Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading

Abstract

Considering the uncertainties of the materials and loads, the nanobeam made of functionally graded materials were investigated based on the non-local elastic theory. The spline finite point method (SFPM) was established to analyze the bending behavior of the nanobeam-based Timoshenko theory. In comparison with finite element method (FEM), SFPM has higher accuracy. Further, the multi-source uncertainties are considered, material properties are quantified as interval parameters, and loads are taken as random parameters. To deal with the problems with two types of uncertainties coexisting, a hybrid uncertain analysis model was established, and the method of polynomial chaos expansion and dimensional wise (PCE–DW) analysis was proposed to predict the response of nanobeam in the hybrid uncertain system. Numerical examples ultimately illustrate the effectiveness of the model and solution techniques, compared with MCS. The results furtherly verify the efficiency and accuracy.

1. Introduction

Nowadays, a new generation of materials with excellent performance has attracted wide attention. Among these materials, functional gradient materials (FGMs) play an important role in engineering [1,2]. FGMs are made of two different materials (such as metals and ceramics) with extraordinary mechanical, electrical, and thermal properties. FGMs have been applied to micro scale systems, such as nanoelectromechanical systems (NEMS) [3].

A nanobeam is the basic component of nanostructures; thus, the mechanical behavior of nanobeam is essential to the analysis of nanostructures. However, the classical continuum theories cannot accurately describe the mechanical behavior of nanostructures, due to the presence of the small-scale effect. Thus, it is imperative to develop new adaptive models to predict the mechanical behavior, considering the size-dependent effect [4]. The nanobeam is the basic component of the NEMS; thus, the prediction of the mechanical behavior of nanobeams is the key of microstructure analysis and design. However, the mechanical behavior can not be accurately described because of the presence of small-scale effect. In order to predict the mechanical behavior, considering the size-dependent effect, some new methods were developed. One popular model is nonlocal elastic theory, proposed by Eringen [5], which states the relationship between stress at a point and strain at each point. The theory was applied in investigating the mechanical behaviors of nanostructures, including vibration [6], bending [7], buckling [8,9,10], wave propagation analysis [11,12], and thermal [13]. In recent years, the research on the mechanical behavior of FGM nanostructures has attracted the attention of many scholars. Asghari [14,15] investigated the vibration of nanobeams, considering the size-dependent effect, based on the modified coupled stress theory. Ke and Wang [16] examined the size effect on dynamic stability of FGM Timoshenko nanobeam. Simsek and Yurtcu [10] analyzed the bending and buckling behaviors of the FGM Euler-Bernoulli and Timoshenko beams, taking the small-scale effect into account. Eltaher et al. [17] derived the governing equation of the free vibration for the FGM nanobeam and computed the structural response utilizing the finite element method (FEM).

The works mentioned above were based on the deterministic model. It can be said that the deterministic model, with deterministic properties, has been well-applied to analyze mechanical behaviors of the microstructures. However, the properties of the model are always uncertain [18,19,20], such as the material properties (Young’s modulus, Poisson’s ratio), geometric dimensions (width and height of section), external load (value and position of load), and boundary conditions (rigid constraint, elastic constraint), etc. For instance, a Stone–Thrower–Wales defect [21] and pentagon-octagon-pentagon defect [22] will lead to uncertainties, regarding the material properties. The uncertainties existing in reality are ignored in the deterministic model, which is simple, but unreasonable. To handle these uncertain problems, the probabilistic method is the most popular technique—by assuming the uncertain parameters, as random variables, obey a certain distribution form. These random problems can be solved by Monte Carlo simulation (MCS) method, which is a simple, but powerful, technique [23]. However, a large number of samples need to generate and calculate the response of the samples, which is very time consuming. The stochastic perturbation method [24] can be alternatively utilized to well-investigate the propagation of the randomness in the uncertain system, which reduces the computing burden. However, it is just as efficient for the problems with mall uncertainty, due to the fact that it is a local approximate method based on first-order Taylor series expansion. The spectral stochastic method [25] is also based on the series expansion, but the response of uncertain system is expanded in the random space, not in a single point. The polynomial chaos expansion (PCE) is an one of the outstanding methods, thanks to its profound mathematical basis [26].

Unfortunately, the distribution information can be accurately known [27] in some cases. The lower and upper bounds of the uncertain parameters is easily obtained in practice, which can be well-quantified by the interval mathematics [28]. Therefore, the convex model, based on interval mathematics, can serve as a powerful supplement to this part [29]. In order to predict the bounds of the response, some methods have been developed, such as the perturbation method [30], Taylor expansion method and gradient method [31], and response surface method [32]. Using the vertex method (VM), the exact solution will be obtained if the monotonicity of the function is guaranteed [33]. The basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations. However, all the possible combination of each uncertain parameter vertex will be calculated, which leads to the computational costs increasing exponentially, due to the increase in the number of uncertain parameters. Alternatively, based on the interval mathematic and perturbation theories, Qiu et al. [34] developed a matrix perturbation method that is widely used to determine the range of the structural response because it is simple, yet efficient. Unfortunately, the accuracy of this approach cannot be guaranteed by augmenting the uncertain parameter number [31]. In order to avoid the problem, Chen et al. [35] presented a parameter perturbation method by using the partial derivatives information. Although it is successful to use the first order interval approach to analyze the response bounds, it will not work well when the uncertain range is not small enough.

The uncertain problems with random and unknown, but bounded, parameters can be well-described by the probabilistic and interval method, respectively. However, the random parameters may always coexist with the unknown, but bounded, ones. Thus, the behavior of defective nanobeam subjected to random loads can be not described via a single type of uncertain parameters. It is necessary to construct a hybrid uncertain model, with both probabilistic and interval parameters, simultaneously, which is one of motivation of this paper.

In the aspect of a solution, the numerical model is utilized to calculate the deterministic analysis, whose accuracy directly affects predicting the statistical properties or bounds of the nanostructure’s response. More precise responses can be obtained by utilizing more accurate numerical model. Moreover, the efficiency of numerical model is another important factor in the prediction of the response. With high accuracy, meshless methods (MMs) were developed to avoid some of the difficulties in FEM, which is an alternative to method to the finite element method (FEM), due to its unique features. Further, Liew [36] provided a brief review on the advances and trends in the MMs formulations. However, the huge amount of computation is still one of the urgent problems for MMS. In order to deal with the problem of computational consumption and accuracy, the spline-based numerical methods, such as the spline strip [37,38,39,40,41] and B-spline finite point (B-sFPM) [42] methods, have been proposed to structural analysis. Moreover, B-sFPM would be more promising to utilize, with high accuracy and efficiency, because the basic function is cubic B-spline function, which has compact support and explicit form. Therefore, the spline finite point method is introduced to conduct the deterministic analysis of nanobeam, considering the size-dependent effect.

To sum up, this is a summary of the work performed in this paper. First, the mechanical behavior of the FGM nanobeam is investigated, considering the size-dependent effect, based on the nonlocal elastic theory. The spline finite point method (SFPM) is proposed to construct the deterministic framework analyzing the bending behavior. In order to be more objective, the material defect and load randomness are taken into account, which are quantified as interval and random parameters, respectively. Furtherly, a hybrid model, including random and interval, is presented to predict the response of the defective FGM nanobeam subjected to random load. To solve this hybrid model, the polynomial chaos expansion and dimension wise methods are combined. At last, three cases of the simple supported nanobeam numerical example are examined to verify the proposed approach, followed by some conclusions.

2. Theory of FGM Timoshenko Nanobeam

In this section, the static analysis theory of FGM Timoshenko beam is introduced, i.e., an FGM Timoshenko nanobeam subjected to the transverse distributed load. The FGM nanobeam model attached to Cartesian coordinate system is shown in Figure 1, including the material composition, geometric dimension, load, and boundary condition.

