Frequency and Buckling Analysis of FG Beams with Asymmetric Material Distribution and Thermal Effect

Abstract

The frequency and buckling characteristics of functional gradient (FG) beams with asymmetric material distribution in the temperature field are analyzed in this paper. Generally, the asymmetrical material distribution of FG beams results in a non-zero neutral axis and non-zero thermal moment. However, some previous studies adopted the treatment of homogeneous beams in which the neutral axis and thermal moment were set as zero. To this end, a comprehensive FG beam model with thermal effect is developed based on the absolute nodal coordinate formulation, in which Euler–Bernoulli beam theory, Lagrangian strain, exact curvature, thermally induced strain, and neutral axis position are considered. For the convenience of comparisons, the presented model can be simplified into three models which do not consider the neutral axis or thermal moment. The numerical results indicate that the influence of the neutral axis on the thermal axial force is minimal while that on the thermal moment is significant. In the case of the high temperature difference, frequency, critical temperature difference, unstable state, and the buckling type of the FG beams are misjudged when the neutral axis or thermal moment is ignored.

1. Introduction

Metal–ceramic functionally gradient (FG) materials possess high temperature resistance and oxidation resistance (provided by ceramics) as well as strength and toughness (provided by metal materials). In addition, density, Young’s modulus, expansion coefficient, and other material properties continuously and smoothly vary from surface to surface in the desired directions. Compared with laminated materials, this continuous and smooth material distribution makes FG materials effectively avoid the stress concentration at the interfaces of two materials. These advantages have made FG materials an appropriate replacement for conventional materials used in spacecraft heat shields, heat exchange tubes, flywheels, reactor vessels, and other engineering structures under high-temperature environments [1].

Since the beam is a common structural element in engineering, the mechanical behavior of FG beams has attracted much attention from numerous scholars. Generally, Euler–Bernoulli theory [2,3,4,5,6,7], Timoshenko theory [7,8,9], and higher-order shear deformation theory [10,11] can describe its mechanical behavior. Since shear deformation is not considered, Euler–Bernoulli theory is more suitable for FG beam structures with a slenderness ratio that is larger than 20 [12]. In the above-mentioned beam theories, the cross-section of the deformed beam is assumed to rotate around the point at the neutral axis. So, the neutral axis is necessary to construct the displacement field of beams. The asymmetric distribution of FG beam materials means that centroid and neutral axes may not be in the same positions. However, many scholars [2,3,4,5,6,7,8,9,10,11,12] have argued that the neutral axis is the same as the centroid one. Eltaher et al. [13] found that the frequency error is more significant than 10% with the use of inappropriate assumptions for the neutral axis. Because of the non-zero neutral axis, the centrifugal force exhibits an eccentric distance, leading to a static bending deformation of FG beams [14]. In this way, the position of the neutral axis is one of the keys to the accurate modeling of FG beams.

The scale effect is one of the keys for models of FG microbeams in electromechanical systems. In general, the above-mentioned beam theories are based on classical continuum theories, which do not describe scale effects. Therefore, these theories should be further modified by extending non-classical continuum theories, such as strain gradient theory [15,16], nonlocal strain gradient theory [17,18,19], nonlocal micropolar theory [20,21,22], and couple stress theory [23,24,25]. Thus, a large number of FG microbeam models have been developed by combining different beam theories and non-classical continuum theories [26,27,28,29,30,31,32,33,34]. Based on these models, the effects of gradient index, slenderness ratio, intrinsic parameters, and boundary conditions on static bending [27,28,30,31], the vibration characteristic [26,27,28,29,30,33], and buckling behavior [28,32,34] of FG microbeams have been analyzed in detail. Some scholars have combined multiple methods for comprehensive research. For example, on the basis of nonlocal strain gradient theory, Alam et al. [17] studied post-critical nonlinear vibration involving surface energy effects; Simsek [18] investigated the nonlinear free vibration of axially functionally graded (AFG) Euler–Bernoulli microbeams with immovable ends in conjunction with the modified couple stress theory. The influences of the length scale parameters, material variation, vibration amplitude, and boundary conditions on vibration responses were examined in detail.

The frequency characteristics and buckling behavior of FG beams under the temperature field are very critical in engineering applications. Zhao et al. [35] investigated thermal post-buckling equilibrium paths of FG beams under uniform and nonlinear temperature loads by using the shooting method. They found that FG beams are more stable than homogeneous material beams for the same temperature difference. Li et al. [36] analyzed thermal buckling and post-buckling response of simply supported FG beams by using Timoshenko theory and the shooting method. They pointed out that the deflection and constrained force of FG beams were smaller than those of two homogeneous material beams under the same temperature conditions. Kiani et al. [37] presented a critical buckling temperature of an FG beam for different boundaries and loadings. Based on first-order shear deformation beam theory and the neutral axis, Ma et al. [38] found that simply supported FG beams will not exhibit bifurcation buckling, which does occur for clamped FG beams. Ebrahimi et al. [39,40] formulated a Navier-type solution for the buckling and frequency of FG nanobeams based on nonlocal theory. In this solution, as the temperature difference increases, the frequencies decrease to zero. Guo and Alam [41] investigated the critical nonlinear thermal post-buckling and bending behavior of a magneto-electro-elastic nonlocal strain gradient beam by involving surface effects. Their achievements provide important theoretical value and practical significance for developing basic design concepts for magneto-electro-elastic nanobeams at different temperatures. Fang et al. [42] investigated the frequency and critical temperature of rotating nonlocal FG beams, in which material distribution is symmetrical with respect to the central axis. Dehrouyeh-Semnani [1] studied the effects of initial curvature and other parameters on the post-buckling equilibrium path of FG beams. Dehrouyeh-Semnani et al. [43] analyzed the moderately large oscillations of FG pipes which convey hot fluid with a lateral harmonic excitation. Attia et al. [44] proposed a bidirectional FG microbeam model based on the higher-order shear deformation theory and neutral axis. They found that the type of the load-deflection paths is affected by boundary conditions and temperature distribution type. Barretta et al. [45] pointed out that the neutral axis plays an essential role in the thermally induced displacement of a microbeam. Dehrouyeh-Semnani [46] developed an accurate model of FG beams with original boundary conditions and observed that the simplification of boundary conditions (neglecting thermal moment) led to the inaccurate prediction of the equilibrium path.

