Buckling Analysis of a Composite Honeycomb Reinforced Sandwich Embedded with Viscoelastic Damping Material

Abstract

In this study, the buckling loads of a composite sandwich structure, which is reinforced by a honeycomb layer and filled with viscoelastic damping material, are analyzed. By applying von Karman anisotropic plate equations for large deflection, the governing equation of the composite sandwich structure is determined, and the deflection of the structure is further defined by a double triangular series. According to the dynamic equivalent effective stiffness obtained by the homogenous asymptotic method and Hill’s generalized self-consistent model based on the Halpin–Tsai model, limiting the dynamic load buckling of the composite honeycomb reinforced sandwich structure embedded with viscoelastic damping material under axial compression can be achieved. The factors that influence the composite sandwich’s buckling loads are discussed and compared, such as the load and geometry parameters, the thickness of the honeycomb reinforcement layer and the honeycomb’s width. Finally, the results obtained by the present method are validated by the existing literature.

1. Introduction

In engineering applications, structure stability evaluation is more important than stiffness and strength in most cases. On most occasions, structures fail due to instability factors, such as the structure’s buckling, and as is known, the critical buckling loads of the structure are far smaller than the allowable loads. Quebec bridge disasters are usually considered as buckling failures and instability problems, as is known in engineering [1]. Today, to achieve high stiffness/hardness, considerable toughening, light-weight properties and composite materials and structures are introduced into engineering applications, which have been capturing more and more researchers’ attention. Therefore, the buckling analysis of the composite structures is performed extensively in many different fields of engineering, such as aviation, navigation and mechanical engineering [2]. Accordingly, the buckling characteristics are taken into consideration. As pointed out by Wang et al. [3], the buckling of the structure is an important consideration in structural design, especially when the structure is slender and lightweight. Sayyad and Ghugal [4] presented a state-of-the-art discussion and critical review on the bending and buckling analysis of composite laminated structures, especially on comparisons of the methodologies and discussion of the composite structures, such as laminate theories and numerical methods. By applying the numerical and experimental methods, Namdar and Darendeliler [5] examined the buckling and failure process of a composite laminated structure made by bi-directional and oven fabric laminae. By employing the Kelvin–Voigt model and basing their work on nonlocal elastic theory, Kolahchi et al. [6] developed the refined Zigzag theory to analyze the dynamic buckling of laminated nano-plates. The factors that influence the structure’s dynamic buckling were thoroughly explored, such as the viscoelastic layer’s damping coefficient, aspect ratio, various boundary conditions, etc. Based on the cell-based smoothed discrete shear gap method mixed-variable differential evolution theory, Ho-Huu et al. [7] proposed a novel numerical optimization procedure with mixed-integer and continuous design variables for the optimal design of laminated composite plates’ buckling loads. To maximally increase the buckling loads of the composite laminated plate, thin composite grid layers were proposed by Ehsani and Rezepazhand [8] by combining the first-order shear deformation and classical laminated plate theories, and a genetic algorithm was utilized to optimize the stacking sequence and pattern composition of grid composite laminate. In the present analysis, as a type of grid, the honeycomb reinforced layer is applied and embedded in the laminated structure. The buckling response characteristic of the grid stiffened laminated composite plates was also analyzed by Huang et al. [9] by the finite element model. By introducing a unique Fourier series function to describe the longitudinal variation of deflection, Chen and Qiao [10] developed a novel semi-analytical finite strip method to predict the buckling properties of the composite laminated structures. Based on the first-order shear deformation theory and a combination of the modified couple stress theory, Arshid et al. [11] analyzed the bending and buckling characteristics of a heterogeneous annular/circular micro sandwich plate located on the Pasternak foundation.

The critical buckling analysis of composite laminated structures is focused on static changes in external loads, which are either very tiny or neglected. However, in mechanical engineering, the external loads and working environments are changeable. Moreover, after the composite material is introduced in engineering, the dynamic characteristic of the materials should be considered for perusing a higher analysis result. Therefore, the dynamic critical buckling properties of the composite structure are required to consider. By considering the honeycomb structure applied and analyzed in engineering structures, such as honeycomb reinforced laminated beams [12,13,14,15], plates [16,17,18,19,20,21,22,23,24] and shells [25,26,27], to enhance the structure’s stiffness or to strengthen it, the honeycomb reinforced layer is introduced in the present analysis. Through a stress function approach, Southward et al. [13] analyzed the buckling response of a composite laminated beam with a honeycomb core on a Winkler foundation. Through a two-scale theory of the updated Lagrangian type, Ohno et al. [17] analyzed the buckling of elastic square honeycomb structures subject to in-plane biaxial compression, and the results were validated by the energy method. Qiu et al. [16] examined the buckling of honeycomb structures under out-of-plane loads. Three types of honeycomb were considered, and finally, the results were validated by numerical and experimental methods. By applying the asymptotic homogenous method and basing their work on laminated plate theory, Zhou et al. [23] examined the dynamic equivalent effective modulus of the composite laminated structures with a honeycomb reinforced core, and factors that affect the structure’s dynamic modulus were qualitatively and quantitatively discussed and compared.