2.1. Material Properties of FGM Beam

It is assumed that the nanobeam is made of FGM materials, where the top and bottom surfaces of the beam are ceramics and metals, respectively. Generally, the FGMs are continuously varied in the thickness (z-axis) direction. The material properties are assumed to vary, as the following simple power law:

$$\begin{array}{l}E\left(z\right)=\left(E_{\mathrm{m}}-E_{\mathrm{c}}\right){\left(\frac{z}{h}\right)}^n+E_{\mathrm{c}}\\ G\left(z\right)=\left(G_{\mathrm{m}}-G_{\mathrm{c}}\right){\left(\frac{z}{h}\right)}^n+G_{\mathrm{c}}\end{array}$$

(1)

where $E$ and $G$ present the Young’s and shear moduli, respectively. In Equation (1), the subscript ‘m’ and ‘c’ stand for metals and ceramics, respectively. $n$ denotes the material variation factor, which is a non-negative number. For instance, a Timoshenko nanobeam is made of FGM, as listed in

Table 1. Material properties of FGM constituents.

Properties	Unit	Aluminum	Aluina (Al2O3)
Young’s modulus/$E$	Gpa	70	393
Shear modulus/$G$	GPa	26	157

. The variation the of the properties is plotted in Figure 2.

2.2. Nonlocal Elastic Theory for FGM Timoshenko Nanobeam

To consider the size-dependent effect, we analyze the FGM Timoshenko nanobeams by nonlocal elastic theory, which assumes that the stress at one point in a continuum is related to the strain of all points in the continuum [43]. Furthermore, the nonlocal stress tensor $\sigma$ at $x$ point was provided by Erigen [5], as follows:

$${\sigma }_{ij}\left(x\right)={\displaystyle {\int }_V\alpha \left(\left|x^{\prime }-x\right|,\tau \right)t_{ij}\left(x^{\prime }\right)}\mathrm{d}{x}^{\prime }$$

(2)

where $t_{ij}\left(x^{\prime }\right)$ is the stress tensor at $x$ point; $\alpha \left(\left|x^{\prime }-x\right|,\tau \right)$ is the kernel function that denotes the nonlocal modulus; $\left|x^{\prime }-x\right|$ presents the distance in Euclidean space; $\tau$ is a material constant. However, there is a tricky problem in Equation (2), where the integral constitutive relation is difficult to solve. To deal with the issue, in Erigen’s work [5], the integral constitutive relation can be converted to the differential equation as follows:

$$L{\sigma }_{ij}=t_{ij}$$

(3)

in which $L$ is a linear differential operator. It is written as:

$$L=1-{\left(e_{0}a\right)}^2{\nabla }^2$$

(4)

where $e_0$ is a parameter to modify the model to coincide with the experimental results, and $a$ is the internal characteristic length. To simplify the formulas, a nonlocal factor $\mu ={\left(e_{0}a\right)}^2$ is defined to describe the size-dependent effect. Thus, the constitutive relation can be presented as:

$$\left(1-\mu {\nabla }^2\right){\sigma }_{ij}=t_{ij}$$

(5)

2.3. The Governing Equation of Timoshenko Beam Considering Size-Dependent Effect

According to the Timoshenko beam theory (TBT), the kinematic relation can be written as:

$$\begin{array}{l}u\left(x,z\right)=u_0\left(x\right)+z\phi \left(x\right)\\ w\left(x,z\right)=w_0\left(x\right)\end{array}$$

(6)

where $u$ and $w$ are axial and transverse displacement at any point in the beam, $u_0$ and $w_0$ are the axial and transverse displacement at neutral-plane, and $\phi$ is the rotation of the cross section. Thus, the strain of the Timoshenko beam, based on TBT, can be calculated as:

$$\begin{array}{l}{\epsilon }_{xx}=\frac{\partial u_0}{\partial x}+z\frac{\partial \phi }{\partial x}={\epsilon }_{xx}^0+z{k}^0\\ {\gamma }_{xz}=\frac{\partial w_0}{\partial x}+\phi \end{array}$$

(7)

where ${\epsilon }_{xx}^0$ and ${\gamma }_{xz}$ are external and transverse shear strain, and $k^0$ is the bending strain.

Due to varying the material properties (Young’s and shear moduli) in the FGM beam, the neutral plane is always not at the midplane. To find the neutral plane, the resultant axial force equals zero, which can be formulated as:

$${\displaystyle {\int }_A{\sigma }_{xx}\mathrm{d}A={\displaystyle {\int }_{A}E\left(\tilde{z}\right){\epsilon }_{xx}\mathrm{d}A=0}}$$

(8)

According to the Equation (8), the position of neutral plane can be obtained as follows

$${\tilde{z}}_{\mathrm{c}}=\frac{{\displaystyle {\int }_{A}E\left(\tilde{z}\right)\tilde{z}\mathrm{d}A}}{{\displaystyle {\int }_{A}E\left(\tilde{z}\right)\mathrm{d}A}}=\frac{h\left(2E_{\mathrm{m}}+n{E}_{\mathrm{c}}\right)\left(n+1\right)}{2\left(E_{\mathrm{m}}+n{E}_{\mathrm{c}}\right)\left(n+2\right)}$$

(9)

where ${\tilde{z}}_{\mathrm{c}}$ is the distance of neutral plane from the bottom, and $\tilde{z}$ describes the z-directional position of the point in the coordinates of ${\tilde{z}}_{\mathrm{c}}$ as the origin.

Furthermore, according to the principle of minimum potential energy, we can obtain the governing equation as follows

$$\begin{array}{l}{\displaystyle {\int }_0^l\left[\left(A_{xx}\frac{\partial u_0}{\partial x}+B_{xx}\frac{\partial \phi }{\partial x}-\mu \frac{\partial f}{\partial x}\right)\frac{\partial \delta u_0}{\partial x}\right.}\\ +\left(B_{xx}\frac{\partial u_0}{\partial x}+D_{xx}\frac{\partial \phi }{\partial x}-\mu \frac{{\partial }^2}{\partial x^2}\left[\frac{\partial }{\partial x}\left(\overline{N}\frac{\partial w_0}{\partial x}\right)-q\right]\right)\frac{\partial \delta \phi }{\partial x}\\ \left.+\left(A_{xz}\left(\frac{\partial w_0}{\partial x}+\phi \right)+\mu \frac{\partial }{\partial x}\left[\frac{\partial }{\partial x}\left(\overline{N}\frac{\partial w_0}{\partial x}\right)-q\right]\right)\left(\frac{\partial \delta w_0}{\partial x}+\delta \phi \right)\right]\mathrm{d}x\\ -{\displaystyle {\int }_0^l\left[\overline{N}\left(\frac{\partial w_0}{\partial x}\frac{\partial \delta w_0}{\partial x}\right)+q\delta w_0+f\delta u_0\right]}\mathrm{d}x+{\left[N_b\delta u_0+\overline{V}\delta w_0+\overline{M}\delta \phi \right]}_0^l=0\end{array}$$

(10)

Then, equilibrium equations can be obtained:

$$\begin{array}{l}\frac{\partial N}{\partial x}+f=0\\ \frac{\partial Q}{\partial x}-\frac{\partial }{\partial x}\left(\overline{N}\frac{\partial w_0}{\partial x}\right)+q=0\\ \frac{\partial M}{\partial x}-Q=0\end{array}$$

(11)

Subject to the boundary conditions:

$$\begin{array}{l}{u}_0 \mathrm{or} N\\ w \mathrm{or} V=Q-\overline{N}\frac{\partial w_0}{\partial x}\\ \phi \mathrm{or} M\end{array}$$

(12)

where

$$N={\displaystyle {\int }_A{\sigma }_{xx}}\mathrm{d}A$$

(13)

$$M={\displaystyle {\int }_A\tilde{z}{\sigma }_{xx}}\mathrm{d}A$$

(14)

$$Q={\displaystyle {\int }_A{k}_s{\sigma }_{xz}}\mathrm{d}A$$

(15)

in which $N$, $M$, and $Q$ stand for induced normal force, bending moment, and shear force, respectively; $k_s$ is the shear correction factor.