In general, the effect of the thermal moment and neutral axis on the mechanical behavior of unsymmetric FG beams is significant. However, ignoring the thermal moment or neutral axis has been common in most studies [35,36,37,39,40,42]. This motivated us to study this subject. In this paper, the displacement field of FG beams is described in the frame of absolute nodal coordinate formulation (ANCF), where the absolute position of nodes and their gradient are used as generalized coordinates. Compared with the traditional structure finite element dynamics, ANCF derives the stress and strain of components based on the complete theory of continuum mechanics without making small deformation assumptions during the modeling process, and it does not require adding incremental units to handle nonlinear deformation problems. ANCF is very suitable for solving mechanics problems of structures with large rotation and deformation, and a large number of ANCF elements based on various beam, plate, and shell theories have been developed [47,48,49,50,51,52]. Therefore, ANCF has the potential to solve problems related to FG material structures with geometric nonlinearity and material nonlinearity. Still, to the best of our knowledge, there is no relevant research on this subject. Using this displacement field description and Euler–Bernoulli beam theory, the tensile strain at an arbitrary point of an FG beam is calculated according to Lagrangian strain, exact curvature, and neutral axis. Then, the stress is calculated based on Hook’s law as well as tensile- and thermal-induced strains. The asymmetry of FG beams is considered in the integral calculation of potential and kinetic energies. Using the Lagrange equation of the second kind, the nonlinear dynamic equations of the system are derived. If the neutral axis or thermal moment is set as zero, the presented model can be simplified and divided into three models. The static state of these models under different temperature difference is obtained by using the arc length method or Newton–Raphson method. The nonlinear equations can be linearized by using Taylor expansion with respect to the static state, and then the linear equations can be transformed into eigenvalue equations to calculate frequencies. The influences of neutral axis and thermal moment on the frequency characteristics and buckling behavior of FG beams are analyzed by comparing the numerical results of four models and existing studies. Finally, several numerical examples are presented to discuss the effects of temperature difference, slenderness ratio parameters, and gradient index on frequency and buckling characteristics.

2. Model of FG Beams in Temperature Field

As shown in Figure 1, the research object of this paper is a slender FG beam, whose material distribution varies along the thickness direction. The varied law of material properties is according to a power law which can be defined by the following [40]:

$$P\left(y\right)=\left(P_u-P_l\right){\left({\displaystyle \frac{y}{h}}+{\displaystyle \frac{1}{2}}\right)}^k+P_l$$

(1)

where the subscripts ‘l’ and ‘u’ denote the lower and upper surfaces of the beam, respectively, and k is the material gradient index. The material properties of ${\mathrm{Al}}_2{\mathrm{O}}_3$ (ceramic) and SUS304 (metal) are determined by temperature and fundamental parameters, in which the relationship can be expressed as follows [40]:

$$P=P_0\left(P_{-1}{T}^{-1}+1+P_{1}T+P_2{T}^2+P_3{T}^3\right)$$

(2)

where the actual temperature T is the reference temperature $T_0$ (300 K) plus temperature difference $\Delta T$. The fundamental parameters of ${\mathrm{Al}}_2{\mathrm{O}}_3$ and SUS304 are listed in

Table 1. The fundamental parameters of ${\mathrm{Al}}_2{\mathrm{O}}_3$ and SUS304.

Material	Properties	${\mathit{P}}_0$	${\mathit{P}}_{-1}$	${\mathit{P}}_1$	${\mathit{P}}_2$	${\mathit{P}}_3$
${\mathrm{Al}}_2{\mathrm{O}}_3$	E (Pa)	$349.55\times {10}^9$	0	$-3.853\times {10}^{-4}$	$4.027\times {10}^{-7}$	$1.673\times {10}^{-10}$
	$\alpha \left({\mathrm{K}}^{-1}\right)$	$6.8269\times {10}^{-6}$	0	$1.838\times {10}^{-4}$	0	0
	$\rho \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	3800	0	0	0	0
$\mathrm{SUS}304$	E (Pa)	$201.04\times {10}^9$	0	$3.079\times {10}^{-4}$	$-6.534\times {10}^{-7}$	0
	$\alpha \left({\mathrm{K}}^{-1}\right)$	$12.330\times {10}^{-6}$	0	$8.086\times {10}^{-4}$	0	0
	$\rho \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	8166	0	0	0	0

.

