}^{(3)} + \frac{h_{3}}{2} N_{x x}^{(3)}) - Q_{x z}^{(3)} = [\begin{array}{l} (I_{1}^{(3)} + \frac{I_{0}^{(3)} h_{3}}{2}) \frac{\partial^{2} u_{0}}{\partial t^{2}} + \frac{h_{2}}{2} (\frac{I_{0}^{(3)} h_{3}}{2} + I_{1}^{(3)}) \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} \\ + (I_{2}^{(3)} + I_{1}^{(3)} h_{3} + \frac{I_{0}^{(3)} h_{3}^{2}}{4}) \frac{\partial^{2} φ^{(3)}}{\partial t^{2}} \end{array}]

(11d)

\frac{\partial}{\partial x} (Q_{x z}^{(1)} + Q_{x z}^{(2)} + Q_{x z}^{(3)}) + (N_{T}^{(1)} + N_{T}^{(2)} + N_{T}^{(3)}) \frac{\partial^{2} w_{0}}{\partial x^{2}} = (I_{0}^{(1)} + I_{0}^{(2)} + I_{0}^{(3)}) \frac{\partial^{2} w_{0}}{\partial t^{2}}

(11e)

2.3. Eringen’s nonlocal theory

To capture the micro/nanoscale size effects, Eringen (2002) proposed a nonlocal elasticity theory which is based on the concepts of physics (atomic theory, lattice dynamics, electromagnetic theory, polaritons, gyrotropic effects, superconductivity, etc.,), that is, the behavior of a material point in a body depends on the state of all points of the body. Accordingly, the stress at a point $x$ in a continuum field is a function of strain in that field and can be expressed in the integral form as

σ_{i j}^{N L} = \int_{V} β (| x^{'} - x |, τ) σ_{i j}^{L} (x^{'}) d V (x^{'})

(12a)

σ_{i j}^{L} (x) = C (x) : ϵ_{i j}

(12b)

where

σ_{i j}^{L}

and

σ_{i j}^{N L}

are the local and nonlocal stresses,

β

(

| x^{'} - x |

τ

) is the nonlocal kernel function,

τ = e_{0} a / L

being the length scale parameter;

e_{0}

is the material constant, and

a

is an internal characteristics length;

C

(

x

) is the fourth-order elasticity tensor. An appropriate choice of kernel function leads to the following nonlocal constitutive equation given by

(1 - μ \nabla^{2}) σ_{i j}^{N L} (x) = C (x) : ϵ_{i j},

(13)

where nonlocal parameter

μ = {(e_{0} a)}^{2}

controls the size effect for nanostructures and

\nabla^{2}

refers for Laplacian operator. For the present FG composite nanobeams, the nonlocal stresses are defined as

(1 - μ \frac{\partial}{\partial x^{2}}) σ_{x x}^{(i)} = E^{(i)} (z, T) ε_{x x}^{(i)}, (1 - μ \frac{\partial}{\partial x^{2}}) σ_{x z}^{(i)} = κ G^{(i)} (z, T) γ_{x z}^{(i)}, i = 1, 2, 3,

where

κ

refers to shear correction factor and

G^{i} (z, T)

is the shear modulus.

The nonlocal normal stresses, shear stresses, and moments resultants for each layer of FG composite nanobeams are expressed in terms of displacements as

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [\begin{array}{l} N_{x x}^{(1)} \\ M_{x x}^{(1)} \end{array}] = [\begin{array}{l} A_{x x}^{(1)} \\ B_{x x}^{(1)} \end{array}] (\frac{\partial u_{0}}{\partial x} - \frac{h_{2}}{2} \frac{\partial φ^{(2)}}{\partial x} - \frac{h_{1}}{2} \frac{\partial φ^{(1)}}{\partial x}) + [\begin{array}{l} B_{x x}^{(1)} \\ C_{x x}^{(1)} \end{array}] \frac{\partial φ^{(1)}}{\partial x}

(14a)

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [\begin{array}{l} N_{x x}^{(3)} \\ M_{x x}^{(3)} \end{array}] = [\begin{array}{l} A_{x x}^{(3)} \\ B_{x x}^{(3)} \end{array}] (\frac{\partial u_{0}}{\partial x} + \frac{h_{2}}{2} \frac{\partial φ^{(2)}}{\partial x} + \frac{h_{3}}{2} \frac{\partial φ^{(3)}}{\partial x}) + [\begin{array}{l} B_{x x}^{(3)} \\ C_{x x}^{(3)} \end{array}] \frac{\partial φ^{(3)}}{\partial x}

(14b)

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [\begin{array}{l} N_{x x}^{(2)} \\ M_{x x}^{(2)} \end{array}] = [\begin{array}{l} A_{x x}^{(2)} \\ B_{x x}^{(2)} \end{array}] \frac{\partial u_{0}}{\partial x} + [\begin{array}{l} B_{x x}^{(2)} \\ C_{x x}^{(2)} \end{array}] \frac{\partial φ^{(2)}}{\partial x}, (1 - μ \frac{\partial^{2}}{\partial x^{2}}) [\begin{array}{l} Q_{x z}^{(1)} \\ Q_{x z}^{(2)} \\ Q_{x z}^{(3)} \end{array}] = \frac{\partial w_{0}}{\partial x} [\begin{array}{l} D_{x z}^{(1)} \\ D_{x z}^{(2)} \\ D_{x z}^{(3)} \end{array}] + [\begin{array}{l} D_{x z}^{(1)} φ^{(1)} \\ D_{x z}^{(2)} φ^{(2)} \\ D_{x z}^{(3)} φ^{(2)} \end{array}]

(14c)

where

[\begin{array}{l} A_{x x}^{(i)} \\ B_{x x}^{(i)} \\ C_{x x}^{(i)} \end{array}] = \int_{a_{i - 1}}^{a_{i}} E^{(i)} (z, T) [\begin{array}{l} 1 \\ z^{(i)} - z_{0} \\ {(z^{(i)} - z_{0})}^{2} \end{array}] d z, D_{x x}^{(i)} = \int_{a_{i - 1}}^{a_{i}} κ G^{(i)} (z, T) d z, i = 1, 2, 3

(14d)

