Free Vibration of FML Beam Considering Temperature-Dependent Property and Interface Slip

Abstract

This paper presents an analytical investigation of the free vibration behavior of fiber metal laminate (FML) beams with three types of boundary conditions, considering the temperature-dependent properties and the interfacial slip. In the proposed model, the non-uniform temperature field is derived based on one-dimensional heat conduction theory using a transfer formulation. Subsequently, based on the two-dimensional elasticity theory, the governing equations are established. Compared with shear deformation theories, the present solution does not rely on a shear deformation assumption, enabling more accurate capture of interlaminar shear effects and higher-order vibration modes. The relationship of stresses and displacements is determined by the differential quadrature method, the state-space method and the transfer matrix method. Since the corresponding matrix is singular due to the absence of external loads, the natural frequencies are determined using the bisection method. The comparison study indicates that the present solutions are consistent with experimental results, and the errors of finite element simulation and the solution based on the first-order shear deformation theory reach 3.81% and 3.96%, respectively. At last, the effects of temperature, the effects of temperature degree, interface bonding and boundary conditions on the vibration performance of the FML beams are investigated in detail. The research results provide support for the design and analysis of FML beams under high-temperature and vibration environments in practical engineering.

1. Introduction

Fiber metal laminates (FMLs), as a class of advanced hybrid composites, combine the high specific stiffness of fiber-reinforced polymers (FRPs) with the excellent ductility and vibration damping capacity of metal alloys [1]. Due to their excellent mechanical performance, FMLs are widely adopted in aerospace, automotive, and civil engineering fields. For example, they are used in bridge deck layers to enhance fatigue performance and vibration control, and in aircraft fuselage panels to improve impact resistance [2]. As FML beams are composed of multiple layers, the interfacial stiffness exerts a significant influence on the overall structural stiffness and vibration behavior. Under thermal environments, elevated temperatures induce variations in the constituent material properties [3], thereby altering the vibrational characteristics of the beam. Such thermal effects can lead to stiffness degradation, which poses challenges to maintaining dynamic stability. Therefore, given the increasing demand for reliable dynamic performance in thermally variable environments, the free vibration behavior of FML beams under temperature variations still requires further in-depth investigation.

Experimental investigation plays an essential role in studying the vibration behavior of FML beams, with extensive research focusing on their dynamic characteristics. Senthil Kumar et al. [4] conducted experimental modal analysis to determine the natural frequencies and modal damping of composite laminates, and further examined how fiber length and weight fraction influence their mechanical properties and free vibration behavior.

Kali et al. [5] measured the natural frequencies of nanohybrid FMLs using a dynamic mechanical analysis, revealing that both the natural frequencies and viscoelastic properties are sensitive to fiber type, nanoparticle reinforcement, and particle size. A series of experiments using fast Fourier transform with the pulse platform determined that edge constraints and boundary conditions significantly influence the natural vibration frequencies of glass FML plates according to Prasad and Sahu [6]. Thomas et al. [7] conducted impact hammer experiments to investigate the effect of crack-type damage on structural integrity and stiffness via frequency response. The effects of temperature variations and boundary conditions on the free vibration characteristics of composite beams were investigated using experimental techniques by Ergun and Alkan [8]. Cai et al. [9] analyzed the effects of temperature on the natural frequencies of simply supported beams within the range of −40 °C to 60 °C using a small ball excitation test and observed a linear negative correlation between the beam’s natural frequencies and temperature. Araki et al. [10] investigated the effects of interface stiffness and delamination on resonant oscillations of metal laminates using a RUS apparatus comprising piezoelectric transducers, a spectrum analyzer with a tracking generator, and a force gauge. For different fiber stacking sequences with distinct inherent properties, Kirubakaran et al. [11] investigated the vibration characteristics of FMLs designed with multi-layer energy-absorbing flax fibers and high-strength basalt fibers stacked on titanium plates through impact hammer testing. The frequency-driven investigation of bi-directional, industry-driven laminated glass/epoxy composite beams was conducted by Das and Sahu [12] through experimental modal analysis. Pai et al. [13] employed impulse hammer excitations to investigate the free vibration responses of FMLs, focusing on the influence of ply arrangements on vibration performance. Functionally graded (FG) materials possess the characteristic of continuously varying material properties. Incorporating them into FML can alleviate interfacial stress concentrations caused by large differences in material properties between layers. Additionally, the continuously varying thermal expansion or thermal conductivity can reduce the thermal stresses induced by temperature gradients, thereby improving reliability in high-temperature environments. Pai et al. [14] demonstrated the novelty of layering FMLs based on the FG of ply shock impedance, and characterized their flexural, tensile, and vibration damping behaviors. The influence of shock-impedance tuning of individual layers on the high-strain-rate behavior of FMLs was investigated by Pai et al. [15], where the variation in shock energy absorption with strain rate was analyzed as a function of transmission loss using regression.