Since the material of the beam is not symmetrically distributed, the neutral axis of the beam does not coincide with the central axis, as shown in Figure 1. Its position can be calculated by

$$h_0={\displaystyle \frac{{\int }_{-h/2}^{h/2}E\left(y\right)y\mathrm{d}y}{{\int }_{-h/2}^{h/2}E\left(y\right)\mathrm{d}y}}$$

(3)

The frame of the absolute nodal coordinate formulation (ANCF) is used to describe the displacement field, and herein, the position vector ${\mathbf{r}}_0$ of an arbitrary point on the central axis of the i-th element can be expressed as follows:

$${\mathbf{r}}_0(x,t)={\left[\begin{array}{cc}{r}_1(x,t)& r_2(x,t)\end{array}\right]}^{\mathrm{T}}=\mathbf{S}{\mathbf{e}}_i$$

(4)

As shown in Figure 2, $r_1$ and $r_2$ are, respectively, the distances between the point and coordinate axes. The subscript “$,x$” denotes the first derivative with respect to the arc length coordinate x. The coordinate vector and shape function matrix can be, respectively, written as

$${\mathbf{e}}_i={\left[\begin{array}{cccc}{\mathbf{r}}_i^{\mathrm{T}}& {\mathbf{r}}_{i,x}^{\mathrm{T}}& {\mathbf{r}}_{i+1}^{\mathrm{T}}& {\mathbf{r}}_{i+1,x}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}$$

(5)

and

$$\mathbf{S}={\left[\begin{array}{cccc}\left(1-3{\xi }^2+2{\xi }^3\right){\mathbf{I}}_{2\times 2}& l\left(\xi -2{\xi }^2+{\xi }^3\right){\mathbf{I}}_{2\times 2}& \left(3{\xi }^2-2{\xi }^3\right){\mathbf{I}}_{2\times 2}& l\left({\xi }^3-{\xi }^2\right){\mathbf{I}}_{2\times 2}\end{array}\right]}^{\mathrm{T}}$$

(6)

where ${\mathbf{I}}_{2\times 2}$ is an identity matrix and $\xi =x/l$.

The cross-section of the deformed beam is assumed to rotate around the point at the neutral axis, and then the position of an arbitrary point in the element is calculated as follows [14]:

$$\mathbf{r}={\mathbf{r}}_0-\left(y-h_0\right){\tilde{\mathbf{I}}}^{\mathrm{T}}{\displaystyle \frac{{\mathbf{r}}_{0,x}}{\left|{\mathbf{r}}_{0,x}\right|}}\approx {\mathbf{r}}_0-\left(y-h_0\right){\tilde{\mathbf{I}}}^{\mathrm{T}}{\mathbf{r}}_{0,x}$$

(7)

where

$${\tilde{\mathbf{I}}}^{\mathrm{T}}=\left[\begin{array}{cc}cos{\displaystyle \frac{\pi }{2}}& -sin{\displaystyle \frac{\pi }{2}}\\ \\ sin{\displaystyle \frac{\pi }{2}}& cos{\displaystyle \frac{\pi }{2}}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],\left|{\mathbf{r}}_{0,x}\right|\approx 1$$

(8)

With the use of the displacement description, the kinetic energy of the system can be derived as

$$\begin{array}{cc}\hfill T& ={\displaystyle \frac{1}{2}}w{\int }_0^L{\int }_{-h/2}^{h/2}\rho \left(y\right){\dot{\mathbf{r}}}^{\mathrm{T}}\dot{\mathbf{r}}\mathrm{d}y\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}w{\int }_0^L\left[{\rho }_0{{\dot{\mathbf{r}}}_0}^2+{\rho }_2{\dot{\mathbf{r}}}_{0,x}^{\mathrm{T}}\tilde{\mathbf{I}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\dot{\mathbf{r}}}_{0,x}-2{\rho }_1{{\dot{\mathbf{r}}}_0}^{\mathrm{T}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\dot{\mathbf{r}}}_{0,x}\right]\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}w\sum _{i=1}^n{\int }_0^l\left[{\rho }_0{{\dot{\mathbf{e}}}_i}^{\mathrm{T}}{\mathbf{S}}^{\mathrm{T}}\mathbf{S}{\dot{\mathbf{e}}}_i+{\rho }_2{{\dot{\mathbf{e}}}_i}^{\mathrm{T}}{\mathbf{S}}_{,x}^{\mathrm{T}}\tilde{\mathbf{I}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\mathbf{S}}_{,x}{\dot{\mathbf{e}}}_i-2{\rho }_1{{\dot{\mathbf{e}}}_i}^{\mathrm{T}}{\mathbf{S}}^{\mathrm{T}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\mathbf{S}}_{,x}{\dot{\mathbf{e}}}_{,x}\right]\mathrm{d}x\hfill \end{array}$$

(9)

where

$${\rho }_0={\int }_{-h/2}^{h/2}\rho \left(y\right)\mathrm{d}y,{\rho }_1={\int }_{-h/2}^{h/2}\rho \left(y\right)\left(y-h_0\right)\mathrm{d}y,{\rho }_2={\int }_{-h/2}^{h/2}\rho \left(y\right){\left(y-h_0\right)}^2\mathrm{d}y$$

(10)

The strain at arbitrary point on the beam can be expressed as

$$\epsilon \left(x,y\right)=\epsilon -{\epsilon }_{\mathrm{T}}={\epsilon }_0-y_1\kappa -\alpha \Delta T={\epsilon }_0-\left(y-h_0\right)\kappa -\alpha \Delta T$$

(11)

where ${\epsilon }_0$, ${\epsilon }_{\mathrm{T}}$, and $\kappa$ are the tensile strain in the neutral axis, thermal-induced strain, and curvature, respectively. The thermal-induced strain can be determined by ${\epsilon }_{\mathrm{T}}=\alpha \Delta T$.