Submitting Equations (13)–(15) to the integral form of Equation (5), we can obtain the relation of the resultant force between with displacement as follows

$$N-\mu \frac{{\partial }^{2}N}{\partial x^2}=A_{xx}\frac{\partial u_0}{\partial x}+B_{xx}\frac{\partial \phi }{\partial x}$$

(16)

$$M-\mu \frac{{\partial }^{2}M}{\partial x^2}=B_{xx}\frac{\partial u_0}{\partial x}+D_{xx}\frac{\partial \phi }{\partial x}$$

(17)

$$Q-\mu \frac{{\partial }^{2}Q}{\partial x^2}=A_{xz}\left(\frac{\partial w_0}{\partial x}+\phi \right)$$

(18)

in which

$$\begin{array}{l}{A}_{xx}={\displaystyle {\int }_{A}E\left(\tilde{z}\right)\mathrm{d}A}\\ B_{xx}={\displaystyle {\int }_{A}E\left(\tilde{z}\right)\left(\tilde{z}-{\tilde{z}}_{\mathrm{c}}\right)\mathrm{d}A}\\ D_{xx}={\displaystyle {\int }_{A}E\left(\tilde{z}\right){\left(\tilde{z}-{\tilde{z}}_{\mathrm{c}}\right)}^2\mathrm{d}A}\\ A_{xz}=k_s{\displaystyle {\int }_{A}G\left(\tilde{z}\right)\mathrm{d}A}\end{array}$$

(19)

Submitting Equation (11) to Equations (16)–(18), the nonlocal resultant force can be derived:

$$\begin{array}{l}N=A_{xx}\frac{\partial u_0}{\partial x}+B_{xx}\frac{\partial \phi }{\partial x}-\mu \frac{\partial f}{\partial x}\\ M=B_{xx}\frac{\partial u_0}{\partial x}+D_{xx}\frac{\partial \phi }{\partial x}-\mu \left[\frac{\partial }{\partial x}\left(\overline{N}\frac{\partial w_0}{\partial x}\right)-q\right]\\ Q=A_{xz}\left(\frac{\partial w_0}{\partial x}+\phi \right)+\mu \frac{\partial }{\partial x}\left[\frac{\partial }{\partial x}\left(\overline{N}\frac{\partial w_0}{\partial x}\right)-q\right]\end{array}$$

(20)

3. Deterministic Static Analysis of FGM Timoshenko Nanobeam

To deal with the differential equation of the FGM Timoshenko beam, we adopt a numerical method, called spline finite point method (SFPM). SFPM can be used as an alternative choice to finite difference and finite element method, thanks to its high accuracy and the capacity to solve the differential equations [42]. We will provide a brief introduction of the SPFM in this section.

3.1. Formulas of SFPM

Consider a FGM Timoshenko nanobeam with length $L$, the beam can be divided by several points in x-axial direction, as shown in Figure 3. We assume the number of the points are $n$ + 1; then, the position of the points can be described as

$$x_i=x_0+i\Delta l \left(i=0,1,2,\dots ,n\right)$$

(21)

where $\Delta l=l/n$ stands for the distance between two adjacent two points, and

$$0=x_0<x_1<x_2<\dots <x_{n-1}<x_n=l$$

(22)

In this work, the cubic B-spline function is adopted to the discrete displacement field, due to its superiority in approximation. The detail expression of the cubic B-spline ${\phi }_3\left(x\right)$ can be written as follows

$${\phi }_3\left(x\right)=\frac{1}{6}\left\{\begin{array}{ll}{\left(x+2\right)}^3& x\in \left[-2,-1\right]\\ {\left(x+2\right)}^3-4{\left(x+1\right)}^3& x\in \left[-1,0\right]\\ {\left(2-x\right)}^3-4{\left(1-x\right)}^3& x\in \left[0,1\right]\\ {\left(2-x\right)}^3& x\in \left[1,2\right]\\ 0& x\notin \left[-2,2\right]\end{array}\right.$$

(23)

A function $F\left(x\right)$ can be expressed as a group of spline basic functions, i.e.,

$$F\left(x\right)={\displaystyle \sum _{i=-1}^{n+1}{\varphi }_i{a}_i}=\left[\varphi \right]\left\{a\right\}$$

(24)

where $\left[\varphi \right]$ and $\left\{a\right\}$ denote the vector of spline basic function and coefficient, respectively. The basic function is constructed by cubic B-spline function (see Figure 4), which can be written as

$$\begin{array}{ll}{\varphi }_i\left(x\right)& =\frac{10}{3}{\phi }_3\left(\frac{x-x_i}{l}-i\right)-\frac{4}{3}{\phi }_3\left(\frac{x-x_i}{l}-i+\frac{1}{2}\right)-\frac{4}{3}{\phi }_3\left(\frac{x-x_i}{l}-i-\frac{1}{2}\right)\\ & +\frac{1}{6}{\phi }_3\left(\frac{x-x_i}{l}-i+1\right)+\frac{1}{6}{\phi }_3\left(\frac{x-x_i}{l}-i-1\right)\end{array}$$

(25)

where ${\varphi }_i$ satisfies ${\varphi }_i\left(x_j\right)={\delta }_{ij}$, and ${\delta }_{ij}$ is the Kronecker delta.

Similarly, the displacement can also be described based on the spline function, as follows

$$\begin{array}{l}{u}_0={\displaystyle \sum _{i=-1}^{n+1}{\varphi }_i{U}_i}\\ w_0={\displaystyle \sum _{i=-1}^{n+1}{\varphi }_i{W}_i}\\ {\phi }_0={\displaystyle \sum _{i=-1}^{n+1}{\varphi }_i{\vartheta }_i}\end{array}$$

(26)

It is more convenient to express via a matrix; to do so, the displacement field can be written as:

$$U={\left[u_0,w_0,\phi \right]}^{\mathrm{T}}=\left[N\right]\left\{C\right\}$$

(27)

where $\left[N\right]$ and $\left\{C\right\}$ are the shape function and the nodal displacement, respectively. $\left[N\right]$ can be written as

$$\left[N\right]=\left[\begin{array}{ccc}\left[\varphi \right]& \left[0\right]& \left[0\right]\\ \left[0\right]& \left[\varphi \right]& \left[0\right]\\ \left[0\right]& \left[0\right]& \left[\varphi \right]\end{array}\right]$$

(28)

where $\left[\varphi \right]$ is the spline basic function (SBF), which is constructed by the cubic B-spline function. To guarantee that the boundary condition of the Timoshenko beam is satisfied at both of the ends, the SBF is determined by the following conditions:

$$\begin{array}{l}{\varphi }_i=0 \left(i\ne -1\right)\\ {\varphi }_i=0 \left(i\ne n+1\right)\\ {{\varphi }^{\prime }}_i=0 \left(i\ne -1,0\right)\\ {{\varphi }^{\prime }}_i=0 \left(i\ne n,n+1\right)\end{array}$$

(29)

Submitting the Equations (26)–(28) to the Equation (10), the equilibrium equations of SFPM can be derived as follows:

$$\left[G\right]\left\{C\right\}=\left\{P\right\}$$

(30)

where $\left[G\right]$ is stiffness matrix, $\left\{C\right\}$ is nodal displacement vector, and $\left\{P\right\}$ is the load vector.