Experimental studies are essential for understanding the free vibration behavior of FMLs, as they provide direct and valuable observations under various conditions. At the same time, these investigations can lead to considerable expenditure of time and resources, especially when investigating multiple design parameters and environmental influences. Alongside experimental investigations, theoretical modeling provides a cost-effective approach for extensively exploring parameter variations and gaining deeper insight into the underlying physical mechanisms. Shear deformation theories, including Euler-Bernoulli beam theory as well as first-order shear deformation theory (FSDT) and higher-order shear deformation theories, incorporate the assumption of transverse shear deformation. This approach reduces the dimensionality of the governing equations while preserving the essential mechanical characteristics, thereby significantly simplifying analysis and computation. Following the framework of various higher-order deformation beam theories, Ameri et al. [16] conducted a study on the fundamental frequency analysis of composite beams using the Rayleigh–Ritz method. Qin et al. [17] developed a unified method for free vibration analysis of graphene platelet–reinforced FG laminated shallow shells with arbitrary boundary conditions, deriving the governing equations through the FSDT coupled with the artificial spring technique. Ghasemi et al. [18] employed the FSDT with the Fourier series method to analyze the free vibration of FML plates, examining the effects of geometrical and material parameters. Maraş and Yaman [19] examined the vibration behavior of FMLs by formulating the governing equations based on classical plate theory and conducting numerical analyses using the generalized differential quadrature method. The free vibration of FML cylindrical shells studied using Love’s first-approximation shell theory and a beam modal function model was investigated by Mohandes et al. [20]. Ghasemi and Mohandes [21] derived the frequencies of micro- and nano-scale FML cylindrical shells based on the modified couple stress theory. Ghashochi-Bargh and Sadr [22] used classical laminated plate theory and the finite strip method to analyze FML composite panels, optimizing their natural frequencies under different boundary conditions via particle swarm optimization. Ellouz et al. [23] introduced a high-order shear deformation-based numerical method that efficiently captures the non-linear multi-physics response of porous FG magnetoelectro-elastic shells. Maraş and Yaman [24] applied the differential quadrature, generalized differential quadrature, and harmonic differential quadrature methods to numerically solve the governing equations of FML composite plates, demonstrating the applicability of these methods in designing FML structures. Based on the first-order shear deformation theory and Hamilton’s principle, Lai et al. [25] conducted a numerical analysis of the free vibration of rotating sandwich beams with functionally graded carbon nanotube face sheets under thermal environments. Zhao et al. [26] proposed a piecewise shear deformation theory for laminated composite and sandwich plates, enabling accurate and efficient forced vibration analysis under thermal environments. A simplified model based on Euler–Bernoulli beam theory and the mode stiffness matrix was proposed to analyze the dynamic behavior of continuous steel-concrete composite beams with interlayer slip, as demonstrated by Fang et al. [27].

To the best of the authors’ knowledge, no analytical model without assumption of transverse shear deformation for the vibration performance of FMLs under temperature effects has been published. In this study, an analytical model is established within the framework of two-dimensional elasticity theory to investigate the free vibration behavior of FML beams considering the orthotropic properties of FRP, the temperature-dependent mechanical parameters of constituent materials, and the influence of interfacial slip. Compared with shear deformation theories, the present analytical model does not rely on a shear deformation assumption, enabling more accurate capture of interlaminar shear effects and higher-order vibration modes. In comparison with the finite element method, it achieves high accuracy with far fewer degrees of freedom and without the need for large-scale meshes or matrix operations, resulting in significantly higher computational efficiency. The non-uniform temperature distribution through the beam thickness is obtained via one-dimensional (1D) heat conduction theory formulation. By employing the state-space approach together with the differential quadrature method, the governing equations are discretized, while the transfer matrix method is applied to ensure interlayer continuity. Subsequently, the proposed model is validated against existing results, after which parametric analyses are conducted to investigate the effects of temperature, interfacial stiffness, and boundary conditions on the vibration behavior of FML beams.

2. Vibration Analysis

As shown in Figure 1, the FML beam studied for free vibration in this paper has a length L and a total thickness H, and is composed of M layers of FRP and metal, in which the FRP is each with a thickness h_m (m = 1,2… M). The FML is positioned within a Cartesian coordinate system o-xy, where the origin is at the beam’s bottom-left corner. The top and bottom surfaces of the FML beam are subjected to temperatures T_a and T_b, respectively, and these temperatures are invariant along the x-direction. The mechanical properties of the constituent materials in the FML plate are temperature-dependent, with ${\xi }_i$ representing the temperature-induced modulus attenuation coefficient. The effect of interlayer slip between adjacent layers is considered, and all interfaces are assumed to have the same interfacial stiffness value s. Three types of boundary conditions are considered, including simply supported, clamped, and cantilevered.

2.1. Temperature Field

As the mechanical properties of the FML’s constituent materials are influenced by temperature, this section begins with the solution of the temperature field. Given the uniform temperature distribution on the top and bottom surfaces of the FML beam, the temperature variation can be considered solely along the y-direction. The heat conduction analysis in this study is based on the following assumptions:

(1)

Since the temperature gradient mainly occurs on the upper and lower surfaces of the FML beam, this study primarily considers 1D heat conduction in the y-direction.