By solving the equations (11a, 11b, 11c, 11d and 11e) and (14a) to (14d), the nonlocal governing equations of motion for FG composite nanobeam under thermal environment are deduced, and given by

[\begin{array}{l} (A_{x x}^{(1)} + A_{x x}^{(2)} + A_{x x}^{(3)}) \frac{\partial^{2} u_{0}}{\partial x^{2}} + (B_{x x}^{(1)} - \frac{h_{1}}{2} A_{x x}^{(1)}) \frac{\partial^{2} φ^{(1)}}{\partial x^{2}} + \\ {B_{x x}^{(2)} + \frac{h_{2}}{2} (A_{x x}^{(3)} - A_{x x}^{(1)})} \frac{\partial^{2} φ^{(2)}}{\partial x^{2}} + (B_{x x}^{(3)} + \frac{h_{3}}{2} A_{x x}^{(3)}) \frac{\partial^{2} φ^{(3)}}{\partial x^{2}} \end{array}] =

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [\begin{array}{l} (I_{0}^{(1)} + I_{0}^{(2)} + I_{0}^{(3)}) \frac{\partial^{2} u_{0}}{\partial t^{2}} + (I_{1}^{(1)} - \frac{h_{1}}{2} I_{0}^{(1)}) \frac{\partial^{2} φ^{(1)}}{\partial t^{2}} + \\ {I_{1}^{(2)} - \frac{h_{2}}{2} (I_{0}^{(1)} - I_{0}^{(3)})} \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} + (I_{1}^{(3)} + \frac{h_{3}}{2} I_{0}^{(3)}) \frac{\partial^{2} φ^{(3)}}{\partial t^{2}} \end{array}]

(15a)

[\begin{array}{l} (B_{x x}^{(1)} - \frac{h_{1}}{2} A_{x x}^{(1)}) \frac{\partial^{2} u_{0}}{\partial x^{2}} + {C_{x x}^{(1)} + h_{1} (\frac{h_{1}}{4} A_{x x}^{(1)} - B_{x x}^{(1)})} \frac{\partial^{2} φ^{(1)}}{\partial x^{2}} \\ + \frac{h_{2}}{2} (\frac{h_{1}}{2} A_{x x}^{(1)} - B_{x x}^{(1)}) \frac{\partial^{2} φ^{(2)}}{\partial x^{2}} - D_{x z}^{(1)} (\frac{\partial w_{0}}{\partial x} + φ^{(1)}) \end{array}] =

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [(I_{1}^{(1)} - \frac{I_{0}^{(1)} h_{1}}{2}) \frac{\partial^{2} u_{0}}{\partial t^{2}} + (I_{2}^{(1)} - I_{1}^{(1)} h_{1} + \frac{I_{0}^{(1)} h_{1}^{2}}{4}) \frac{\partial^{2} φ^{(1)}}{\partial t^{2}} + \frac{h_{2}}{2} (\frac{I_{0}^{(1)} h_{1}}{2} - I_{1}^{(1)}) \frac{\partial^{2} φ^{(2)}}{\partial t^{2}}]

(15b)

[\begin{array}{l} (B_{x x}^{(2)} + \frac{h_{2}}{2} (A_{x x}^{(3)} - A_{x x}^{(1)})) \frac{\partial^{2} u_{0}}{\partial x^{2}} + \frac{h_{2}}{2} (\frac{h_{1}}{2} A_{x x}^{(1)} - B_{x x}^{(1)}) \frac{\partial^{2} φ^{(1)}}{\partial x^{2}} - D_{x z}^{(2)} (\frac{\partial w_{0}}{\partial x} + φ^{(2)}) \\ + (C_{x x}^{(2)} + \frac{h_{2}^{2}}{4} (A_{x x}^{(3)} + A_{x x}^{(1)})) \frac{\partial^{2} φ^{(2)}}{\partial x^{2}} + \frac{h_{2}}{2} (B_{x x}^{(3)} + \frac{h_{3}}{2} A_{x x}^{(3)}) \frac{\partial^{2} φ^{(3)}}{\partial x^{2}} \end{array}] =

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [\begin{array}{l} {I_{1}^{(2)} + \frac{h_{2}}{2} (I_{0}^{(3)} - I_{0}^{(1)})} \frac{\partial^{2} u_{0}}{\partial t^{2}} + \frac{h_{2}}{2} (\frac{I_{0}^{(1)} h_{1}}{2} - I_{1}^{(1)}) \frac{\partial^{2} φ^{(1)}}{\partial t^{2}} + \\ {I_{2}^{(2)} + \frac{h_{2}^{2}}{4} (I_{0}^{(1)} + I_{0}^{(3)})} \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} + \frac{h_{2}}{2} (I_{1}^{(3)} + \frac{I_{0}^{(3)} h_{3}}{2}) \frac{\partial^{2} φ^{(3)}}{\partial t^{2}} \end{array}]

(15c)

[\begin{array}{l} (B_{x x}^{(3)} + \frac{h_{3}}{2} A_{x x}^{(3)}) \frac{\partial^{2} u_{0}}{\partial x^{2}} + \frac{h_{2}}{2} (B_{x x}^{(3)} + \frac{h_{3}}{2} A_{x x}^{(3)}) \frac{\partial^{2} φ^{(2)}}{\partial x^{2}} + \\ {C_{x x}^{(3)} + h_{3} (\frac{h_{3}}{4} A_{x x}^{(3)} + B_{x x}^{(3)})} \frac{\partial^{2} φ^{(3)}}{\partial x^{2}} - D_{x z}^{(3)} (\frac{\partial w_{0}}{\partial x} + φ^{(3)}) \end{array}] =

(1 - μ \frac{\partial^{2}}{\partial x^{2}}) [(I_{1}^{(3)} + \frac{I_{0}^{(3)} h_{3}}{2}) \frac{\partial^{2} u_{0}}{\partial t^{2}} + \frac{h_{2}}{2} (\frac{I_{0}^{(3)} h_{3}}{2} + I_{1}^{(3)}) \frac{\partial^{2} φ^{(2)}}{\partial t^{2}} + (I_{2}^{(3)} + I_{1}^{(3)} h_{3} + \frac{I_{0}^{(3)} h_{3}^{2}}{4}) \frac{\partial^{2} φ^{(3)}}{\partial t^{2}}]