3.2. Material Parametric Study of the Bending Response

To verify the pressed method, SFPM is a good alternative choice to FEM, in terms of accuracy, and a numerical example of FGM Timoshenko nanobeam subject to uniform load is investigated. An FGM Timoshenko nanobeam with rectangle section is examined, where $b/h=1$ and $L/h=10$. Then, the displacement results are compared with results of previous published results, which are defined in dimensionless, i.e.,

$$w_{\max }=100{\delta }_{\max }\frac{EI}{q_0{L}^4}$$

(31)

The model describing the FGM nanobeam can be degenerated into that to describe the nanobeam made up of single material, i.e., n = 0. The parameters of this case are as follows, Young’s modulus $E=30\times {10}^6 \mathrm{Pa}$, Poisson’s ratio $\upsilon =0.3$, and $G=11.538 \times { 10}^6 \mathrm{Pa}$. Actually, ${\delta }_{\max }$ in (31) is the max deflection of the nanobeam, which is related to the Young’s modulus $E$. The deflection ${\delta }_{\max }$ is inversely proportional to Young’s modulus $E$. Thus, $w_{\max }$ is independent on Young’s modulus $E$. To illustrate the effectiveness and accuracy, a simple supported nanobeam, made up of a single material, is investigated, and the results are listed in

Table 2. Comparison of the results of SFPM with those of analytical method and FEM.

$\mathit{\mu }$	Analytical [44]	FEM [17]		SFPM
	Value	Value	Error	Value	Error
0	1.3483	1.3264	−1.62%	1.3346	−1.02%
1	1.4937	1.4514	−2.83%	1.4596	−2.28%
2	1.6391	1.5764	−3.83%	1.5846	−3.33%
3	1.7845	1.7014	−4.66%	1.7096	−4.20%
4	1.9299	1.8264	−5.36%	1.8346	−4.94%
5	2.0754	1.9514	−5.97%	1.9596	−5.58%

. It is observed that the dimensionless deflection at mid-span of the nanobeam, obtained by SFPM and FEM, have a good agreement with those calculated by analytical method. Furthermore, it can be seen that SFPM had higher accuracy, compared with FEM.

Further, an FGM Timoshenko nanobeam is investigated by SFPM and FEM. The dimensionless deflection of the FGM Timoshenko nanobeam obtained by SFPM and FEM are listed in

Table 3. Comparison of the dimensionless deflection of the FGM nanobeam, calculated by FEM and SFPM.

$\mathit{\mu }$	Method	n = 0	n = 0.2	n = 0.5	n = 1	n = 2	n = 5
0	FEM	1.3357	0.7681	0.5871	0.4795	0.3984	0.3234
	SFPM	1.3357	0.7681	0.5871	0.4795	0.3984	0.3234
1	FEM	1.4607	0.8401	0.6422	0.5246	0.4359	0.3538
	SFPM	1.4608	0.8401	0.6422	0.5246	0.4359	0.3538
2	FEM	1.5857	0.9120	0.6974	0.5698	0.4734	0.3842
	SFPM	1.5858	0.9121	0.6974	0.5698	0.4734	0.3842
3	FEM	1.7107	0.9840	0.7525	0.6149	0.5109	0.4146
	SFPM	1.7108	0.9840	0.7525	0.6149	0.5109	0.4146
4	FEM	1.8357	1.0560	0.8077	0.6600	0.5484	0.4450
	SFPM	1.8358	1.0560	0.8077	0.6600	0.5484	0.4450
5	FEM	1.9607	1.1280	0.8628	0.7051	0.5859	0.4755
	SFPM	1.9608	1.1280	0.8628	0.7051	0.5859	0.4755

. As the results show, the deflection calculated by the two methods have a good agreement. The results of SFPM list in Table 3 are calculated according to the following formula,

$${\tilde{w}}_{\max }=100{\delta }_{\max }\frac{E_{\mathrm{m}}I}{q_0{L}^4}$$

(32)

To further study the effect of the material property on the bending response of the nanobeam, the Young’s and shear moduli are taken as uncertain parameters to discuss the effect they have on the dimensionless deflection. Figure 5 demonstrates the variation of the deflection at the midspan point under the varying Young’s and shear moduli in the range $\left[p-\gamma p, p+\gamma p\right]$, where $p=\left(E,G\right)$, and $\gamma$ is a concept to describe the variation magnitude of material properties. As Figure 5 and Figure 6 shown, some traits can be summarized as follows.

(a)

The dimensionless deflection of the nanobeam decreases with the increase of the Young’s and shear moduli. The reduction of the dimensionless deflection is because the beam becomes stiffer, as contributed by the Young’s and shear moduli. It means that the dimensionless deflection of the nanobeam with uncertain material properties is indeterministic. Thus, it is necessary to analyze the response of the nanobeam with defects.

(b)

Comparatively, the shear modulus has less impact on dimensionless deflection than Young’s modulus. Dimensionless deflection is mainly determined by Young’s modulus. Thus, the impact of shear modulus on the static response can be ignored during the uncertainty analysis in problems with too many uncertain parameters.

4. Uncertain Analysis of FGM Timoshenko Nanobeam with Defects

4.1. Sources of Uncertainty

The aforementioned theory and method are under the deterministic framework, i.e., the materials are perfect, and the loads are deterministic. However, in the real world, the material property of nanostructures is always uncertain, due to the material defect, and the external load always cannot keep as a constant value. As the simple analysis in Section 3, the uncertain propagation analysis is necessary. Thus, considering the uncertainty, the framework of the defective FGM Timoshenko nanobeam subject to random loads is introduced in this section.

4.2. The Uncertainty Model of the Defective Nanobeam Subjected to Random Loads

In FGM Timoshenko nanobeam analysis, uncertain parameters with specific probability distribution can be quantified as a random vector $\mathbf{\alpha }=\left[{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_s\right]$, and those without sufficient statistical information are modeled as interval vectors $\mathbf{\beta }=\left[{\beta }_1,{\beta }_2,\dots ,{\beta }_t\right]$. Considering the two types of uncertain parameters, a hybrid uncertain system was established, where uncertain parameters in $\mathbf{\alpha }$ and $\mathbf{\beta }$ are assumed to be independent. The expectation and variance vector of $\mathbf{\alpha }$ can be written as

$${\mathbf{\mu }}_{\mathbf{\alpha }}={\left({\mu }_1,{\mu }_1,\dots ,{\mu }_s\right)}^{\mathrm{T}}, {\mathbf{\sigma }}_{\mathbf{\alpha }}^2={\left({\sigma }_1^2,{\sigma }_2^2,\dots ,{\sigma }_s^2\right)}^{\mathrm{T}}$$

(33)

where

$${\mu }_i=\mu \left({\alpha }_i\right), {\sigma }_i^2={\sigma }^2\left({\alpha }_i\right), \left(i=1,2,\dots ,s\right)$$

(34)

The interval vector is described as

$$\mathbf{\beta }\in \left[\underset{\_}{\mathbf{\beta }} , \overline{\mathbf{\beta }}\right]$$

(35)

where

$$\underset{\_}{\mathbf{\beta }} ={\left({\underset{\_}{\beta }}_1,{\underset{\_}{\beta }}_2,\dots ,{\underset{\_}{\beta }}_t\right)}^{\mathrm{T}}, \overline{\mathbf{\beta }} ={\left({\overline{\beta }}_1,{\overline{\beta }}_2,\dots ,{\overline{\beta }}_t\right)}^{\mathrm{T}}$$