In this paper, tensile strain and curvature are calculated by Green–Lagrange strain and the exact curvature formula, which can be written by

$${\epsilon }_0={\displaystyle \frac{1}{2}}\left({\mathbf{r}}_{,x}^{\mathrm{T}}{\mathbf{r}}_{,x}-1\right)={\displaystyle \frac{1}{2}}\left({\mathbf{e}}_i^{\mathrm{T}}{\mathbf{S}}_{,x}^{\mathrm{T}}{\mathbf{S}}_{,x}{\mathbf{e}}_i-1\right)={\displaystyle \frac{1}{2}}\left({\mathbf{e}}_i^{\mathrm{T}}\mathbf{N}{\mathbf{e}}_i-1\right)$$

(12)

$$\kappa ={\displaystyle \frac{\left|{\mathbf{r}}_{,x}\times {\mathbf{r}}_{,xx}\right|}{{\left|{\mathbf{r}}_{,x}\right|}^3}}={\displaystyle \frac{{\mathbf{e}}_i^{\mathrm{T}}\mathbf{T}{\mathbf{e}}_i}{{{\mathbf{e}}_i^{\mathrm{T}}\mathbf{N}{\mathbf{e}}_i}^{3/2}}}$$

(13)

where the subscript “$,xx$” denotes the second derivative with respect to the arc length coordinate x and

$$\mathbf{N}={\mathbf{S}}_{,x}^{\mathrm{T}}{\mathbf{S}}_{,x},\mathbf{T}={\mathbf{S}}_{,xx}^{\mathrm{T}}\tilde{\mathbf{I}}{\mathbf{S}}_{,x}+{\mathbf{S}}_{,x}^{\mathrm{T}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\mathbf{S}}_{,xx}$$

(14)

For the large deformation problem, these strain calculations are more nonlinear and suitable than the von Karman strain.

The strain energy of the system can be obtained by

$$\begin{array}{cc}\hfill U& ={\displaystyle \frac{1}{2}}{\int }_0^L{\int }_A\sigma \left(\epsilon -{\epsilon }_{\mathrm{T}}\right)\mathrm{d}x\mathrm{d}A\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_0^L\left[E_0{{\epsilon }_0}^2-2E_1\kappa {\epsilon }_0+E_2{\kappa }^2\right]\mathrm{d}x+{\int }_0^L\left[-N_{\mathrm{T}}{\epsilon }_0+M_{\mathrm{T}}\kappa \right]\mathrm{d}x+{\displaystyle \frac{1}{2}}{\int }_0^L{\int }_A{\alpha }^2\Delta T^2\mathrm{d}x\mathrm{d}A\hfill \\ \hfill & =U_1+U_2+U_3\hfill \end{array}$$

(15)

where

$$U_1={\displaystyle \frac{1}{2}}{\int }_0^L\left[E_0{{\epsilon }_0}^2-2E_1\kappa {\epsilon }_0+E_2{\kappa }^2\right]\mathrm{d}x$$

(16)

$$U_2={\int }_0^L\left[-N_{\mathrm{T}}{\epsilon }_0+M_{\mathrm{T}}\kappa \right]\mathrm{d}x$$

(17)

$$U_3={\displaystyle \frac{1}{2}}{\int }_0^L{\int }_A{\alpha }^2\Delta T^2\mathrm{d}x\mathrm{d}A$$

(18)

$$\left[\begin{array}{c}{E}_0\\ E_1\\ E_2\end{array}\right]=w{\int }_{-h/2}^{h/2}E\left(y\right)\left[\begin{array}{c}1\\ y-h_0\\ {(y-h_0)}^2\end{array}\right]\mathrm{d}y$$

(19)

The case of uniform temperature rise is considered in this paper. The thermal force $N_{\mathrm{T}}$ and moment $M_{\mathrm{T}}$ on the cross-section can be defined by

$$\left[\begin{array}{c}{N}_{\mathrm{T}}\\ M_{\mathrm{T}}\end{array}\right]={\int }_{A}E(y,T)\alpha (y,T)\Delta T\left[\begin{array}{c}1\\ y-h_0\end{array}\right]\mathrm{d}A$$

(20)

For homogeneous beams, $h_0$ and $M_{\mathrm{T}}$ are both equal to zero because of their uniform material distribution. However, some previous studies adopted this treatment of homogeneous beams. Obviously, this treatment method, where $h_0$ and $M_{\mathrm{T}}$ are set as zero, is inappropriate.