(15d)

(D_{x z}^{(1)} + D_{x z}^{(2)} + D_{x z}^{(3)}) \frac{\partial^{2} w_{0}}{\partial x^{2}} + D_{x z}^{(1)} \frac{\partial φ^{(1)}}{\partial x} + D_{x z}^{(2)} \frac{\partial φ^{(2)}}{\partial x} + D_{x z}^{(3)} \frac{\partial φ^{(3)}}{\partial x} +

(N_{T}^{(1)} + N_{T}^{(2)} + N_{T}^{(3)}) (1 - μ \frac{\partial^{2}}{\partial x^{2}}) \frac{\partial^{2} w_{0}}{\partial x^{2}} = (1 - μ \frac{\partial^{2}}{\partial x^{2}}) [(I_{0}^{(1)} + I_{0}^{(2)} + I_{0}^{(3)}) \frac{\partial^{2} w_{0}}{\partial t^{2}}]

(15e)

Using the harmonic solutions and non-dimensional variables $X = x / L$ , $H_{i} = h_{i} / L$ , $u_{0} (x, t) = L U (X) e^{i ω t}$ , $φ^{(i)} (x, t) = ϕ^{(i)} (X) e^{i ω t}$ , $w_{0} (x, t) = L W (X) e^{i ω t}$ , $λ = \sqrt{μ} / L$ , $Ω = ω L^{2} \sqrt{ρ_{0}^{c} / E_{0}^{c}}$ , $D_{x z}^{(i)} = L H_{i} A_{1}^{(i)} E_{0}^{c}$ , $N_{T}^{(i)} = L H_{i} {\bar{N}}_{T}^{(i)} E_{0}^{c}$ , $[\begin{array}{l} A_{x x}^{(i)} & B_{x x}^{(i)} & C_{x x}^{(i)} \end{array}] = [\begin{array}{l} L H_{i} D_{1}^{(i)} & L^{2} H_{i}^{2} D_{2}^{(i)} & L^{3} H_{i}^{3} D_{3}^{(i)} \end{array}] E_{0}^{c}$ , $[\begin{array}{l} I_{0}^{(i)} & I_{1}^{(i)} & I_{2}^{(i)} \end{array}] = [\begin{array}{l} L H_{i} {\tilde{I}}_{0}^{(i)} & L^{2} H_{i}^{2} {\tilde{I}}_{1}^{(i)} & L^{3} H_{i}^{3} {\tilde{I}}_{2}^{(i)} \end{array}] ρ_{0}^{c}$ , i = 1, 2, 3, and equations (15a)–(15e) are transform to the non-dimensional coupled ordinary differential equations as

[\begin{array}{l} (\sum_{i = 1}^{3} H_{i} D_{1}^{(i)}) \frac{d^{2} U}{d X^{2}} + H_{1}^{2} (D_{2}^{(1)} - \frac{D_{1}^{(1)}}{2}) \frac{d^{2} ϕ^{(1)}}{d X^{2}} \\ + H_{2} (H_{2} D_{2}^{(2)} - \frac{H_{1} D_{1}^{(1)}}{2} + \frac{H_{3} D_{1}^{(3)}}{2}) \frac{d^{2} ϕ^{(2)}}{d X^{2}} \\ + H_{3}^{2} (D_{2}^{(3)} + \frac{D_{1}^{(3)}}{2}) \frac{d^{2} ϕ^{(3)}}{d X^{2}} \end{array}] = Ω^{2} (1 - λ^{2} \frac{d^{2}}{d X^{2}}) [\begin{array}{l} (\sum_{i = 1}^{3} H_{i} {\tilde{I}}_{0}^{(i)}) U + H_{1}^{2} ({\tilde{I}}_{1}^{(1)} - \frac{{\tilde{I}}_{0}^{(1)}}{2}) ϕ^{(1)} - \\ {H_{2} {\tilde{I}}_{1}^{(2)} + \frac{1}{2} (H_{3} {\tilde{I}}_{0}^{(3)} - H_{1} {\tilde{I}}_{0}^{(1)})} H_{2} ϕ^{(2)} \\ + H_{3}^{2} ({\tilde{I}}_{1}^{(3)} + \frac{{\tilde{I}}_{0}^{(3)}}{2}) ϕ^{(3)} \end{array}]

(16a)

H_{1} {(D_{2}^{(1)} - \frac{D_{1}^{(1)}}{2}) \frac{d^{2} U}{d X^{2}} - H_{1} (D_{2}^{(1)} - D_{3}^{(1)} - \frac{D_{1}^{(1)}}{4}) \frac{d^{2} ϕ^{(1)}}{d X^{2}}} - \frac{H_{2} H_{1}}{2} (D_{2}^{(1)} - \frac{D_{1}^{(1)}}{2}) \frac{d^{2} ϕ^{(2)}}{d X^{2}} - A_{1}^{(1)} (\frac{d W}{d X} + ϕ^{(1)}) =

H_{1} Ω^{2} (1 - λ^{2} \frac{d^{2}}{d X^{2}}) [({\tilde{I}}_{1}^{(1)} - \frac{{\tilde{I}}_{0}^{(1)}}{2}) U + \frac{H_{2}}{2} (\frac{{\tilde{I}}_{0}^{(1)}}{2} - {\tilde{I}}_{1}^{(1)}) ϕ^{(2)} + H_{1} ({\tilde{I}}_{2}^{(1)} - {\tilde{I}}_{1}^{(1)} + \frac{{\tilde{I}}_{0}^{(1)}}{4}) ϕ^{(1)}]

(16b)