(36)

in which $\underset{\_}{\mathbf{\beta }}$ and $\overline{\mathbf{\beta }}$ denotes the lower and upper bound vector, respectively. Additionally, the midpoint and radius of $\mathbf{\beta }$ is defined as

$${\mathbf{\beta }}^{\mathrm{c}}=\frac{\left(\overline{\mathbf{\beta }}+\underset{\_}{\mathbf{\beta }} \right)}{2}, {\mathbf{\beta }}^{\mathrm{r}}=\frac{\left(\overline{\mathbf{\beta }}-\underset{\_}{\mathbf{\beta }} \right)}{2}$$

(37)

Therefore, the SFPM equation of the FGM Timoshenko nanobeam, considering the two types of uncertain parameters, is convert to uncertain system, as follows

$$\left[G\left(\mathbf{\alpha },\mathbf{\beta }\right)\right]\left\{C\left(\mathbf{\alpha },\mathbf{\beta }\right)\right\}=\left\{P\left(\mathbf{\alpha },\mathbf{\beta }\right)\right\}$$

(38)

Thus, a hybrid uncertain framework to analyze the size-dependent mechanical behaviors of defective FGM nanobeams is subjected to random loading. The method to solve the hybrid uncertain system is introduced in next subsection.

4.3. Hybrid Method of Polynormal Chaos Expansion and Dimension-Wise Analysis (PCE-WD)

In the PCE-WD, the lower and upper bounds of the expectation and variance of the uncertain system are calculated by the hybrid method, in which the expectation and variance of the random response are determined by the polynomial chaos expansion. Then, the bounds of the bounds of the two statistical parameters are predicted by the dimension-wise (DW) analysis.

4.3.1. Formulas of Polynormal Chaos Expansion (PCE)

The random field is constructed by the general function w.r.t. independent random variables $\mathbf{\xi }$. The polynomial chaos expansion is used to deal with the random parameters $\mathbf{\alpha }$. Thus, the random parameters $\mathbf{\alpha }$ should be transformed to the standard one $\mathbf{\xi }$ first. Then, expansion is conducted in the transformed space. For the different types of random distribution of the random variables, the respective polynomial can be chosen as basic to expanded, in order to approximate the random field. For instance, for uniform, exponential, beta, and gamma random parameters, the random field can be constructed by the Legendre, Jacobi, Laguerre, and generalized Laguerre polynomials [45], respectively. In this work, we assume the random variables $\mathbf{\alpha }$ as the normal ones, and it can be standardized to $\mathit{\xi }={\left({\xi }_1,{\xi }_2,\dots ,{\xi }_m\right)}^{\mathrm{T}}$ by a linear transformation, i.e.,

$${\xi }_i=\frac{{\alpha }_i-\mu \left({\alpha }_i\right)}{\sigma \left({\alpha }_i\right)} i=1,2,\dots ,m$$

(39)

Thus, the random field is expanded by the Hermite polynomials, which are expressed as

$$\Phi \left(\mathbf{\xi }\right)={\left({\Phi }_0\left(\mathbf{\xi }\right),{\Phi }_1\left(\mathbf{\xi }\right),{\Phi }_2\left(\mathbf{\xi }\right),\dots \right)}^{\mathrm{T}}$$

(40)

In this context, the random and interval parameters are assumed to be independent. Hence, in the procedure of the PCE, the interval parameters $\mathbf{\beta }$ are taken as constant ones, and the response $\mathit{C}\left(\mathbf{\alpha },\mathbf{\beta }\right)$ can be converted to the following form

$$\mathit{C}\left(\mathbf{\alpha },\mathbf{\beta }\right)=\mathit{C}\left(\mathbf{\xi },\mathbf{\beta }\right)=\mathit{H}\left(\mathbf{\beta }\right)\Phi \left(\mathbf{\xi }\right)$$

(41)

where $\mathit{H}\left(\mathbf{\beta }\right)$ denotes the coefficient matrix, which can be written as

$$\mathit{H}\left(\mathbf{\beta }\right)=\left[\begin{array}{cccc}{H}_{1,0}\left(\mathbf{\beta }\right)& H_{1,1}\left(\mathbf{\beta }\right)& H_{1,2}\left(\mathbf{\beta }\right)& \cdots \\ H_{2,0}\left(\mathbf{\beta }\right)& H_{2,1}\left(\mathbf{\beta }\right)& H_{2,2}\left(\mathbf{\beta }\right)& \cdots \\ ⋮& ⋮& ⋮& \ddots \\ H_{N,0}\left(\mathbf{\beta }\right)& H_{N,1}\left(\mathbf{\beta }\right)& H_{N,2}\left(\mathbf{\beta }\right)& \dots \end{array}\right]$$

(42)

In the paper, the p-order complete basis of Hermite polynomials are used to construct the surrogate model of the system response. The system response is approximated by S truncated terms, the number of which can be determined by the following formula

$$S=\frac{\left(n+p\right)!}{n!p!}$$

(43)

At a constant interval input vector, thus, $\mathit{H}\left(\mathbf{\beta }\right)$ can be obtained via the linear regression analysis [46]. In this context, the spare grid collocation (SGC) strategy (see details in Ref. [45]) is adopted to generate the samples, which is a powerful tool to enhance the computational efficiency. The number of samples $m_{\xi }$ are recommended more than twice that of the PCE terms $S$, i.e., $m_{\xi }\ge 2S$. The $m_{\xi }$ samples are denoted as ${\xi }^{\left(i_{\xi }\right)},i_{\xi }=1,2,\dots ,m_{\xi }$, and the random parameters of mechanical system at the sample points can be expressed by the standard ones, as follows

$${\mathbf{\alpha }}^{\left(i_{\xi }\right)}=\mu \left(\mathbf{\alpha }\right)+\sigma \left(\mathbf{\alpha }\right)\circ {\xi }^{\left(i_{\xi }\right)}$$

(44)

where ‘$\circ$’ denotes the operator to multiply the elements of the vector or matrix, correspondingly. Assuming the interval is constant, the value of the basic function at the sample points of standard random parameter spaces can be obtained.

$${\mathit{\Phi }}_{\mathrm{out}}=\left[\begin{array}{cccc}{\Phi }_0\left({\xi }^{\left(1\right)}\right)& {\Phi }_0\left({\xi }^{\left(2\right)}\right)& \cdots & {\Phi }_0\left({\xi }^{\left(m_{\xi }\right)}\right)\\ {\Phi }_1\left({\xi }^{\left(1\right)}\right)& {\Phi }_1\left({\xi }^{\left(2\right)}\right)& \cdots & {\Phi }_1\left({\xi }^{\left(m_{\xi }\right)}\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\Phi }_S\left({\xi }^{\left(1\right)}\right)& {\Phi }_S\left({\xi }^{\left(2\right)}\right)& \cdots & {\Phi }_S\left({\xi }^{\left(m_{\xi }\right)}\right)\end{array}\right]$$

(45)

The random parameters in the real mechanical system at the sample points are taken as the input variables of the static bending analysis, and the response of the nanobeam can be calculated by deterministic analysis, based on SFPM, i.e.,

$${\mathit{U}}_{\mathrm{out}}\left(\mathbf{\beta }\right)=\left[\mathit{U}\left({\mathbf{\alpha }}^{\left(1\right)},\mathbf{\beta }\right),\mathit{U}\left({\mathbf{\alpha }}^{\left(2\right)},\mathbf{\beta }\right),\dots ,\mathit{U}\left({\mathbf{\alpha }}^{\left(m_{\xi }\right)},\mathbf{\beta }\right)\right]$$

(46)