(2)

The temperature dependence of thermal conductivity is neglected because, even at high temperatures, the metal in the FML beam has a thermal conductivity roughly two orders of magnitude higher than that of the FRP layer, so its effect on the temperature field is negligible.

(3)

There is no internal heat generation, and that convective and radiative heat losses are neglected.

(4)

The material in each layer is homogeneous.

(5)

The analysis considers only steady-state heat conduction.

The heat conduction equation, derived from the energy conservation principle and Fourier’s law, is the fundamental governing equation of heat transfer [28,29], which can be given by

$${\rho }_m{c}_m{\displaystyle \frac{\partial T_m(y,t)}{\partial t}}+{\displaystyle \frac{\partial F_y^m(y,t)}{\partial y}}=Q_m$$

(1)

$$F_y^m(y,t)=-k_y^m\frac{\partial T_m(y,t)}{\partial y}$$

(2)

where ${\rho }_m$, $c_m$, $T_m$, $F_y^m$, $k_y^m$ and $Q_m$ denote the density, the specific heat capacity, the temperature, the y-direction heat flux, the y-direction thermal conductivity and the internal heat source of the m-th layer, respectively. With the above assumptions, $T_m$ is the function of y, $F_y^m$ is constant, and the y-direction heat conduction equation for the m-th layer becomes

$${\displaystyle \frac{d^2{T}_m(y)}{d{y}^2}}=0.$$

(3)

The temperature and y-direction heat flux are continuous at the interface, one has

$$T_m(d_m)=T_{m+1}(d_m), F_y^m=F_y^{m+1}, m=1,2\dots M-1$$

(4)

where $d_m={\displaystyle {\sum }_{p=1}^m{h}_p}$. By integrating Equation (2) through the m-th layer from its bottom surface d_m-1 to the top surface d_m

$$T_m(d_m)=T_m(d_{m-1})-{\displaystyle \frac{F_y^m{h}_m}{k_y^m}}.$$

(5)

The surficial temperatures are known

$$T_1(0)=T_b,T_M(H)=T_a.$$

(6)

Inserting Equation (6) into Equation (5), the recursive progression from the bottom to the top surface yields the matrix form below

$$\left[\begin{array}{c}{T}_a\\ F_y^m\end{array}\right]=\left[\begin{array}{cc}1& -{\displaystyle \sum _{p=1}^M\frac{h_p}{k_y^p}}\\ 0& 1\end{array}\right]\left[\begin{array}{c}{T}_b\\ F_y^m\end{array}\right].$$

(7)

For an arbitrary position within the m-th layer, the temperature distribution is derived from Equations (5) and (6) by accumulating the single-layer relation, yielding

$$T_m=T_b-[F_y^m(y){\displaystyle \sum _{p=1}^{m-1}\frac{h_p}{k_y^p}+\frac{y-d_{i-1}}{k_y^m}}]$$

(8)

Substituting the $F_y^m$ obtained in (7) into the above equation, the final expression is obtained as

$$T_m={\displaystyle \frac{1}{{\displaystyle \sum _{\lambda =1}^M\frac{h_{\lambda }}{k_{\lambda }}}}}[{\displaystyle \frac{T_a-T_b}{k_y^m}}(y-d_{m-1})+{\displaystyle \sum _{\lambda =1}^{m-1}\frac{h_{\lambda }}{k_{\lambda }}}{T}_a+{\displaystyle \sum _{\lambda =m}^M\frac{h_{\lambda }}{k_{\lambda }}}{T}_b], m=1,2\dots M$$

(9)

2.2. State Space Equations

For FML beams, the effect of temperature on the natural frequencies is mainly manifested in two aspects: thermal strain and the temperature dependence of material properties. For most free or partially constrained beams, the influence of temperature-dependent material properties on the natural frequencies is greater than that of thermal strain [30]. Therefore, this study considers only the effect of temperature-dependent material properties.

Owing to the variation in temperature through the thickness, the elastic constants vary correspondingly. As illustrated in Figure 2, to account for this variation, the m-th layer of the FML beam is divided into p_m sublayers with the same thickness, with i denoting the sublayer index (i = 1,2… P, $P={\displaystyle {\sum }_{m=1}^M{p}_m}$). Each sublayer is made sufficiently thin, and the material within the sublayer is assumed to be independent of y.