[\begin{array}{l} (H_{2} D_{2}^{(2)} - \frac{H_{1}}{2} D_{1}^{(1)} + \frac{H_{3}}{2} D_{1}^{(3)}) \frac{d^{2} U}{d X^{2}} \\ + \frac{H_{1}^{2}}{2} (\frac{D_{1}^{(1)}}{2} - D_{2}^{(1)}) \frac{d^{2} ϕ^{(1)}}{d X^{2}} + \\ H_{2} (H_{2} D_{3}^{(2)} + H_{1} D_{1}^{(1)} + H_{3} \frac{D_{1}^{(3)}}{4}) \frac{d^{2} ϕ^{(2)}}{d X^{2}} + \\ \frac{H_{3}^{2}}{2} (\frac{D_{1}^{(3)}}{2} + D_{2}^{(3)}) \frac{d^{2} ϕ^{(3)}}{d X^{2}} - A_{1}^{(2)} (\frac{d W}{d X} + ϕ^{(2)}) \end{array}] = Ω^{2} (1 - λ^{2} \frac{d^{2}}{d X^{2}}) [\begin{array}{l} {H_{1} {\tilde{I}}_{1}^{(2)} + \frac{1}{2} (H_{3} {\tilde{I}}_{0}^{(3)} - H_{1} {\tilde{I}}_{0}^{(1)})} U + \\ \frac{H_{1}^{2}}{2} (\frac{{\tilde{I}}_{0}^{(1)}}{2} - {\tilde{I}}_{1}^{(1)}) ϕ^{(1)} + \frac{H_{3}^{2}}{2} ({\tilde{I}}_{1}^{(3)} + \frac{{\tilde{I}}_{0}^{(3)}}{2}) ϕ^{(3)} \\ + H_{2} {H_{2} {\tilde{I}}_{2}^{(2)} + \frac{1}{4} (H_{1} {\tilde{I}}_{0}^{(1)} + H_{3} {\tilde{I}}_{0}^{(3)})} ϕ^{(2)} \end{array}]

(16c)

[\begin{array}{l} H_{3} {(D_{2}^{(3)} + \frac{D_{1}^{(3)}}{2}) \frac{d^{2} U}{d X^{2}} + \frac{H_{2}}{2} (D_{2}^{(3)} + \frac{D_{1}^{(3)}}{2}) \frac{d^{2} ϕ^{(2)}}{d X^{2}}} + \\ H_{3}^{2} (D_{2}^{(3)} + D_{3}^{(3)} + \frac{D_{1}^{(3)}}{4}) \frac{d^{2} ϕ^{(3)}}{d X^{2}} - A_{1}^{(3)} (\frac{d W}{d X} + ϕ^{(3)}) \end{array}] = H_{3} Ω^{2} (1 - λ^{2} \frac{d^{2}}{d X^{2}}) [\begin{array}{l} ({\tilde{I}}_{1}^{(3)} + \frac{{\tilde{I}}_{0}^{(3)}}{2}) U + \frac{H_{2}}{2} (\frac{{\tilde{I}}_{0}^{(3)}}{2} + {\tilde{I}}_{1}^{(3)}) ϕ^{(2)} \\ + H_{3} ({\tilde{I}}_{2}^{(3)} + {\tilde{I}}_{1}^{(3)} + \frac{{\tilde{I}}_{0}^{(3)}}{4}) ϕ^{(3)} \end{array}]

(16d)

(\sum_{i = 1}^{3} H_{i} A_{1}^{(i)}) \frac{d^{2} W}{d X^{2}} + (\sum_{i = 1}^{3} H_{i} A_{1}^{i} \frac{d ϕ^{(i)}}{d X}) + (\sum_{i = 1}^{3} H_{i} {\bar{N}}_{T}^{(i)}) (1 - λ^{2} \frac{d^{2}}{d X^{2}}) \frac{d^{2} W}{d X^{2}} = Ω^{2} (1 - λ^{2} \frac{d^{2}}{d X^{2}}) {(\sum_{i = 1}^{3} H_{i} {\tilde{I}}_{0}^{(i)}) W}

(16e)

2.4. Nonlocal forces

As the governing equations are derived for Eringen’s nonlocal theory, the moments and shear forces for the boundary conditions must be modified accordingly in the nonlocal form (Li et al., 2020). Therefore, for the present formulation, the nonlocal moments and shear forces are obtained and given as:

M_{x x}^{(1)} = B_{x x}^{(1)} (\frac{d u_{0}}{d x} - \frac{h_{2}}{2} \frac{d φ^{(2)}}{d x}) + (C_{x x}^{(1)} - \frac{h_{1}}{2} B_{x x}^{(1)}) \frac{d φ^{(1)}}{d x} + μ {B_{x x}^{(1)} (\frac{d^{3} u_{0}}{d x^{3}} - \frac{h_{2}}{2} \frac{d^{3} φ^{(2)}}{d x^{3}}) + (C_{x x}^{(1)} - \frac{h_{1}}{2} B_{x x}^{(1)}) \frac{d^{3} φ^{(1)}}{d x^{3}}},

M_{x x}^{(2)} = B_{x x}^{(2)} \frac{d u_{0}}{d x} + C_{x x}^{(2)} \frac{d φ^{(2)}}{d x} + μ {B_{x x}^{(2)} \frac{d^{3} u_{0}}{d x^{3}} + C_{x x}^{(2)} \frac{d^{3} φ^{(2)}}{d x^{3}}},

M_{x x}^{(3)} = B_{x x}^{(3)} (\frac{d u_{0}}{d x} + \frac{h_{2}}{2} \frac{d φ^{(2)}}{d x}) + (C_{x x}^{(3)} + \frac{h_{3}}{2} B_{x x}^{(3)}) \frac{d φ^{(3)}}{d x} + μ {B_{x x}^{(3)} (\frac{d^{3} u_{0}}{d x^{3}} + \frac{h_{2}}{2} \frac{d^{3} φ^{(2)}}{d x^{3}}) + (C_{x x}^{(3)} + \frac{h_{3}}{2} B_{x x}^{(3)}) \frac{d^{3} φ^{(3)}}{d x^{3}}}

Q_{x x} = \frac{d w_{0}}{d x} + φ^{(1)} + φ^{(2)} + φ^{(3)} + μ (\frac{d^{3} w_{0}}{d x^{3}} + \frac{d^{2} φ^{(1)}}{d x^{2}} + \frac{d^{2} φ^{(2)}}{d x^{2}} + \frac{d^{2} φ^{(3)}}{d x^{2}})

Being the variable coefficients in governing differential equations (16a)–(16e), their closed-form solutions are not possible, except for certain values of various parameters. Hence, the approximate solutions are obtained using numerical techniques.