The generalized coordinate vector can be defined by

$$\mathbf{q}={\left[\begin{array}{ccccc}{\mathbf{e}}_{\mathbf{1}}& ...& {\mathbf{e}}_{\mathbf{i}}& ...& {\mathbf{e}}_{\mathbf{n}}\end{array}\right]}^{\mathrm{T}}$$

(21)

According to the Lagrange equation of the second kind, the dynamic equation can be derived as

$$\mathbf{M}\ddot{\mathbf{q}}+{\mathbf{Q}}_{\mathrm{E}}-{\mathbf{Q}}_{\mathrm{T}}=\mathbf{0}$$

(22)

where

$$\mathbf{M}=w{\int }_0^L\left({\rho }_0{\mathbf{S}}^{\mathrm{T}}\mathbf{S}+{\rho }_2{\mathbf{S}}_x^{\mathrm{T}}{\dot{\mathbf{r}}}_{0,x}^{\mathrm{T}}\tilde{\mathbf{I}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\mathbf{S}}_{,x}-2{\rho }_1{\mathbf{S}}^{\mathrm{T}}{\tilde{\mathbf{I}}}^{\mathrm{T}}{\mathbf{S}}_{,x}\right)\mathrm{d}x$$

(23)

$${\mathbf{Q}}_{\mathrm{E}}={\left({\displaystyle \frac{\partial U_1}{\partial \mathbf{q}}}\right)}^{\mathrm{T}}=\sum _{i=1}^n{\int }_0^l\left\{E_0{\epsilon }_0{\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}-E_1\kappa {\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}-E_1{\epsilon }_0{\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}+E_2\kappa {\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}\right\}\mathrm{d}x$$

(24)

$${\mathbf{Q}}_{\mathrm{T}}={\left({\displaystyle \frac{\partial U_2}{\partial \mathbf{q}}}\right)}^{\mathrm{T}}=\sum _{i=1}^n{\int }_0^l\left\{-N{\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}+M{\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}\right\}\mathrm{d}x$$

(25)

and some partial derivatives can be found in Appendix A.

To calculate the natural frequency of the system, the linearization of the dynamics in Equation (22) is very necessary. The dynamics vibration displacement $\mathbf{q}$ of FG beams can be regarded as the superposition of a static displacement ${\mathbf{q}}_{\mathrm{s}}$ and a small disturbance displacement ${\mathbf{q}}_{\delta }$. The relationship can be written as

$$\mathbf{q}={\mathbf{q}}_{\mathrm{s}}+{\mathbf{q}}_{\delta }$$

(26)

The static state of the system is the static equilibrium state of the beam under the temperature load, and then there are the following conditions:

$$\left\{\begin{array}{c}{\mathbf{q}}_{\mathrm{s}}={\dot{\mathbf{q}}}_{\mathrm{s}}={\mathbf{q}}_{\delta }={\dot{\mathbf{q}}}_{\delta }={\ddot{\mathbf{q}}}_{\delta }=\mathbf{0}\hfill \\ \mathbf{q}={\mathbf{q}}_{\mathrm{s}}\hfill \end{array}\right.$$

(27)

$${\mathbf{Q}}_{\mathrm{E}}\left({\mathbf{q}}_{\mathrm{s}}\right)-{\mathbf{Q}}_{\mathrm{T}}\left({\mathbf{q}}_{\mathrm{s}}\right)=\mathbf{0}$$

(28)

where the nonlinear static given by Equation (28) can be solved by using the arc length method or Newton–Raphson method for obtaining the static state.

Substituting Equation (26) into Equation (22) and using first-order Taylor expansion with respect to ${\mathbf{q}}_{\mathrm{s}}$, the vibration equation about ${\mathbf{q}}_{\delta }$ can be obtained as

$$\mathbf{M}{\ddot{\mathbf{q}}}_{\delta }+{\mathbf{K}}_{\mathrm{E}}{\mathbf{q}}_{\delta }-{\mathbf{K}}_{\mathrm{T}}{\mathbf{q}}_{\delta }=\mathbf{0}$$

(29)

where

$${\mathbf{K}}_{\mathrm{E}}={\left({\displaystyle \frac{{\partial }^2{U}_1}{\partial {\mathbf{q}}^2}}\right)}^{\mathrm{T}}=\sum _{i=1}^n{\int }_0^l\left\{\begin{array}{c}{E}_0{\epsilon }_0{\left({\displaystyle \frac{{\partial }^2{\epsilon }_0}{\partial {\mathbf{e}}_i^2}}\right)}^{\mathrm{T}}+E_0{\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)-E_1\kappa {\left({\displaystyle \frac{{\partial }^2{\epsilon }_0}{\partial {\mathbf{e}}_i^2}}\right)}^{\mathrm{T}}\hfill \\ -E_1{\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)-E_1{\epsilon }_0{\left({\displaystyle \frac{{\partial }^2\kappa }{\partial {\mathbf{e}}_i^2}}\right)}^{\mathrm{T}}-E_1{\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}\left({\displaystyle \frac{\partial {\epsilon }_0}{\partial {\mathbf{e}}_i}}\right)\hfill \\ +E_2\kappa {\left({\displaystyle \frac{{\partial }^2\kappa }{\partial {\mathbf{e}}_i^2}}\right)}^{\mathrm{T}}+E_2{\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)}^{\mathrm{T}}\left({\displaystyle \frac{\partial \kappa }{\partial {\mathbf{e}}_i}}\right)\hfill \end{array}\right\}\mathrm{d}x$$