Based on the framework of two-dimensional elastic theory [31], the natural frequencies and mode shapes of the FML beam under temperature effect are developed in this section. Considering that the constituent material FRP is orthotropic, the constitutive equations of the i-th sublayer are given by

$$\left[\begin{array}{c}{\sigma }_x^i\\ {\sigma }_y^i\\ {\tau }_{xy}^i\end{array}\right]=\left[\begin{array}{ccc}{c}_{11}^i& c_{12}^i& 0\\ c_{21}^i& c_{22}^i& 0\\ 0& 0& c_{66}^i\end{array}\right]\left[\begin{array}{c}{\epsilon }_x^i\\ {\epsilon }_y^i\\ {\gamma }_{xy}^i\end{array}\right],$$

(10)

where ${\sigma }_x^i$, ${\sigma }_y^i$ and ${\tau }_{xy}^i$ denote the normal and shear stresses, respectively, and c represents the elastic constants, which can be determined by the modulus and Poisson’s ratio:

$$c_{11}^i={\displaystyle \frac{{(E_1^i)}^2}{E_1^i-{({\mu }_{12})}^2{E}_2^i}}{\xi }_i, c_{12}^i={\displaystyle \frac{{\mu }_{12}{E}_1^i{E}_2^i}{E_1^i-{({\mu }_{12})}^2{E}_2^i}}{\xi }_i, c_{22}^i={\displaystyle \frac{E_1^i{E}_2^i}{E_1^i-{({\mu }_{12})}^2{E}_2^i}}{\xi }_i, c_{66}^i=G_{12}^i{\xi }_i,$$

(11)

where E, G and µ represent the elastic modulus, the shear modulus and Poisson’s ratio, respectively; the subscripts 1 and 2 mean the x- and y-directions of the elastic constants, respectively. The strain-displacement relationships are given by

$${\epsilon }_x^i={\displaystyle \frac{\partial u^i}{\partial x}}, {\epsilon }_y^i={\displaystyle \frac{\partial v^i}{\partial y}}, {\gamma }_{xy}^i={\displaystyle \frac{\partial v^i}{\partial x}}+{\displaystyle \frac{\partial u^i}{\partial y}},$$

(12)

in which, $u^i$ and $v^i$ denote the longitudinal and transverse displacements. In the absence of body forces, the equilibrium equations are

$${\displaystyle \frac{\partial {\sigma }_x^i}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{xy}^i}{\partial y}}={\rho }^i{\displaystyle \frac{{\partial }^2{u}^i}{\partial t^2}}, {\displaystyle \frac{\partial {\tau }_{xy}^i}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_y^i}{\partial y}}={\rho }^i{\displaystyle \frac{{\partial }^2{v}^i}{\partial t^2}}.$$

(13)

The simply supported (S), clamped (C), and free (F) boundary conditions are, respectively, expressed as

$${\sigma }_x^i=0, v^i=0, \mathrm{at} x=0 \mathrm{or} L, \mathrm{for\; S\; boundary},$$

(14)

$$u^i=0, v^i=0, \mathrm{at} x=0 \mathrm{or} L, \mathrm{for\; C\; boundary},$$

(15)

$${\sigma }_x^i=0, {\tau }_{xy}^i=0, \mathrm{at} x=0 \mathrm{or} L, \mathrm{for\; F\; boundary}.$$

(16)

For the governing equations of elasticity, the state-space method (SSM) is an effective solution technique. Under a harmonic motion, the stresses and displacements can be expressed in a separated variable form, as follows

$$\left[\begin{array}{c}{\sigma }_x^i(x,y,t)\\ {\sigma }_y^i(x,y,t)\\ {\tau }_{xy}^i(x,y,t)\\ u^i(x,y,t)\\ v^i(x,y,t)\end{array}\right]=\left[\begin{array}{c}{\overline{\sigma }}_x^i(x,y)\\ {\overline{\sigma }}_y^i(x,y)\\ {\overline{\tau }}_{xy}^i(x,y)\\ {\overline{u}}^i(x,y)\\ {\overline{v}}^i(x,y)\end{array}\right]e^{j\omega t},$$

(17)

where ω denotes the natural frequency and $j=\sqrt{-1}$. By substituting Equation (17) into the governing equations of Equations (10)–(13) and employing the SSM, the out-of-plane variables can be arranged into a matrix differential equation

$${\displaystyle \frac{\partial }{\partial y}}{\Psi }^i(x,y)=N^i{\Psi }^i(x,y),$$

(18)

where ${\Psi }^i$ and $N^i$ are, respectively, the state vector and the system matrix, which are given by

$${\Psi }^i=\left[\begin{array}{c}{\overline{\sigma }}_y^i(x,y)\\ {\overline{u}}^i(x,y)\\ {\overline{v}}^i(x,y)\\ {\overline{\tau }}_{xy}^i(x,y)\end{array}\right], N^i=\left[\begin{array}{cccc}0& 0& -{\rho }^i{\omega }^2& -{\displaystyle \frac{\partial }{\partial x}}\\ 0& 0& -{\displaystyle \frac{\partial }{\partial x}}& {\displaystyle \frac{1}{c_{66}^i}}\\ {\displaystyle \frac{1}{c_{22}^i}}& -{\displaystyle \frac{c_{12}^i}{c_{22}^i}}{\displaystyle \frac{\partial }{\partial x}}& 0& 0\\ -{\displaystyle \frac{c_{12}^i}{c_{22}^i}}{\displaystyle \frac{\partial }{\partial x}}& -{\beta }^i{\displaystyle \frac{{\partial }^2}{\partial x^2}}-{\rho }^i{\omega }^2& 0& 0\end{array}\right],$$

and ${\beta }^i=c_{11}^i-{(c_{12}^i)}^2/c_{22}^i$. According to the governing equations of Equations (10)–(13), the corresponding in-plane variable, can be expressed in terms of out-of-plane ones

$${\overline{\sigma }}_x^i={\displaystyle \frac{c_{12}^i}{c_{22}^i}}{\overline{\sigma }}_y^i+{\beta }^i{\displaystyle \frac{\partial {\overline{u}}^i}{\partial x}},$$