3. Chebyshev collocation technique

By assuming a new independent variable $y = 2 X - 1$ to transform the range of the problem in the applicability range of the technique, that is, (0, 1)–(−1, 1), the equations (16a)–(16e) are modified. The fourth-order derivative of the modified equations can be written as a linear combination of Chebyshev polynomials (Gupta and Lal, 1979), that is,

\frac{d^{4} f (y)}{d y^{4}} = \sum_{i = 0}^{m - 5} c_{i + 5} T_{i},

(17)

where c_i’s (i = 5, 6, …,

m

) are unknown constants and T_i’s (i = 0, 1, …, m − 5) are the Chebyshev polynomials.

The functional value can be obtained by integrating equation (17) as

f (y) = c_{1} + c_{2} T_{1} + c_{3} T_{1}^{(1)} + c_{4} T_{1}^{(2)} + \sum_{i = 0}^{m - 5} c_{i + 5} T_{i}^{(4)}, where, T_{i}^{(j)} represents the j^{t h} integral of T_{i} .

Substitution of $U_{i}$ , $ϕ_{i}^{(1)}$ , $ϕ_{i}^{(2)}$ , $ϕ_{i}^{(3)}$ , W_i, and their derivatives with respect to $y$ in the modified equations give simultaneous equations in terms of Chebyshev polynomials and unknowns. The satisfaction of the resultant equations at the roots of Chebyshev polynomials provides a set of 5 ( $m - 2$ ) linear equations in terms of $5 m$ unknowns, which can be written in matrix form as

[M - Ω^{2} N] [Y] = 0,

(18)

where

M

and

N

are the matrices of size 5

(m - 2) \times 5 n

Y

being the column vector of independent variables.

Equation (18) together with corresponding boundary conditions provides an eigenvalue problem which can be represented as

[\begin{array}{l} M - Ω^{2} N \\ M^{B C} \end{array}] [Y] = 0 .

(19)

For the fundamental frequency, the determinant of the frequency equation (19) must vanish, and hence

| \begin{array}{l} M - Ω^{2} N \\ M^{B C} \end{array} | = 0 .

(20)

The equation (20) is solved by MATLAB to obtain the dimensionless fundamental frequency Ω of FG composite nanobeams.

4. Results and discussion

In this section, the convergence, comparison, and a theoretical discussion on the computed numerical results are presented for the well-defined values of volume fraction index, nonlocal parameter, temperature difference, and thickness ratios of each layer for CC, CS, SS, and CF boundary conditions considering Type 1 and Type 2 composite FG nanobeams. The material properties for ceramic and metal constituents as Si₃N₄ and SUS304, respectively, are reported in Table 1 (Pandey and Pradyumna, 2015). The values of parameters are taken as:

Nonlocal parameter $λ$ = 0.1, 0.2, 0.3, 0.4, Volume fraction index $n$ = 0.5, 1, 2, 4

Temperature difference $∆ T$ = 100, 300, 500 K, $T_{0}$ = 300 K.

The facings have same thickness so the thickness ratios of layers are taken as 1-2-1 and 2-1-2 for nanobeams. The total dimensionless thickness of the nanobeam, that is, $H_{1}$ + $H_{2}$ + $H_{3}$ is assumed to be 0.4, and the shear correction factor is 5/6.

4.1. Convergence and comparison study

In order to choose an accurate numerical technique with less number of collocation points, a computer program is developed for CC, CS, SS, and CF (cantilever) composite FG nanobeams under thermal environment using Chebyshev collocation technique and DQM (Shu, 2000). It is observed that the Chebyshev collocation technique required a smaller number of collocation points as compared to DQM for the accuracy of four decimal places, and the corresponding results are given in Table 2. The convergence of both types of FG composite nanobeams with layer thickness ratio 2-1-2 is shown in Table 3 for all boundary conditions. Accordingly, 18 collocation points are fixed for the computation of frequency parameter in fundamental mode. A composite beam with metallic core and FGM face sheets under thermal environment is considered for the comparison of the results. The values of frequency parameter obtained by present formulation are compared with those obtained by DQM using two-dimensional theory (Setoodeh et al., 2012), FSDT (Xiang and Yang, 2008) and reported in Table 4.

Table 2.

Number of collocation points required for the accuracy of four decimal places, $n$ = 2, $∆ T$ = 300 K, $λ$ = 0.2.

		Chebyshev method				DQM
		CC	CS	SS	CF	CC	CS	SS	CF
Type 1	1-2-1	8	11	10	15	9	11	10	13
Type 1	2-1-2	7	12	10	16	7	14	10	17
Type 2	1-2-1	8	11	10	13	8	11	10	14
Type 2	2-1-2	8	10	10	14	9	10	10	15

Table 3.

Convergence of frequency parameter, thickness ratio 2-1-2, $n$ = 2, $∆ T$ = 300 K, $λ$ = 0.2.

$m$ ↓	Type 1				Type 2
$m$ ↓	CC	CS	SS	CF	CC	CS	SS	CF
5	0.6840	0.5768	0.5164	0.4913	0.7687	0.6322	0.5613	0.5273
6	0.6862	0.6045	0.5528	0.4991	0.7711	0.668	0.6046	0.5458
7	0.6833	0.6095	0.5571	0.5369	0.7672	0.6753	0.6098	0.5732
8	0.6833	0.6055	0.5522	0.5204	0.7671	0.6695	0.6037	0.5712
9	—	0.6049	0.552	0.5277	0.7671	0.6689	0.6033	0.5715
10	—	0.6052	0.5521	0.5219	—	0.6692	0.6035	0.5723
11	—	0.6051	0.5521	0.5255	—	0.6692	0.6035	0.5729
12	—	0.6052	—	0.5234	—	—	—	0.5731
13	—	0.6052	—	0.5245	—	—	—	0.5733
14	—	—	—	0.5239	—	—	—	0.5734
15	—	—	—	0.5242	—	—	—	0.5734
16	—	—	—	0.5241	—	—	—	—
17	—	—	—	0.5241	—	—	—	—

Table 4.

Comparison of frequency parameter for type 1 FG composite beam under thermal environment.