Compared to the buckling analysis of the traditional analysis, the external energy applied to the laminated structure is transformed into strain energy and potential energy, and the energies are stored or transformed in the structure, which do not dissipate. Consequently, instability happens when the loads reach critical buckling, and finally, failure may occur. Moreover, the flows of material and tiny fractures of the composite structure may appear microscopically, which can also absorb and dissipate a large amount of mechanical energy. For improving the critical buckling loads of the laminated structures, viscoelastic materials are applied. Viscoelastic materials can transform and dissipate the mechanical energy and improve the friction coefficient of the laminated structure in microscopically [28,29,30,31,32,33]. Moreover, the structure’s toughness is enhanced by the viscoelastic material’s visco- and friction-characteristic and mechanical energy dissipation properties. Aiming towards energy dissipation and vibration control, Jung and Aref [34] proposed a combined polymer composite damping system, and the structure consists of a polymer honeycomb and a viscoelastic solid material. Considering that the constrained layer dampers are extensively applied for passive vibration damping in fields of engineering, a novelty honeycomb structure filled with viscoelastic damping material was proposed by Aumjaud et al. [35], which can improve the modal loss factor of the composite laminated structure efficiently.

Following the structures constructed by Zhou et al. [23], in the present analysis, honeycomb structures and viscoelastic material are applied. The laminate’s stiffness and hardness can be significantly enhanced by honeycomb reinforcement layers, and its toughening and energy dissipation characteristics are improved at the same time by the viscoelastic material. Considering that the dynamic effective equivalent stiffness of the honeycomb structure with viscoelastic material is strongly influenced by the external load’s frequency, the structure’s buckling loads are affected by frequency too. Consequently, the dynamic critical buckling loads of honeycomb reinforced laminated structures with viscoelastic material are examined here. Moreover, the geometry and external load factors that affect the dynamic buckling load of composite structures are researched and compared in detail.

2. Mathematical Modeling

A mathematical model of the honeycomb reinforced composite laminated plate with coupling in-plane forces N_x and N_y on the x- and the y-axis is shown in the following figures. The length and width of the composite laminate are denoted as a and b. The middle layer is composed of a honeycomb reinforcement structure and viscoelastic material fillers. The honeycomb structure is made of elastic material, and the fillers are viscoelastic material. The thickness of the three-layered composite laminated structure is illustrated in Figure 1c, and the laminated structure in the present analysis is considered as the symmetrical composite structure. The thickness of the two face layers is t₁ and t₃, and the honeycomb reinforced layer is t₂.

3. Methodologies and Theoretical Derivation

In this analysis, the critical buckling load determination mainly contains two sections: the determination of the critical buckling equation and the prediction of the effective dynamic stiffness modulus of the composite sandwich. The determination of the critical buckling equation is performed based on large deflection theory, and the effective dynamic stiffness modulus of the composite sandwich is obtained through the homogenous asymptotic theory. More detail is presented in the following subsections.

3.1. Critical Buckling Equation Determination

According to von Karman’s anisotropic plate equations for large deflection [36,37], the governing equation of the composite sandwich structure under two directions of in-plane forces can be expressed as

$$\begin{array}{l}{D}_{11}\frac{{\partial }^{4}w}{\partial x^4}+4D_{16}\frac{{\partial }^{4}w}{\partial x^3\partial y}+2\left(D_{12}+2D_{66}\right)\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+4D_{26}\frac{{\partial }^{4}w}{\partial x\partial y^3}\\ +D_{22}\frac{{\partial }^{4}w}{\partial y^4}=N_x\frac{{\partial }^{2}w}{\partial x^2}+2N_{xy}\frac{{\partial }^{2}w}{\partial x\partial y}+N_y\frac{{\partial }^{2}w}{\partial y^2}-p_x\frac{\partial w}{\partial x}-p_y\frac{\partial w}{\partial y}-p_z\end{array}$$