(30)

$${\mathbf{K}}_{\mathrm{T}}={\left({\displaystyle \frac{{\partial }^2{U}_2}{\partial {\mathbf{q}}^2}}\right)}^{\mathrm{T}}=\sum _{i=1}^n{\int }_0^l\left\{N_{\mathrm{T}}{\left({\displaystyle \frac{{\partial }^2{\epsilon }_0}{\partial {\mathbf{e}}_i^2}}\right)}^{\mathrm{T}}-M_{\mathrm{T}}{\left({\displaystyle \frac{{\partial }^2\kappa }{\partial {\mathbf{e}}_i^2}}\right)}^{\mathrm{T}}\right\}\mathrm{d}x$$

(31)

some vectors and matrices in the stiffness matrices ${\mathbf{K}}_E$ and ${\mathbf{K}}_T$ can be found in Appendix A.

The solution of Equation (29) can generally be assumed as

$${\mathbf{q}}_{\delta }=Z{e}^{j\varpi t}$$

(32)

Substituting Equation (32) into Equation (29) yields

$$\left(-{\varpi }^2\mathbf{M}+{\mathbf{K}}_{\mathrm{E}}-{\mathbf{K}}_{\mathrm{T}}\right)\mathbf{Z}=\mathbf{0}$$

(33)

by solving the eigenvalue problem of Equation (33), the frequency ${\varpi }^2$ and vibration mode $\mathbf{Z}$ of FG beams can be obtained.

3. Thermal Buckling Analysis

In order to better investigate the influence of neutral axis and thermal moment, according to different conditions listed in

Table 2. Conditions for four models.

Models	Model 1	Model 2	Model 3	Model 4
Conditions	$h_0=0,M_T=0$	$h_0\ne 0,M_T=0$	$h_0=0,M_T\ne 0$	$h_0\ne 0,M_T\ne 0$

, the models proposed in this paper can be divided into four models:

1. Model 1 neglects the influences of the neutral axis and thermal moment;

2. Model 2 considers the neutral axis but ignores the thermal moment;

3. Model 3 considers the thermal moment but neglects the neutral axis;

4. Model 4 considers the neutral axis and thermal moment.

The size parameters of the FG beam analyzed in this paper are $L=2$ m, $b=0.4$ m, and $h=0.1$ m, which denote the length, width, and height of the beam, respectively. For the convenience of comparison, the calculated frequency can be re-expressed in a dimensionless form. The dimensionless relationship is defined by

$${\omega }_i={\varpi }_i{L}^2\sqrt{{\displaystyle \frac{12{\rho }_L}{E_L{h}^2}}}$$

(34)

Table 3. $\sqrt{{\varpi }_1}$ of simply supported FG beams with different gradient index and ratio of elastic modulus ($\Delta T=0$).

${\mathit{E}}_{\mathit{U}}/{\mathit{E}}_{\mathit{L}}$	$\mathit{k}=1$			$\mathit{k}=2$
	Models 1 and 3	Models 2 and 4	Ref. [14]	Models 1 and 3	Models 2 and 4	Ref. [14]
1	3.1400	3.1400	3.1400	3.1400	3.1400	3.1400
2	3.4421	3.4423	3.4423	3.3765	3.3767	3.3767
4	3.8234	3.8239	3.8239	3.6485	3.6492	3.6492
8	4.3216	4.3225	4.3225	3.9754	3.9768	3.9768
${\mathit{E}}_{\mathit{U}}/{\mathit{E}}_{\mathit{L}}$	$\mathit{k}=\mathbf{4}$			$\mathit{k}=\mathbf{10}$
	Models 1 and 3	Models 2 and 4	Ref. [14]	Models 1 and 3	Models 2 and 4	Ref. [14]
1	3.1400	3.1400	3.1400	3.1400	3.1400	3.1400
2	3.3330	3.3332	3.3332	3.2746	3.2726	3.2726
4	3.5539	3.5546	3.5546	3.4544	3.4547	3.4547
8	3.7823	3.7840	3.7840	3.6627	3.6639	3.6639

lists the square root of the first frequency for simply supported FG beams with a different gradient index and ratio of elastic modulus ($\Delta T=0$). The results obtained by models 2 and 4 are entirely consistent with those obtained in Ref. [14], which can validate the presented models. The frequencies obtained by models 1 and 3 are slightly smaller than those obtained from models 2 and 4, in which the maximum difference is only 0.0027 (0.07%) for the case of $E_U/E_L=8$ and $k=4$. This shows that the frequency calculation without an accurate neutral axis leads to a lower numerical result, but the error is very small.

Table 4. First frequencies of hinged-hinged FG beams with temperature difference and different gradient index ($L/h=20$).