(T_u T_l) →	(0 0)°C			(50 50)°C			(150 50)°C
n →	0.2	2	10	0.2	2	10	0.2	2	10
CC-beam
Present	0.7716	0.6819	0.6290	0.7678	0.6786	0.6259	0.7640	0.6753	0.6228
LWDQ^a	—	0.6332	—	—	0.6233	—	—	0.6140	—
ABAQUS^a	—	0.6315	—	—	0.6218	—	—	0.6129	—
$FSDT$ ^b	0.715	0.655	0.620	0.706	0.644	0.609	0.697	0.634	0.598
SS-beam
Present	0.3983	0.3475	0.3174	0.3918	0.3418	0.3120	0.3851	0.3360	0.3065
LWDQ^a	—	0.2942	—	—	0.2942	—	—	0.2942	—
ABAQUS^a	—	0.2933	—	—	0.2927	—	—	0.2920	—
$FSDT$ ^b	0.336	0.306	0.289	—	0.306	—	—	—	—
Cantilever beam
Present	0.2423	0.2110	0.1924	0.2384	0.2076	0.1892	0.2344	0.2041	0.1859
$FSDT$ ^b	0.121	0.110	0.104	—	0.110	—	—	—	—

^aSetoodeh et al., 2012.

^bXiang and Yang, 2008.

4.2. Parametric study

As discussed earlier, two types of composite FG nanobeams are studied here. Type 1 composite nanobeam has a metallic core and FGM as facings. The upper and lower surfaces of the top FG layer are ceramic and metal-rich, respectively, while the bottom layer has reversed material distribution at the surfaces. The top and bottom layers of Type 2 composite nanobeam are made of pure ceramic and metallic constituents, respectively. However, its core is made of FGM with upper and lower surface as ceramic and metal-rich, respectively. Therefore, the gradation of Type 1 FG nanobeam is symmetric about the mid-plane of the structure but for Type 2, it is asymmetric. Hence, the contribution of the physical neutral plane is included only for the analysis of Type 2 composite nanobeams. Further, the displacement and slope at the boundaries are independent of the size scale parameter for the nanobeams. Moreover, the computed modified nonlocal moment and shear forces at the boundaries are considered for the implementation of simply-supported and free boundary conditions. The effect of these nonlocal boundaries on the frequency parameter Ω is shown in Table 5. It is observed that the contribution of size parameter is marginal for CS and SS boundary conditions while it is more effective for cantilever nanobeams. Moreover, as the size parameter has the tendency to decrease the stiffness of the structure, the difference between the values of frequency parameter for local and nonlocal boundary conditions increases as the value of nonlocal parameter increases.

Table 5.

Frequency parameter for composite FG nanobeams $∆ T$ = 300 K, L (boundary conditions without nonlocal effect), N (boundary conditions with nonlocal effect).

	1-2-1				2-1-2
λ →	0.1		0.4		0.1		0.4
n →	0.5	2	0.5	2	0.5	2	0.5	2
Type 1
CS	0.7195^L	0.6303^L	0.4642^L	0.4059^L	0.8333^L	0.6825^L	0.5390^L	0.4406^L
CS	0.7195^NL	0.6303^NL	0.4723^NL	0.4101^NL	0.8333^NL	0.6825^NL	0.5397^NL	0.4439^NL
SS	0.6610^L	0.5759^L	0.4271^L	0.3715^L	0.7620^L	0.6234^L	0.4929^L	0.4025^L
SS	0.6610^NL	0.5759^NL	0.4271^NL	0.3716^NL	0.7620^NL	0.6234^NL	0.4930^NL	0.4026^NL
CF	0.6600^L	0.5764^L	0.4539^L	0.3967^L	0.7593^L	0.6164^L	0.5263^L	0.4290^L
CF	0.6508^NL	0.5683^NL	0.2751^NL	0.2400^NL	0.7486^NL	0.6077^Nl	0.3175^NL	0.2603^NL
Type 2
CS	0.8239^L	0.7186^L	0.5364^L	0.4653^L	0.7935^L	0.7502^L	0.5181^L	0.4878^L
CS	0.8245^NL	0.7189^NL	0.5314^NL	0.4664^NL	0.7951^NL	0.7514^NL	0.5109^NL	0.4838^NL
SS	0.7466^L	0.6545^L	0.4831^L	0.4231^L	0.7202^L	0.6819^L	0.4659^L	0.4405^L
SS	0.7469^NL	0.6548^NL	0.4928^NL	0.4298^NL	0.7203^NL	0.6817^NL	0.4750^NL	0.4427^NL
CF	0.7520^L	0.6550^L	0.5240^L	0.4544^L	0.7156^L	0.6773^L	0.5038^L	0.4748^L
CF	0.7414^NL	0.6459^NL	0.3115^NL	0.2716^NL	0.7047^NL	0.6671^NL	0.3052^NL	0.2885^NL

To examine the effect of nonlocal (size) parameter, volume fraction index, and temperature difference on the frequency parameter Ω for FG composite nanobeams, the results are reported in Tables 6 –13 for Type 1 and Type 2, respectively. It can be seen that the total volume of Type 1 and Type 2 nanobeams are same. For thickness ratio as 1-2-1, the volume of the metal constituent remains the same in Type 1 and Type 2 FG nanobeams with the increasing values of volume fraction index

n

. However, for thickness ratio 2-1-2, it is different and the variation with the increasing values of volume fraction index is shown in Figure 2. It shows that the volume of metal constituent is higher in Type 2 nanobeams as compared to Type 1 nanobeams for smaller values of

n

. As Young’s modulus of the metal constituent is smaller to ceramic, the value of frequency parameter is found to be less for Type 2 nanobeams. Similarly, the behavior of frequency parameter can be justified for higher values of volume fraction index that leads to lower Young’s modulus. Further, for

n =

0.5, the volume of the metal constituent is same for both types of nanobeams but it can be seen that the values of frequency parameter are higher for Type 1 FG composite nanobeams as compared to those for the Type 2 ones, keeping other parameters fixed. It is due to the temperature applied to the surfaces. Both the surfaces of the Type 1 nanobeams are fully ceramic rich which has low thermal conductivity while in Type 2 nanobeam, the lower surface is fully metallic which leads to the higher thermal conductivity in the composite nanobeam and then to the lower frequencies. The value of frequency parameter is found to be in the order of Ω_CC > Ω_CS > Ω_SS > Ω_CF.