(19)

For differential equations, differential quadrature method (DQM) is an effective approach that represents the unknown functions and their derivatives at a set of discrete points, as demonstrated below

$${\displaystyle \frac{{\partial }^{n}f(x,y)}{\partial x^n}}\left|\begin{array}{c}\\ x=x_k\end{array}\right.={\displaystyle \sum _{r=1}^N{g}_{kr}^{(n)}}f(x_r,y), n=1,2\dots N-1, k, r =1,2\dots N$$

(20)

where

$$x_k={\displaystyle \frac{1}{2}}\left[1-\cos \left({\displaystyle \frac{k-1}{N-1}}\pi \right)\right],$$

$$g_{kr}^{(1)}=\left\{\begin{array}{c}{\displaystyle \frac{{\displaystyle \prod _{q=1,q\ne k,r}^N{x}_k-x_q}}{{\displaystyle \prod _{q=1,q\ne r}^N{x}_r-x_q}}},k\ne r\\ {\displaystyle \sum _{q=1,q\ne k}^N\frac{1}{x_k-x_q},k=r}\end{array}\right., g_{kr}^{(n)}=\left\{\begin{array}{c}n\left[g_{kk}^{(n-1)}{g}_{kr}^{(1)}-{\displaystyle \frac{g_{kr}^{(n-1)}}{x_k-x_r}}\right],k\ne r\\ -{\displaystyle \sum _{r=1,r\ne k}^N{g}_{kr}^{(n)}},k=r\end{array}\right., n=2,3\dots N-1$$

in which f (x, y) is a continuous function which represents the stress or the displacement; $g_{kr}^{(n)}$ and x_k, respectively, represent the weight coefficient and the Lobatto discrete point [32]. By applying the DQM to Equations (18) and (19), the state space equation at the discrete points is derived as

$${\displaystyle \frac{d}{dy}}{\Phi }^i(y)=M^i{\Phi }^i(y),$$

(21)

$${\overline{\sigma }}_{x,k}^i(y)={\displaystyle \frac{c_{12}^i}{c_{22}^i}}{\overline{\sigma }}_{y,k}^i(y)+{\beta }^i{\displaystyle \sum _{r=1}^N{g}_{ikr}^{(1)}}{\overline{u}}_r^i(y),$$

(22)

where ${\Phi }^i$ and $M^i$ are, respectively, the state vector and the system matrix after being substituted into the DQM formulation, which are given by

$${\Phi }^i(y)=\left[\begin{array}{c}{\overline{\sigma }}_y^i(y)\\ {\overline{u}}^i(y)\\ {\overline{v}}^i(y)\\ {\overline{\tau }}_{xy}^i(y)\end{array}\right], M^i=\left[\begin{array}{cccc}{M}_{11}^i& M_{12}^i& M_{13}^i& M_{14}^i\\ M_{21}^i& M_{22}^i& M_{23}^i& M_{24}^i\\ M_{31}^i& M_{32}^i& M_{33}^i& M_{34}^i\\ M_{41}^i& M_{42}^i& M_{43}^i& M_{44}^i\end{array}\right]$$

and the non-zero elements of M are given below

$$\begin{gathered} M_{13}^{ikr}=-{\delta }_{kr}{\rho }^i{\omega }^2, M_{14}^{ikr}=-g_{ikr}^{(1)}, M_{23}^{ikr}=-g_{ikr}^{(1)}, M_{24}^{ikr}={\delta }_{kr}{\displaystyle \frac{1}{c_{66}^i}}, M_{31}^{ikr}={\delta }_{kr}{\displaystyle \frac{1}{c_{22}^i}}, \\ {M}_{32}^{ikr}=-{\displaystyle \frac{c_{12}^i}{c_{22}^i}}{g}_{ikr}^{(1)}, M_{41}^{ikr}=-{\displaystyle \frac{c_{12}^i}{c_{22}^i}}{g}_{ikr}^{(1)}, M_{42}^{ikr}=-{\beta }^i{g}_{ikr}^{(2)}-{\delta }_{kr}{\rho }^i{\omega }^2, {\delta }_{kr}=\left\{\begin{array}{c}1,k=r\\ 0,k\ne r\end{array}\right. \end{gathered}$$