	$\Delta \mathit{T}=0$ K			$\Delta \mathit{T}=20$ K			$\Delta \mathit{T}=40$ K
	$\mathit{k}=\mathbf{0}$	$\mathit{k}=\mathbf{0}.\mathbf{2}$	$\mathit{k}=\mathbf{5}$	$\mathit{k}=\mathbf{0}$	$\mathit{k}=\mathbf{0}.\mathbf{2}$	$\mathit{k}=\mathbf{5}$	$\mathit{k}=\mathbf{0}$	$\mathit{k}=\mathbf{0}.\mathbf{2}$	$\mathit{k}=\mathbf{5}$
Ref. [40]	9.8594	8.6846	6.0498	9.5065	8.3093	5.6476	9.1374	7.9105	5.2021
Model 1	9.8595	8.6939	6.0756	9.5093	8.3187	5.6720	9.1425	7.9226	5.2278
Model 2	9.8595	8.6849	6.0497	9.5093	8.3092	5.6442	9.1425	7.9128	5.1973
Model 3	9.8595	8.6839	6.0756	9.5080	8.3150	5.6678	9.1420	7.9145	5.2170
Model 4	9.8595	8.6849	6.0497	9.5080	8.3099	5.6457	9.1420	7.9166	5.2073

gives the first frequencies ${\varpi }_i$ of hinged-hinged FG beams with a different temperature difference and gradient index. The results obtained from the four models are in agreement with those presented by Ref. [40]. It should be noted that the model presented by Ref. [40] neglected the influences of the neutral axis and thermal moment. Nevertheless, in the case of the low temperature difference, the influence of the neutral axis and thermal moment on the frequency is relatively small, and that of the thermal force is dominated. Accordingly, this result comparison can validate the presented models for considering thermal effect.

Figure 3 shows the first frequency’s variation with the temperature difference increase. In the case of homogeneous material beams ($k=0$; see Figure 3a), the results obtained from the four models and Ref. [40] are almost the same, in which the frequencies decrease and reach zero with increasing temperature difference. The reason for this is that the neutral axis and thermal moment are both equal to zero, and then these four models are precisely the same in this case. In those cases of FG beams ($k\ne 0$), the results obtained from model 1 and Ref. [40] are almost the same as those obtained by model 2, in which the frequencies also decrease and reach zero. This shows that the influence of the neutral axis on thermal axial force is very small. The results obtained by models 1 and 2 ignoring the thermal moment are different from those obtained by models 3 and 4 considering the thermal moment in the cases of high temperature difference. However, these show a good agreement in the cases of low temperature difference. This indicates that the frequency is mainly dominated by the thermal force, and the thermal moment can be neglected in the case of the low temperature difference. However, the effect of the thermal moment on frequencies is significant in the case of high temperature differences. As the temperature difference increases, the correct result first decreases to the lowest non-zero value and then increases. Regarding the key transitions, this is because, when considering the influence of thermal moment, an increase in temperature will increase the thermal moment, gradually offsetting the effect of the thermal axial force. When these two are relatively balanced, the frequency reaches its lowest value. As the temperature rises, the influence of thermal moment is larger than the thermal axial force, leading to an increase in frequency. However, the incorrect results still monotonically decrease to 0. When the frequency is equal to 0, the structure is unstable. This indicates that the frequency and unstable state of the FG beams with a high temperature difference are misjudged when the thermal moment is ignored. It is worth noting that the results of models 3 and 4 are also evidently different, which shows that the neutral axis plays a vital role in the thermal moment.

Figure 4 gives deflection curves of the midpoint for hinged-hinged FG beams in the thermal pre-buckling state. The deflections calculated by models 1 and 2 are identically equal to zero, while those obtained by models 3 and 4 increase along the negative direction of the y-axis with the rise in the temperature difference. This indicates that the initial configuration is mistakenly used as the equilibrium position in models 1 and 2. However, the initial configuration is unstable where the beam should be bent by the thermal moment. The deformations obtained from model 4 are larger than those calculated by model 3. Chen et al. [14] pointed out that the frequencies of beams with large curvatures are higher than those for cases of small curvatures. Therefore, the frequencies obtained from model 4 are larger than those calculated by model 3 (see Figure 3). In other words, frequency and bending deformation are easily underestimated by ignoring the neutral axis.

Figure 5 shows the deflection variation for the midpoint for hinged-hinged FG beams with respect to the gradient index in the case of $\Delta T=20$ K. The deflections calculated by models 1 and 2 remain at zero, while those obtained from models 3 and 4 increase and then decrease along the negative direction of the y-axis with the rise in the gradient index. The deflections obtained by model 3 are less than those obtained by model 4 while $k\ne 0$. As shown in Figure 6, the variation of the thermal moment with gradient index is similar to that of the deflection. This shows that the thermal moment is the main factor causing the bending deformation. Here, the reason for the change in thermal moment is the difference in heat resistance and thermal expansion coefficient between these two materials, which causes a difference in bending deformations. The error for model 3 should be induced by ignoring the neutral axis in the calculation of thermal moment.