According to the Equations (41), (45), and (46), the coefficient matrix $\mathit{C}\left(\beta \right)$ can be easily obtained, i.e.,

$$\mathit{C}\left(\beta \right)={\mathit{U}}_{\mathrm{out}}\left(\beta \right){\mathit{\Phi }}_{\mathrm{out}}^{\mathrm{T}}{\left({\mathit{\Phi }}_{\mathrm{out}}{\mathit{\Phi }}_{\mathrm{out}}^{\mathrm{T}}\right)}^{-1}$$

(47)

Then, the expectation $\mu \left(\mathbf{\beta }\right)$ and variance ${\sigma }^2\left(\mathbf{\beta }\right)$ can be derived and expressed as follows

$$\begin{array}{l}\mu \left(\mathbf{\beta }\right)=\mu \left(\mathit{U}\left(\mathbf{\xi },\mathbf{\beta }\right)\right)={\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\right)\\ {\sigma }^2\left(\mathbf{\beta }\right)={\sigma }^2\left(\mathit{U}\left(\mathbf{\xi },\mathbf{\beta }\right)\right)=\mathit{C}\left(\mathbf{\beta }\right)\circ \mathit{C}\left(\mathbf{\beta }\right)\langle \mathit{\Phi }\left(\mathbf{\xi }\right)\circ \mathit{\Phi }\left(\mathbf{\xi }\right)\rangle -{\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\right)\circ {\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\right)\end{array}$$

(48)

where ${\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\right)$ is the first column of $\mathit{C}\left(\mathbf{\beta }\right)$, and $\langle •\rangle$ is a norm operator. What should be noted is that the expectation and variance are the intervals, and the lower and upper bounds can be predicted dimension-wise, which will be introduced in next subsection.

4.3.2. Formulas of Dimension-Wise (DW) Analysis

Aiming to determine the bounds of the expectation and variance of the response for the nanobeam, the dimension-wise approach is adopted to integrate with the PCE. The idea of the WD is that separating the variables from a high-dimensional problem and converting the original problem to that combining univariate function, i.e.,

$$f\left(x_1,x_2,\dots ,x_n\right)=f_0+{\displaystyle \sum _{i=1}^n{f}_i\left(x_i\right)}$$

(49)

where $f_0$ is the zeroth-order effect, and $f_i\left(x_i\right)$ provides the first-order effect associated with $x_i$, acting independently upon the output. $f_i\left(x_i\right)$ is a univariate function, which can be considered as the dimension-wise cut along the i-th input dimensionality. It is noted that the hierarchical expansion of the response function in Equation (49) is just to interpret the idea of DW, rather than utilized as the surrogate model. To identify the min/max point, the Legendre polynomials are adopted, which are orthogonal to each other in the interval $\left[-1, 1\right]$. The interval parameters should be standardized with the following rule

$$\mathbf{\zeta }=\frac{\mathbf{\beta }-{\mathbf{\beta }}^{\mathrm{c}}}{{\mathbf{\beta }}^{\mathrm{r}}}$$

(50)

In order to be convenient, the first two statistical moments are denoted as

$$\mathit{w}\left(\mathbf{\beta }\right)=\left[\begin{array}{l}\mu \left(\mathbf{\beta }\right)\\ {\sigma }^2\left(\mathbf{\beta }\right)\end{array}\right]$$

(51)

Furthermore, the i-th dimensional cut of $\mathit{w}\left(\mathbf{\beta }\right)$ can be expressed as

$$\mathit{w}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)=\left[\begin{array}{l}{\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)\\ \mathit{C}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)\circ \mathit{C}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)\langle \mathit{\Phi }\left(\mathbf{\xi }\right)\circ \mathit{\Phi }\left(\mathbf{\xi }\right)\rangle -{\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)\circ {\mathit{C}}_{1\mathrm{col}}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)\end{array}\right]$$

(52)

where $\mathbf{\beta }\left({\zeta }_i\right)$ is defined as

$$\mathbf{\beta }\left({\zeta }_i\right)={\left[{\beta }_1^{\mathrm{c}},\dots {\beta }_{i-1}^{\mathrm{c}},{\beta }_i^{\mathrm{c}}+{\beta }_i^{\mathrm{r}}{\zeta }_i,{\beta }_{i+1}^{\mathrm{c}},\dots ,{\beta }_n^{\mathrm{c}}\right]}^{\mathrm{T}}, {\zeta }_i\in \left[-1, 1\right]$$

(53)

To approximate the synthesis vector, $\mathit{w}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)$ can be expressed as

$$\mathit{w}\left(\mathbf{\beta }\left({\zeta }_i\right)\right)={\mathit{T}}^{\left(i\right)}\left({\zeta }_i\right)=\left[T_1^{\left(i\right)}\left({\zeta }_i\right),T_2^{\left(i\right)}\left({\zeta }_i\right),\dots ,T_{2N}^{\left(i\right)}\left({\zeta }_i\right)\right]={\mathit{D}}^{\left(i\right)}\mathit{L}\left({\zeta }_i\right)$$

(54)

where $T_1^{\left(i\right)}\left({\zeta }_i\right)$ denotes the approximate polynomial of the i-th dimension-wise cut of the j-the component in $\mathit{w}\left(\mathbf{\beta }\right)$; ${\mathit{D}}^{\left(i\right)}$ is the coefficient matrix; $\mathit{L}\left({\zeta }_i\right)$ is a vector composed of the first n_L-order Legendre polynomials, i.e.,

$${\mathit{D}}^{\left(i\right)}=\left[\begin{array}{cccc}{d}_{1,0}^{\left(i\right)}& d_{1,1}^{\left(i\right)}& \cdots & d_{1,n_L}^{\left(i\right)}\\ d_{2,0}^{\left(i\right)}& d_{2,1}^{\left(i\right)}& \cdots & d_{2,n_L}^{\left(i\right)}\\ ⋮& ⋮& \ddots & ⋮\\ d_{2N,0}^{\left(i\right)}& d_{2N,1}^{\left(i\right)}& \cdots & d_{2N,n_L}^{\left(i\right)}\end{array}\right]$$

(55)

$$\mathit{L}\left({\zeta }_i\right)={\left[L_0\left({\zeta }_i\right),L_1\left({\zeta }_i\right),\dots ,L_{n_L}\left({\zeta }_i\right)\right]}^{\mathrm{T}}.$$

(56)

The elements of ${\mathit{D}}^{\left(i\right)}$ can be obtained in the following way

$$d_{j,k}^{\left(i\right)}=\left(2k+1\right){\displaystyle \sum _{l=1}^s\frac{L_k\left({\zeta }_l^{\mathrm{int}}\right)w_j\left(\mathbf{\beta }\left({\zeta }_l^{\mathrm{int}}\right)\right)}{\left(1-{\left({\zeta }_l^{\mathrm{int}}\right)}^2\right){\left[{L^{\prime }}_s\left({\zeta }_l^{\mathrm{int}}\right)\right]}^2}}$$

(57)

It is the Gauss integral formula, in which ${\beta }_l^{\mathrm{int}}$ is the l-th integral point; $w_j\left({\beta }_l^{\mathrm{int}}\right)$ is the j-th component of $\mathit{w}\left(\mathbf{\beta }\right)$; ${\zeta }_i^{\mathrm{int}}\left(l=1,2,\dots ,s\right)$ is the zero-roots of the s-order Legendre polynomial. The zero-roots can be obtained by

$$\frac{\mathrm{d}{T}_j^{\left(i\right)}\left({\zeta }_i\right)}{\mathrm{d}{\zeta }_i}=0$$

(58)