Considering that the x-coordinates of the two external boundaries correspond to the first and last discrete points, i.e., x₁ = 0 and x_N = L, and with reference to Equations (14)–(16), the S–S, C–C, and C–F boundary conditions can be expressed as follows

$${\overline{v}}^i={\overline{\sigma }}_x^i=0, \mathrm{at} x=x_1, {\overline{v}}^i={\overline{\sigma }}_x^i=0, \mathrm{at} x = x_N, (S–S)$$

(23)

$${\overline{u}}^i={\overline{v}}^i=0, \mathrm{at} x=x_1, {\overline{u}}^i={\overline{v}}^i=0, \mathrm{at} x = x_N, (C–C)$$

(24)

$${\overline{u}}^i={\overline{v}}^i=0, \mathrm{at} x=x_1, {\overline{\tau }}^i={\overline{\sigma }}_x^i=0, \mathrm{at} x = x_N. (C–F)$$

(25)

By substituting the in-plane stress condition ${\overline{\sigma }}_x^i=0$ in Equations (23) and (25) into Equation (22), one has

$${\displaystyle \frac{c_{12}^i}{c_{22}^i}}{\overline{\sigma }}_y^i(y)=-{\beta }^i{\displaystyle \sum _{r=1}^N{g}_{ikr}^{(1)}}{\overline{u}}_r^i(y), x = x_1 \mathrm{or} x_{\mathrm{N}}.$$

(26)

Substituting Equations (23)–(26) into Equation (21), yields the matrix elements of M under the three boundary conditions, which are given in Appendix A. The general solutions for the stresses and displacements at the discrete points can be obtained by solving the state-space equation of Equation (21)

$${\Phi }^i(y)=e^{{{\rm M}}^i(y-d_{i-1})}{\Phi }^i(d_{i-1}), d_{d-1} \le y \le d_i$$

(27)

2.3. Natural Frequencies and Mode Shapes

To determine the natural frequencies of the FML beam, the relationship of the stresses and displacements between layers are established based on the general solutions of Equation (27) and interface continuity conditions as described below

$${\overline{\sigma }}_y^{i+1}(d_i)={\overline{\sigma }}_y^i(d_i), s_i[{\overline{u}}^{i+1}(d_i)-{\overline{u}}^i(d_i)]={\overline{\tau }}_{xy}^i(d_i)$$

$${\overline{v}}^{i+1}(d_i)={\overline{v}}^i(d_i), {\overline{\tau }}_{xy}^{i+1}(d_i)={\overline{\tau }}_{xy}^i(d_i), i=1,2\dots P-1$$

(28)

where $s_i=s$ ($i={\displaystyle {\sum }_{\lambda =1}^m{p}_{\lambda }}$), $s_i=\infty$ ($i\ne {\displaystyle {\sum }_{\lambda =1}^m{p}_{\lambda }}$). The interface continuity conditions can be rearranged into matrix form:

$${\Phi }^{i+1}(d_i)=S_i{\Phi }^i(d_i),.$$

(29)

where

$$S_i=\left[\begin{array}{cccc}I& 0& 0& 0\\ 0& I& 0& (1/s_i)I\\ 0& 0& I& 0\\ 0& 0& 0& I\end{array}\right]$$

and $I$ represents identity matrix. By alternately employing Equations (27) and (29), a relationship of the stresses and displacements between the first and i-th layers are obtained:

$${\Phi }^i(y)=P^i(y){\Phi }^1(0), {\Omega }^i(y)=e^{{{\rm M}}^i(y-d_{i-1})}{\displaystyle \prod _{l=i-1}^1{S}_l{e}^{{{\rm M}}^l{h}_l}},$$

(30)

Taking i = P and y = H in Equation (30), yields the relationship between the top and bottom surfaces:

$${\Phi }^P(H)={\Omega }^P(H){\Phi }^1(0).$$

(31)

For the free vibration problem, the FML beam is not subjected to any external load, i.e.,

$${\overline{\sigma }}_y^P(H)={\overline{\tau }}_{xy}^P(H)={\overline{\sigma }}_y^1(0)={\overline{\tau }}_{xy}^1(0)=0.$$

(32)

Substituting the above equation into Equation (31), a homogeneous matrix equation for ${\overline{u}}^1(0)$ and ${\overline{v}}^1(0)$ is derived:

$$\left[\begin{array}{cc}{\Omega }_{12}^P(H)& {\Omega }_{13}^P(H)\\ {\Omega }_{42}^P(H)& {\Omega }_{43}^P(H)\end{array}\right]\left\{\begin{array}{c}{\overline{u}}^1(0)\\ {\overline{v}}^1(0)\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}.$$

(33)

where ${\Omega }_{kl}^P(H)$ denotes the submatrix located at the k-th row and l-th column of matrix ${\Omega }^P(H)$ after partitioning it into a 4 × 4 block structure. According to linear algebra [33], the existence of nontrivial solutions to Equation (33) requires that the coefficient matrix be singular, i.e.,

$$\left|\begin{array}{cc}{\Omega }_{12}^P(H)& {\Omega }_{13}^P(H)\\ {\Omega }_{42}^P(H)& {\Omega }_{43}^P(H)\end{array}\right|=0.$$

(34)

Expanding the above determinant yields a polynomial in terms of ω, from which the natural frequencies of various orders can be obtained using the bisection method [34]. For each obtained natural frequency, the displacement mode at the beam’s bottom surface is determined through normalization using Equation (33), while the displacement modes at other positions are obtained by substituting this bottom-surface mode into Equation (30).