Table 6.

Frequency parameter for CC composite FG nanobeams for type 1.

ΔT (K) ↓	n↓\λ→	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ→	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.8201	0.7256	0.6210	0.5286	0.9542	0.8444	0.7228	0.6155
	1	0.7710	0.6819	0.5832	0.4962	0.8632	0.7637	0.6536	0.5564
	2	0.7254	0.6413	0.5481	0.4660	0.7855	0.6948	0.5944	0.5057
	4	0.6910	0.6106	0.5216	0.4431	0.7303	0.6457	0.5521	0.4695
500	0.5	0.7940	0.7000	0.5947	0.5001	0.9267	0.8175	0.6954	0.5858
	1	0.7444	0.6557	0.5563	0.4669	0.8361	0.7372	0.6265	0.5270
	2	0.6981	0.6143	0.5204	0.4358	0.7584	0.6682	0.5672	0.4764
	4	0.6628	0.5828	0.4930	0.4121	0.7026	0.6186	0.5246	0.4399

Table 7.

Frequency parameter for CS composite FG nanobeams for type 1.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.7302	0.6481	0.5568	0.4843	0.8449	0.7502	0.6444	0.5510
	1	0.6840	0.6072	0.5217	0.4520	0.7634	0.6779	0.5824	0.5028
	2	0.6409	0.5690	0.4890	0.4219	0.6932	0.6156	0.5290	0.4556
	4	0.6081	0.5400	0.4641	0.3988	0.6427	0.5708	0.4906	0.4212
500	0.5	0.7063	0.6249	0.5337	0.4585	0.8197	0.7258	0.6203	0.5269
	1	0.6596	0.5835	0.4982	0.4260	0.7386	0.6538	0.5586	0.4765
	2	0.6159	0.5446	0.4648	0.3954	0.6684	0.5916	0.5053	0.4294
	4	0.5823	0.5148	0.4392	0.3717	0.6174	0.5464	0.4665	0.3949

Table 8.

Frequency parameter for SS composite FG nanobeams for type 1.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.6713	0.5954	0.5112	0.4368	0.7731	0.6858	0.5889	0.5032
	1	0.6273	0.5564	0.4777	0.4081	0.6984	0.6195	0.5319	0.4545
	2	0.5861	0.5198	0.4462	0.3812	0.6337	0.5620	0.4825	0.4122
	4	0.5544	0.4917	0.4220	0.3605	0.5864	0.5201	0.4464	0.3814
500	0.5	0.6488	0.5738	0.4902	0.4161	0.7494	0.6630	0.5668	0.4816
	1	0.6044	0.5344	0.4563	0.3871	0.6750	0.5971	0.5102	0.4333
	2	0.5624	0.4971	0.4243	0.3598	0.6102	0.5395	0.4608	0.3911
	4	0.5299	0.4682	0.3995	0.3386	0.5624	0.4972	0.4244	0.3601

Table 9.

Frequency parameter for cantilever composite FG nanobeams for type 1.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.6602	0.5679	0.4246	0.2808	0.7587	0.6542	0.4900	0.3236
	1	0.6174	0.5313	0.3973	0.2627	0.6824	0.5889	0.4422	0.2928
	2	0.5775	0.4972	0.3718	0.2457	0.6169	0.5328	0.4010	0.2660
	4	0.5471	0.4711	0.3522	0.2327	0.5698	0.4924	0.3712	0.2466
500	0.5	0.6394	0.5485	0.4081	0.2684	0.7376	0.6344	0.4729	0.3104
	1	0.5962	0.5115	0.3805	0.2500	0.6614	0.5692	0.4253	0.2798
	2	0.5557	0.4767	0.3545	0.2326	0.5957	0.5129	0.3840	0.2530
	4	0.5245	0.4500	0.3343	0.2191	0.5480	0.4721	0.3538	0.2333

Table 10.

Frequency parameter for CC composite FG nanobeams for type 2.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.9718	0.8583	0.7327	0.6221	0.9390	0.8290	0.7075	0.6006
	1	0.9029	0.7980	0.6818	0.5793	0.9116	0.8050	0.6872	0.5835
	2	0.8323	0.7360	0.6293	0.5352	0.8798	0.7771	0.6636	0.5636
	4	0.7854	0.6948	0.5944	0.5057	0.8565	0.7566	0.6462	0.5490
500	0.5	0.9475	0.8338	0.7068	0.5931	0.9173	0.8074	0.6847	0.5750
	1	0.8797	0.7746	0.6570	0.5517	0.8905	0.7839	0.6649	0.5585
	2	0.8101	0.7138	0.6059	0.5090	0.8593	0.7565	0.6419	0.5392
	4	0.7640	0.6734	0.5718	0.4806	0.8364	0.7365	0.6249	0.5249

Table 11.

Frequency parameter for CS composite FG nanobeams for type 2.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.8347	0.7454	0.6434	0.5425	0.8038	0.7185	0.6202	0.5203
	1	0.7831	0.6979	0.6016	0.5104	0.7835	0.6997	0.6032	0.5077
	2	0.7283	0.6482	0.5582	0.4768	0.7597	0.6776	0.5835	0.4930
	4	0.6913	0.6148	0.5292	0.4542	0.7420	0.6613	0.5690	0.4821
500	0.5	0.8143	0.7245	0.6217	0.5199	0.7862	0.7005	0.6014	0.5007
	1	0.7631	0.6776	0.5807	0.4883	0.7662	0.6819	0.5848	0.4882
	2	0.7089	0.6287	0.5382	0.4552	0.7426	0.6602	0.5654	0.4737
	4	0.6723	0.5958	0.5098	0.4331	0.7250	0.6441	0.5512	0.4630

Table 12.