The solutions are denoted as ${\zeta }_{j,i}^{\left(l\right)}\left(l=1,2,\dots ,n_L-1\right)$ and translated into $\left[-1, 1\right]$, according to the definition domain of Legendre polynomials, as follows:

$${\zeta }_{j,i}^{\left(l\right)}=\left\{\begin{array}{ccc}1& \mathrm{if} \mathrm{Im}\left({\zeta }_{j,i}^{\left(l\right)}\right)\ne 0& \\ -1& \mathrm{if} \mathrm{Im}\left({\zeta }_{j,i}^{\left(l\right)}\right)\ne 0\&\left|\mathrm{Re}\left({\zeta }_{j,i}^{\left(l\right)}\right)\right|>1& l=1,2,\dots ,n_L-1\\ {\zeta }_{j,i}^{\left(l\right)}& \mathrm{if} \mathrm{Im}\left({\zeta }_{j,i}^{\left(l\right)}\right)\ne 0\&\left|\mathrm{Re}\left({\zeta }_{j,i}^{\left(l\right)}\right)\right|<1& \end{array}\right.$$

(59)

where $\mathrm{Im}\left(•\right)$ and $\mathrm{Re}\left(•\right)$ are the imaginary and real parts, respectively. Thus, the min/max points of $w_j\left(\beta \left({\zeta }_i\right)\right)$ are expressed as ${\zeta }_{j,i}^{\min }$ and ${\zeta }_{j,i}^{\max }$, which can be obtained by

$$T_j^{\left(i\right)}\left({\zeta }_{j,i}^{\mathrm{sym}}\right)={\mathrm{sym}}_{{\zeta }_j\in {\zeta }_{j,i}^{\mathrm{ext}}}{T}_j^{\left(i\right)}\left({\zeta }_j\right), \mathrm{sym}\triangleq \min , \max$$

(60)

where

$${\zeta }_{j,i}^{\mathrm{ext}}=\left[-1,1,{\zeta }_{j,i}^{\left(l\right)},\dots ,{\zeta }_{j,i}^{\left(n_{\mathrm{L}}-1\right)}\right]$$

(61)

Therefore, the min/max points of the j-th component in $\mathit{w}\left(\beta \right)$ can be assembled by traversing i from 1 to n parallelly, i.e.,

$${\beta }_j^{\mathrm{sym}}=\left[{\beta }_{j,1}^{\mathrm{sym}},{\beta }_{j,2}^{\mathrm{sym}},\dots ,{\beta }_{j,n}^{\mathrm{sym}}\right], j=1,2,\dots ,2N; \mathrm{sym}\triangleq \min , \max$$

(62)

Then, $\mathit{w}\left({\beta }_j^{\mathrm{sym}}\right)$ can be determined by the PCE, i.e.,

$${\mathit{w}}_{j\mathrm{-out}}^{\mathrm{sym}}=\left[\begin{array}{l}{\mathit{C}}_{1\mathrm{col}, j\mathrm{-out}}^{\mathrm{sym}}\\ {\mathit{C}}_{j\mathrm{-out}}^{\mathrm{sym}}\circ {\mathit{C}}_{j\mathrm{-out}}^{\mathrm{sym}}〈\mathit{\Phi }\left(\mathbf{\xi }\right)\circ \mathit{\Phi }\left(\mathbf{\xi }\right)〉-{\mathit{C}}_{1\mathrm{col}, j\mathrm{-out}}^{\mathrm{sym}}\circ {\mathit{C}}_{1\mathrm{col}, j\mathrm{-out}}^{\mathrm{sym}}\end{array}\right], \mathrm{sym}\triangleq \min , \max$$

(63)

To obtain the bounds of $\mathit{w}\left(\mathbf{\beta }\right)$, the min/max matrixes can be assembled by traversing j from 1 to 2 N,

$${\mathit{w}}_{\mathrm{mat}}^{\mathrm{sym}}=\left[w_{1\mathrm{-out}}^{\mathrm{sym}},w_{2\mathrm{-out}}^{\mathrm{sym}},\dots ,w_{2N\mathrm{-out}}^{\mathrm{sym}}\right]$$

(64)

Then, the bounds of $\mathit{w}\left(\mathbf{\beta }\right)$ can be obtained as follows

$$w=\left[\underset{\_}{w},\overline{w}\right]=\left[\mathrm{diag}\left({\mathit{w}}_{\mathrm{mat}}^{\min }\right),\mathrm{diag}\left({\mathit{w}}_{\mathrm{mat}}^{\max }\right)\right]$$

(65)

Combining the PCE and DW methods, the hybrid uncertainty propagation analysis method, namely PCE–DW, is established. Its execution process is shown in Figure 7.

5. Numerical Examples

To demonstrate the influence of the uncertainties (including random and interval uncertain parameters) of the nanobeam on the static bending response, a simply supported nanobeam subject to distributed load (see Figure 1) is investigated. The material properties of nanobeam are defective; thus, the Young’s $E_i$ and shear $G_i$ moduli are modeled as the interval parameters, where $i=\mathrm{m}, \mathrm{c}$. In the real physical world, the position, direction, and value of the load are often uncertain, and the uncertainty of the load will have a significant impact on the mechanical behaviors of the nano-beam. Therefore, the uncertainty of the load should be considered in the mechanical response analysis of the nano-structures. In this work, it is considered that the load consists of two parts, i.e., the nominal value and dispersion of the load, which are expressed by means of statistics, namely random load. The load is undetermined, so that the distributed load $q$ is quantified as the normal random parameter. Combining the two types of uncertainties, three cases (list in

Table 4. Hybrid uncertain parameters for the nanobeam.

Parameters (Unit)	Case 1	Case 2	Case 3
$E_{\mathrm{m}}$(GPa)	$70$	$\left[66.5, 73.5\right]$	$\left[66.5, 73.5\right]$
$G_{\mathrm{m}}$(GPa)	$26$	$\left[24.7,27.3 \right]$	$\left[24.7,27.3 \right]$
$E_{\mathrm{c}}$(GPa)	$393$	$\left[373.35, 412.65 \right]$	$\left[373.35, 412.65 \right]$
$G_{\mathrm{c}}$(GPa)	$157$	$\left[149.15, 164.85 \right]$	$\left[149.15, 164.85 \right]$
$q$(N)	$N\left(1, {0.05}^2\right)$	$1$	$N\left(1, {0.05}^2\right)$
$N$(N)	$N\left(1, {0.05}^2\right)$	$1$	$N\left(1, {0.05}^2\right)$

) are calculated. As Table 4 shown, case1~case 3 are three conditions, considered random, interval, and hybrid parameters, respectively.

The proposed method (PC-DWM) is compared with MCS for uncertainty problems. The MCS obtains the boundary and statistical characteristics of the mechanical response of the nanobeam, through a large number of samples, and the MCS results are considered as reference values to verify the effectiveness of the proposed method.

5.1. Case 1

In this case, only random parameters are taken as uncertainties, i.e., only the random load is considered. While, the material properties are assumed deterministic, the values of the material properties are taken as its midpoint. Utilizing the proposed PCE, we can calculate the results, including the expectation and variance of the deflection at midspan of the simple supported nanobeam.

To give a convincing verification, the response of deflection under varying nonlocal factor $\mu$ is investigated, where the material variation factor $n=1$. The results calculated by PCE are compared with those of Monte Carlo simulation (MCS), which are listed in

Table 5. The expectation calculated by PC and MCS.