3. Results and Discussion

In the following analysis, unless otherwise specified, the default research object is a three-layer FML beam is a three-layer FML beam with h_m = 15 mm and L = 4500 mm, composed of two CFRP facial layers and an aluminum alloy (AA) core layer. The material properties of CFRP and AA at room temperature T₀ = 20 °C is given in

Table 1. Material properties.

Parameter	CFRP	AA
ρ (kg/m3)	1600	2770
E1 (GPa)	153	70
E2 (GPa)	10.3	70
G12(GPa)	6	26.3
µ12	0.3	0.33
k2 (W/m°C)	0.8	121

and their temperature-induced attenuation coefficients [35] are

$${\xi }_{CFRP}=0.475\tanh [-8.68\times {10}^{-3}(T-367.41)]+0.475.$$

(35)

$${\xi }_{AA}=\{1-\tanh [0.062(T-273)]\}/2.$$

(36)

3.1. Convergence and Comparison Study

To verify the accuracy of the solutions in this study, a convergence analysis of the natural frequencies with respect to the number of Lobatto discretization points N is performed, and the results are compared with those from finite element (FE) solutions, experimental results [36], and the solution based on the FSDT [37]. Here, the boundary condition, the geometric and material parameters of the laminated beam are taken from Ref. [36]. For the FE simulation, the model is established with 27,000 CPS4R elements. The reference points are assigned at both beam ends and kinematically coupled to the corresponding cross-sections to impose the prescribed boundary conditions. The natural frequencies are determined using the subspace eigenvalue extraction method.

Table 2. Comparison of natural frequencies from the present solution and other solutions.

Solutions		1st	2nd	3rd	4th	5th
Present	N = 8	81.9828	505.660	1428.06	2785.90	4493.40
	N = 9	82.1605	512.533	1422.12	2756.61	4487.16
	N = 11	82.1703	512.331	1422.71	2755.32	4487.15
	N = 12	82.1701	512.320	1422.72	2755.34	4487.14
	N = 14	82.1701	512.320	1422.73	2755.34	4487.14
	N = 15	82.1700	512.32	1422.73	2755.34	4487.15
Experiment		82.50	511.3	1423	2784	4365
FE		82.25	512.8	1425	2860	4500
FSDT		85.40	531.5	1472	2839	/

lists the results of the natural frequencies obtained from the present solution with different N, the experiment, the FE simulation, and the FSDT solution. It can be found from Table 2 that (i) the present solution tends to converge as N increases and can achieve a convergence accuracy of at least six significant digits. (ii) The present solution agrees with the other experimental results, with the maximum relative error of 2.80% occurring at the fifth mode. The errors of FE and FSDT solutions reach 3.81% and 3.96%, respectively. In addition, the present solution is more efficient than FE solution, achieving high accuracy with few discretization points, while FE solution requires large matrices and high computational cost, especially for high-order modes. The FSDT solution is based on the linear shear assumption and cannot fully capture shear effects, which leads to certain errors.

3.2. Effect of Temperature

The variation in the first five natural frequencies with uniform temperature under three boundary conditions, as well as the corresponding x-direction mode shape of v at temperature T₀ shown in Figure 3, while Figure 4 shows the variation in the first five natural frequencies with non-uniform temperature.

The following conclusions can be drawn: (i) with increasing temperature, the frequency exhibits a moderate decrease in the range of 0 °C to 200 °C, followed by a sharp decline between 200 °C and 300 °C. This can be primarily attributed to the degradation in the elastic modulus of both CFRP and AA, in which the CFRP undergoes a transition from a glassy to a rubbery state. (ii) The natural frequencies are highest for the C–C condition, followed by S–S and then C–F. Full restraint in the C–C case yields the highest stiffness and frequency. Though S–S and C–F impose equal constraints, the symmetric support in S–S ensures a more uniform stiffness distribution, leading to a higher frequency than C–F. (iii) As the mode order increases, the spacing between adjacent natural frequencies gradually widens. This phenomenon arises because the system’s equivalent stiffness, which resists higher-order deformations, grows nonlinearly at a much faster rate than the equivalent mass. (iv) For the S–S case, the modes exhibit sinusoidal patterns with displacement nodes at both ends. The C–C beam shows more confined shapes due to additional slope constraints at the supports. In contrast, the C–F beam presents asymmetric modes, with maximum deflection always occurring at the free end. These results clearly demonstrate the influence of boundary conditions on mode shape distribution and symmetry. (v) Under non-uniform temperature fields, the variation trend of the frequencies is similar to that observed in uniform temperature environments. However, the decrease in natural frequency with rising temperature is significantly mitigated. This difference is mainly due to the low thermal conductivity of the CFRP layer, which restricts heat transfer to lower layers and thus mitigates the reduction in natural frequencies. (vi) Temperature distribution significantly affects the natural frequency, with a maximum reduction of 33.7% potentially increasing the risk of resonance. To address this, appropriate material selection and lay-up optimization can be employed to control temperature gradients and enhance dynamic stability.