Frequency parameter for SS composite FG nanobeams for type 2.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.7569	0.6720	0.5783	0.5033	0.7285	0.6462	0.5559	0.4831
	1	0.7113	0.6314	0.5433	0.4715	0.7105	0.6300	0.5416	0.4678
	2	0.6638	0.5892	0.5069	0.4393	0.6894	0.6110	0.5250	0.4508
	4	0.6321	0.5610	0.4826	0.4179	0.6740	0.5971	0.5126	0.4385
500	0.5	0.7370	0.6526	0.5590	0.4823	0.7118	0.6299	0.5396	0.4659
	1	0.6921	0.6126	0.5246	0.4514	0.6941	0.6140	0.5256	0.4507
	2	0.6454	0.5712	0.4890	0.4200	0.6734	0.5954	0.5092	0.4337
	4	0.6142	0.5435	0.4651	0.3990	0.6582	0.5817	0.4970	0.4215

Table 13.

Frequency parameter for cantilever composite FG nanobeams for type 2.

ΔT (K) ↓	n↓\λ →	1-2-1				2-1-2
ΔT (K) ↓	n↓\λ →	0.1	0.2	0.3	0.4	0.1	0.2	0.3	0.4
100	0.5	0.7505	0.6486	0.4838	0.3171	0.7126	0.6142	0.4610	0.3103
	1	0.7030	0.6073	0.4530	0.2971	0.6947	0.5983	0.4487	0.3024
	2	0.6541	0.5647	0.4213	0.2765	0.6743	0.5800	0.4344	0.2933
	4	0.6216	0.5364	0.4003	0.2630	0.6598	0.5667	0.4239	0.2868
500	0.5	0.7324	0.6312	0.4688	0.3059	0.6968	0.5996	0.4486	0.2999
	1	0.6856	0.5905	0.4386	0.2863	0.6794	0.5841	0.4365	0.2921
	2	0.6374	0.5486	0.4076	0.2663	0.6596	0.5664	0.4226	0.2832
	4	0.6053	0.5208	0.3870	0.2530	0.6456	0.5536	0.4125	0.2768

Figure 2.

Graph for normalized volume of the metal constituent versus volume fraction index n for thickness ratio 2-1-2.

For Type 1 FG composite nanobeams, it is observed that the value of frequency parameter Ω decreases with the increase in volume fraction index $n$ for all four boundary conditions. Figure 3 shows that the volume of the metal constituent for 1-2-1 nanobeams is always higher as compared to the 2-1-2 nanobeams, whatever be the value of the volume fraction index. Accordingly, the value of frequency parameter for 2-1-2 nanobeams is higher corresponding to 1-2-1 nanobeams. As the nonlocal parameter $λ$ increases, the value of frequency parameter decreases. This effect is higher for 2-1-2 composite nanobeams as compared to 1-2-1 ones. The value of Ω decreases with the increase in the temperature difference at the surfaces of the composite nanobeams with CC, CS, SS, and CF boundary conditions.

Figure 3.

Graph for normalized volume of the metal constituent versus volume fraction index n for Type 1.

For Type 2 FG composite nanobeams, it is evident from Figure 4 that the volume of metal constituent is high for 2-1-2 nanobeams as compared to 1-2-1 nanobeams when $n <$ 1 while this order is reversed when $n >$ 1 and so the value of frequency parameter Ω changes accordingly. Further, the value of Ω decreases with the increasing values of volume fraction index $n$ for CC, CS, SS, and CF boundary conditions, for the same set of values of other parameters. The value of frequency parameter is found to decrease with the increase in nonlocal parameter, keeping other parameters fixed for CC, CS, SS, and CF boundary conditions. It is also observed that the frequency parameter decreases with the increase in temperature profile. To show the behavior of frequency parameter at the intermediate values for the nonlocal parameter, the curves have been plotted for Type 1 and Type 2 composite nanobeams with thickness ratio 1-2-1, 2-1-2, ( $Δ T$ = 300 K) and shown in Figures 5 and 6, respectively. It can be observed from the figures that the value of frequency parameter decreases with the increasing values of nonlocal parameter $λ$ , particularly, for smaller values of $λ (\leq 0.2)$ in cantilever nanobeams that overcame the paradoxes available in the literature (Asemi et al., 2014; Shafiei et al., 2019).

Figure 4.

Graph for normalized volume of the metal constituent versus volume fraction index n for Type 2.

Figure 5.

Graph for frequency parameter Ω versus nonlocal parameter λ for type 1: red line-CC, green line-CS, pink line-SS, black line-CF, solid line-n = 0.5, dashed line-n = 2, ΔT = 300 K.

Figure 6.

Graph for frequency parameter Ω versus nonlocal parameter λ for type 2: red line-CC, green line-CS, pink line-SS, black line-CF, solid line-n = 0.5, dashed line-n = 2, ΔT = 300 K.

5. Conclusions

This investigation presents the vibration analysis for FG-metal-FG and ceramic-FG-metal type composite nanobeams under nonlinearly varying temperature in the thickness direction. A layer-wise theory that accounts for shear deformation and inertia at each layer together with their mutual interaction effect has been adopted to develop the governing equation based on nonlocal theory, nonlocal boundary conditions, and physical neutral plane. The values of fundamental frequencies have been obtained for symmetric/asymmetric gradation of the FG composite nanobeams with CC, CS, SS, and CF (cantilever) boundary conditions using the Chebyshev collocation technique. The variation of the frequency parameter is explained in terms of the volume of the metal constituent from the total volume of FG composite nanobeams with different gradation in layers and thickness ratios. The volume of metal constituent increases with the increase in volume fraction index which reduces the stiffness of the nanobeams and then to the frequencies. Further, the value of frequency parameter decreases with the increase in the values of nonlocal parameter and temperature profile for CC, CS, SS, and CF boundary conditions. The present formulation resolves the issue of paradox behavior for cantilever nanobeam discussed in Asemi et al. (2014); Shafiei et al. (2019) by imposing nonlocal boundary conditions as defined in Li et al. (2020). The results obtained by the present formulation and method have been verified and can be used by engineers as a benchmark.

Footnotes

Acknowledgments

The authors are grateful for the constructive comments of the learned referees to improve the quality of the article. The financial support provided by the Indian Institute of Technology Delhi, India, is gratefully acknowledged by first author to carry out this research work.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Institute Post Doctoral Fellowship. Indian Institute of Technology Delhi.

ORCID iD

Rahul Saini

Data availability statement

All generated or analyzed during this study are included in this article.

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