$\mathit{\mu }$	Method
	PC	MCS (100)	MCS (1000)	MCS (10,000)
0	8.89 × 10−9	8.93 × 10−9	8.87 × 10−9	8.88 × 10−9
1	9.66 × 10−9	9.60 × 10−9	9.65 × 10−9	9.64 × 10−9
2	1.04 × 10−8	1.04 × 10−8	1.04 × 10−8	1.04 × 10−8
3	1.12 × 10−8	1.13 × 10−8	1.12 × 10−8	1.12 × 10−8
4	1.20 × 10−8	1.21 × 10−8	1.20 × 10−8	1.20 × 10−8
5	1.28 × 10−8	1.27 × 10−8	1.27 × 10−8	1.27 × 10−8

and

Table 6. The standard variance calculated by PC and MCS.

$\mathit{\mu }$	Method
	PC	MCS (100)	MCS (1000)	MCS (10,000)
0	7.67 × 10−10	7.47 × 10−10	7.86 × 10−10	7.66 × 10−10
1	8.43 × 10−10	7.52 × 10−10	8.62 × 10−10	8.47 × 10−10
2	9.20 × 10−10	9.13 × 10−10	9.06 × 10−10	9.09 × 10−10
3	9.97 × 10−10	9.50 × 10−10	9.59 × 10−10	9.97 × 10−10
4	1.07 × 10−9	1.01 × 10−9	1.05 × 10−9	1.08 × 10−9
5	1.15 × 10−9	1.29 × 10−9	1.15 × 10−9	1.16 × 10−9

. The load values are sampled according to the mean value and variance of the random loads. The mean value and variance of the mechanical response of each sample point are considered as the results calculated by MCS. In the MCS, different number of samples are generated, and the deterministic analysis by SFPM are conducted repeatedly. From the two tables, we can see that generating 10³ samples is a good choice to balance the computational cost and accuracy. The results calculated by PC have good agreement with those obtained by MCS. However, the efficiency of PC has a great advantage over that of MCS. Further, the response of deflection under varying material variation factor $n$ is examined, where the nonlocal factor $\mu = 1$. The results obtained by PC and MCS (with 10², 10³, 10⁴ samples) are plotted in Figure 8, which shows that the results obtained by PC have a good agreement with those obtained by MCS.

5.2. Case 2

In this case, only interval parameters are uncertain, i.e., the material properties are taken as interval parameters, considering the material defect of the nanobeam, while the loads are considered deterministic, and the value of the loads are taken as their expectation. By using DW, we can predict the bounds of the deflection at midspan point of the simple supported nanobeam. To give a convincing verification, the response of deflection under varying nonlocal factors are investigated, where the material variation factor $n=1$. Further, the results calculated by DW were compared with those of Monte Carlo simulation (MCS), which are plotted in Figure 9. The material parameters were sampled within the upper and lower bounds of the interval parameters to calculate the corresponding mechanical response of each sample point. The maximum and minimum values of the mechanical response of all sample points were treated as the upper and lower bounds of the results calculated by MCS. In the MCS, the number of samples was 1000, which is a compromise choice between computational accuracy and computational efficiency. As shown in Figure 9, the results calculated by DW have a good agreement with those obtained by MCS. Further, the response of deflection under varying material variation factors was examined, where the nonlocal factor $\mu = 1$. The results obtained by DW and MCS are plotted in Figure 9 and Figure 10, which shows that DW has good calculation accuracy.

5.3. Case 3

In this case, the nanobeams were treated as a hybrid uncertain model with random and interval parameters. In this hybrid uncertainty model, the material properties were quantified as interval parameters, and the concentrated load and uniformed uniformly distributed load were modeled as random parameters. The uncertainty model was solved by the combination of the PCE and DW, i.e., PCE–DW. The interval expectation and standard variance of the nanobeam deflection are calculated to investigate the variation of response. To illustrate the accuracy of PCE–DW, MCS were conducted with 10⁶ samples, where the number of samples for the random and interval were both 10³. The lower and upper bounds of the statistical moment were calculated and listed in

Table 7. The bounds of the expectation calculated by PCE–DW and MCS.

$\mathit{\mu }$	Lower Bound			Upper Bound
	PCE–DW (m)	MCS (m)	Error	PCE–DW (m)	MCS (m)	Error
0	8.46 × 10−9	8.38 × 10−9	0.924%	9.35 × 10−9	9.43 × 10−9	−0.851%
1	9.20 × 10−9	9.11 × 10−9	0.921%	1.02 × 10−9	1.02 × 10−9	−0.399%
2	9.94 × 10−9	9.85 × 10−9	0.838%	1.10 × 10−9	1.12 × 10−9	−1.609%
3	1.07 × 10−8	1.06 × 10−8	0.412%	1.18 × 10−9	1.20 × 10−8	−1.900%
4	1.14 × 10−8	1.13 × 10−8	0.622%	1.26 × 10−8	1.28 × 10−8	−1.275%
5	1.21 × 10−8	1.21 × 10−8	0.687%	1.34 × 10−8	1.34 × 10−8	0.470%

and

Table 8. The bounds of the variance calculated by PCE–DW and MCS.

$\mathit{\mu }$	Lower Bound			Upper Bound
	PCE–DW (m)	MCS (m)	Error	PCE–DW (m)	MCS (m)	Error
0	7.31 × 10−10	7.19 × 10−10	1.57%	8.08 × 10−10	8.88 × 10−10	−9.09%
1	8.03 × 10−10	7.49 × 10−10	7.28%	8.88 × 10−10	9.63 × 10−10	−7.84%
2	8.76 × 10−10	7.90 × 10−10	10.95%	9.68 × 10−10	1.08 × 10−9	−10.20%
3	9.49 × 10−10	8.27 × 10−10	14.74%	1.05 × 10−9	1.17 × 10−9	−10.31%
4	1.02 × 10−9	9.51 × 10−10	7.44%	1.13 × 10−9	1.26 × 10−9	−10.69%
5	1.10 × 10−9	1.04 × 10−9	5.12%	1.21 × 10−9	1.32 × 10−9	−8.34%

. From the two tables, we can see that the expectation of the deflection had a good agreement between PCE–DW and MCS, which illustrates that PCE–DW has high accuracy to predict the bounds of expectation.

In terms of efficiency, the computational time of PCE–DW was 121.31 s, while that of MCS was 2941.17 s on the same computer with Intel(R) Core (TM) i7-6700 CPU. It is obvious that PCE–DW has a great advantage over MCS in efficiency, where the accuracy of the results obtained by the two methods are on the same level. Thus, it is suggested that PCE–DW is good choice, due its superior merit, with balance in accuracy and efficiency.

6. Conclusions

Owing to superior mechanical properties, nanostructures have become a hot research area. Further, some of the nanostructures are made of FGMs, due to the superior material properties. Currently, most of the previous work has been performed on the deterministic framework. However, the properties of nanostructures are always uncertain, due to the material defect of the nanostructures. Anyway, the external load applied on nanostructures is always uncertain because of the randomness. Due to multi-source uncertainties existing, the uncertain propagation in nanostructures is necessary to analyze.

In this paper, the nanobeam made of FGM is investigated, considering the size-dependent effect, based on non-local elastic theory. The SFPM is established to analyze the bending behavior of the nanobeam-based Timoshenko theory. In comparison with FEM, SFPM has higher accuracy, taking analytical solution as a benchmark. Considering the multi-source uncertainties, the material properties are quantified as interval parameters, due to the material defect, and loads are taken as random parameters. To deal with the problems with two types uncertainties coexisting, a hybrid uncertain analysis model was established, and the PCE–DW was proposed to predict the response of nanobeam in the hybrid uncertain system. Numerical examples ultimately illustrate the effectiveness of the model and solution techniques, compared to MCS. The results furtherly verify the efficiency and accuracy.

Locking behavior is a numerical problem that often occurs in the process of finite element analysis. This paper focuses on modeling and solving the uncertainty of the material defects and load dispersion of nanobeams. This article did not focus on locking behavior. The authors will further carry out relevant research in future work.