Figure 7 shows the variation of the first frequency for hinged-hinged FG beams in the cases of $L/h=50$, 100, 150, and 200. The dimensionless frequencies for beams with different slenderness ratios are the same in the case of $\Delta T=0$, but different in the cases of $\Delta T\ne 0$. These frequencies all first decrease and then increase with the increase in the temperature difference, where the speed of decreasing or increasing is faster in the case of a larger slenderness ratio than those in the cases of a smaller slenderness ratio. In addition, the frequencies all decrease to the lowest frequency and then increase, where the lowest frequency is almost the same for beams with different slender ratios. The temperature differences in cases of the lowest frequency can be recorded as ${\Delta T}_{low}$. Obviously, the temperature differences ${\Delta T}_{low}$ are different for the cases of different slender ratios. As shown in Figure 8, the lowest frequency hardly varies with the increase in the slenderness ratio, but decreases with the increase in the gradient index. As shown in Figure 9, the temperature differences ${\Delta T}_{low}$ first decrease and then increase with the increase in the gradient index, and decrease with the increase in the slenderness ratio. The understanding of the lowest frequency and ${\Delta T}_{low}$ has crucial guiding significance for the corresponding engineering structure design.

Figure 10 shows the load–displacement curve of the hinged-hinged FG beam under pre- and post-buckling with $L/h=200$ and gradient index $k=1$. The load–displacement curves before the critical temperature $\Delta T_{CT}$ represents pre-buckling, and after $\Delta T_{CT}$ post-buckling. The bifurcation buckling occurs at $\Delta T_{CT}\approx 1.82$ K for results obtained by models 1 and 2 while the snap-through buckling appears, respectively, at $\Delta T_{CT}\approx 2.63$ K and 3.15 K for the results obtained by models 3 and 4. This shows that models that ignore the thermal moment incorrectly predict the buckling type of FG beams. On the other hand, the critical temperature difference $\Delta T_{CT}$ obtained from model 3 is less than that obtained by model 4, which shows that models which ignore the neutral axis underestimate the critical temperature difference. It should be noted that in the post-buckling state ($\Delta T>\Delta T_{CT}$), black solid and red dotted lines in the figure, respectively, represent stable and unstable states of the FG beam. In the numerical calculation of this paper, the stable and unstable solutions of post-buckling for FG beams are obtained by using the arc length method and Newton–Raphson method, respectively. Since the stiffness matrix of the solution obtained by Newton–Raphson method is singular in the post-buckling state, the corresponding frequency is equal to zero. According to this condition, the critical temperature difference under different working conditions can be calculated. As shown in Figure 11, the critical temperature difference decreases with the increase in the gradient index for the four models. In addition, the critical temperature difference is underestimated by models that ignore the neutral axis or thermal moment, which supports the conclusion obtained in Figure 10. The critical temperature for thermal buckling of the structure can be analyzed to guide the optimization of the selection and gradient distribution of structural materials under different thermal environmental conditions and the choice of material combinations that can remain stable at high temperatures to ensure that buckling does not occur at high temperatures. At the same time, it can help optimize the geometrical parameters of the beams to increase the thermal buckling temperature and enhance stability, e.g., to help adjust the slenderness ratio of the beams at high temperatures to avoid buckling due to temperature increase. It also ensures the stresses are within safe limits at the thermal buckling temperature. By fully considering the influence of the neutral axis and thermal moment, a higher critical temperature is obtained, which will help reduce size and weight in designs and lower costs.

Table 5. Computational time (in seconds) for four models.

Models	Model 1	Model 2	Model 3	Model 4
Times	73.06	73.69	190.51	316.28

shows the computational time (in seconds) for four models and 20 elements using the same arc length and Newton–Raphson methods. The tolerance set for the convergence was $\Delta \epsilon$ = $1\times {10}^{-9}$. There is no significant difference between model 1 and model 2. But model 3 and model 4 necessitate a longer computational time than model 1 and model 2. Especially for model 4, the computational cost significantly increases.

4. Results

In this paper, a comprehensive model of FG beams is presented based on the frame of the absolute nodal coordinate formulation, which has the potential advantage of solving the problem of large rotation and deformation. Considering the significant influence of thermal moment and neutral axis on the mechanical behavior of FG material beams with asymmetric distribution, the presented model fully considers the influence of both the thermal moment and neutral axis. In order to qualitatively discuss the influence of the neutral axis and thermal moment in detail, another three models are obtained by simplifying the presented model. The results obtained from these four models show good agreement with previous research results in the case of low temperature difference, which can validate the presented models. Based on some parameter analysis, several meaningful findings are as follows:

(1) In the case of the low temperature difference, the thermal force mainly dominates the frequency, and the thermal moment can be neglected.

(2) The influence of the neutral axis on the thermal axial force is very small, while that on the thermal moment is significant.

(3) In the case of the high temperature difference, the frequency and unstable state of the FG beams are misjudged when the thermal moment is ignored.

(4) The lowest frequency is almost the same for different slenderness ratios. However, the lowest frequency decreases with the increase in the gradient index.

(5) The temperature differences in cases of the lowest frequency first decrease and then increase with the increase in the gradient index, and decrease with the increase in the slenderness ratio.

(6) Models that ignore the thermal moment will incorrectly predict the buckling type of FG beams. Models ignoring the neutral axis or thermal moment underestimate the critical temperature difference.

This paper adopts absolute nodal coordinate formulation to model the mechanical properties of FG beams under thermal environments, considering the influence of the neutral axis and thermal moment. The influences of temperature difference, material gradient index, size effect, and other factors on the structure’s natural frequency and thermal buckling behavior are analyzed. The relevant results help to understand the influence of the neutral axis and thermal moment on the mechanical properties of functionally graded beams and provide a reference for the rational and practical design of FG beams under thermal environments.