3.3. Effect of Interface

A parametric investigation is conducted on the FML beam to examine the influence of interfacial stiffness s on its natural frequencies. In addition, the incorporation of FG layers as transitions between AA and CFRP is also explored.

Figure 5 shows the variation in the natural frequencies with different interfacial shear stiffnesses under four temperature conditions and three types of boundary constraints. At room temperature, the interfacial stiffness of common adhesives for FMLs is generally on the order of 10⁴ or higher, whereas at elevated temperatures—especially when the adhesive exceeds its glass transition temperature T_g—the interfacial stiffness typically falls to the order of 10² or lower. These two ranges of interfacial stiffness are also marked in the figure. It can be found from Figure 5: (i) within the range of approximately 10 MPa < s < 10⁴ MPa, the natural frequency exhibits a pronounced upward trend with increasing s, while outside this range, the variation tends to level off. A shear stiffness s below 10 MPa corresponds to a fully unbonded interface, whereas s above 10⁴ MPa represents a perfectly bonded interface. (ii) As temperature increases, the natural frequency tends to stabilize at lower values of interfacial stiffness s. This is due to the reduction in elastic modulus of the material layers, which decreases the overall stiffness of the FML beam. Consequently, the relative contribution of interfacial stiffness increases, causing the natural frequency to reach equilibrium at a smaller s. (iii) Under C–C boundary conditions, the natural frequency tends to stabilize at lower s values compared to S–S and C–F conditions, yet continues to increase at higher s. This overall trend is primarily attributed to the higher structural stiffness associated with C–C boundaries. This figure can provide a basis for the design of FML interface bonding in practical engineering under both ambient and high-temperature conditions.

With the overall thickness of the FML beam kept constant, FG interlayers are incorporated between the AA and CFRP layers, wherein the elastic modulus is assumed to vary linearly from that of AA to CFRP. The thicknesses of CFRP, AA and FG layers become h₁ = h₃ = 15 − 0.5h_FG, h₂ = 15 − h_FG, while h_FG is variable. Figure 6 shows the normalized stress distribution corresponding to the first-mode shape of the S–S FML beam for different layer thicknesses, in which h_FG = 0 means no FG layer. It can be seen from the figure that, as the thickness of the FG layers increases, both normal and shear stresses exhibit a smoother transition across the interfaces. In practical engineering, this can moderately mitigate the risk of stress concentrations and delamination between layers.

4. Conclusions

The present study aims to develop an analytical model to investigate the free vibration of FML beams under thermal effects, considering boundary conditions and interfacial slip. The main findings are as follows:

• The present solution shows overall agreement with the experimental results, FE simulations, and the FSDT solution. Moreover, it is more efficient than the FE method, as the latter requires large matrices and incurs high computational cost, particularly for higher-order modes. In contrast, the FSDT solution relies on a linear shear assumption and thus cannot fully capture shear effects, resulting in certain errors.
• The natural frequencies decrease gradually from 0 °C to 200 °C, followed by a sharper drop between 200 °C and 300 °C, mainly due to the temperature-dependent degradation of the elastic modulus in both the aluminum and CFRP layers.
• The C–C configuration exhibits the highest natural frequencies, due to full displacement and rotation restraint. Frequencies in the S–S configuration are slightly higher than in C–F, reflecting the more uniform stiffness distribution under symmetric support.
• The natural frequencies of FML beams are significantly influenced by interfacial stiffness, with the most pronounced effect occurring between 10–10⁴ MPa. An interfacial stiffness higher than 10⁴ represents the typical range for FML interfaces at ambient temperature, whereas a stiffness lower than 10² corresponds to the reduced interfacial stiffness after the adhesive surpasses its glass transition temperature. Higher temperatures reduce frequencies and lower the required interfacial stiffness for stabilization. This can provide a basis for the design of FML interface bonding in practical engineering under both ambient and high-temperature conditions.
• By introducing an FG interlayer between the FML layers, both normal and shear stresses transition more smoothly across the interfaces. Furthermore, without altering the total thickness, this effect becomes increasingly pronounced as the FG interlayer thickness increases, moderately mitigating the risk of stress concentrations and interfacial delamination.

The present model can only consider the steady-state temperature field and neglects thermal strains in the FML beam. In future work, the model will be extended to transient temperature fields and include thermal strains to enable a more accurate analysis of the vibrational behavior of FML beams under thermal effects.