(1)

where w is the out-plane deflection of the composite anisotropic laminate, N_xy is the shear force, p_x, p_y and p_z are the distributed loads and D_ij is the bending stiffness. In the present analysis, the sandwich is considered a rectangle with four edges that are simply supported. Furthermore, we assume that there is an absence of shear and distributed loads, so the bending stiffness D₁₆ = D₂₆ = 0 can be considered. Moreover, only the axis loads are considered, and the shear forces on the laminated structure are not considered. Accordingly, the constitutive equation of the current problem can be obtained as

$$D_{11}\frac{{\partial }^{4}w}{\partial x^4}+2\left(D_{12}+2D_{66}\right)\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+D_{22}\frac{{\partial }^{4}w}{\partial y^4}=N_x\frac{{\partial }^{2}w}{\partial x^2}+N_y\frac{{\partial }^{2}w}{\partial y^2}$$

(2)

For the simply supported rectangular plate, as shown in Figure 1c, the boundary conditions can be expressed as

$$w=M_x=-D_{11}\frac{{\partial }^{2}w}{\partial x^2}-D_{12}\frac{{\partial }^{4}w}{\partial y^4}=0\mathrm{at}x=0,a$$

(3)

$$w=M_y=-D_{12}\frac{{\partial }^{2}w}{\partial x^2}-D_{22}\frac{{\partial }^{4}w}{\partial y^4}=0\mathrm{at}y=0,b$$

(4)

To satisfy the boundary conditions as provided in the above equations, double sinusoidal series can be applied to express the deflection of the honeycomb reinforced composite laminated structure according to [38].

$$w={\displaystyle \sum _m^{\infty }{\displaystyle \sum _n^{\infty }{A}_{mn}\sin \alpha x\sin \beta x}}$$

(5)

where A_mn denotes the amplitude of w, α and β are coefficients of the sinusoidal function and α = mπ/a, β = nπ/b. By substituting Equations (3)–(5) into Equation (2), the following expression can be yielded:

$${\pi }^2{A}_{mn}\left[D_{11}{m}^4+2\left(D_{12}+2D_{66}\right)m^2{n}^2{\kappa }_0+D_{22}{n}^4{\kappa }_a{}^4\right]=-A_{mn}{a}^2\left[N_x{m}^2+N_y{n}^2{k}_a{}^2\right]$$

(6)

When critical buckling appears on composite laminated structures, the in-plane deflection w is not zero. Accordingly, A_mn in the above equation can be deleted, and the buckling load of the honeycomb reinforced viscoelastic-material-composed sandwich structure can be achieved as

$$N_{\mathrm{b}}=-\frac{{\pi }^2\left[D_{11}{m}^4+2\left(D_{12}+2D_{66}\right)m^2{n}^2{\kappa }_0+D_{22}{n}^4{\kappa }_a{}^4\right]}{a^2\left(m^2+k_n{n}^2{\kappa }_a{}^2\right)}$$

(7)

where k_n = N_x/N_y, and the minus symbol in Equation (7) can be considered the compress force/load. We can find that the critical buckling load is independent of the amplitude of deflection, and it is correlated with the dynamical equivalent stiffness D_ij, the terms of the sinusoidal series m and n, the loading parameter k_n and the geometry parameter k_a = a/b.

3.2. Equivalent Effective Modulus Determination

Considering that the laminated structures proposed in this analysis are periodically distributed in the x-y direction, as shown in Figure 1, and that a representative unit cell is shown according to the three-dimensional elastic theory [39], the elastic mechanics equilibrium equations of the laminated composites can be written as

$$\{\begin{array}{c}{\sigma }_{ij,j}=f_i\\ {\sigma }_{ij}=\frac{1}{2}{C}_{ijkl}\left({\alpha }_1,{\alpha }_2,\gamma \right)\left(u_{k,l}+u_{l,k}\right)\\ {\sigma }_{ij}{n}_j^{\pm }=p_j^{\pm }\end{array}$$

(8)

where ${(•)}_{,j}=\partial {(•)}_{,{\alpha }_j}$, C_ijkl is the fourth-order elasticity tensor of the material, f_i is the body force on the laminated structure, $p_j^{\pm }$ represents the surface tractions on the top and bottom surfaces of the representative unit cell and u_l(k) denotes displacements of the representative unit cell at the mid-plane in the ${\alpha }_{1\left(2\right)}$ direction. $n_j^{\pm }$ is the normal vector of the representative unit cell’s top and bottom surfaces and can be determined by $n^{\pm }=\frac{\left(\mp S^{\mp }{}_{,{\alpha }_1},\mp S^{\mp }{}_{,{\alpha }_2},1\right)}{\sqrt{{\left(S^{\mp }{}_{,{\alpha }_1}\right)}^2+{\left(S^{\mp }{}_{,{\alpha }_2}\right)}^2+1}}$.

3.3. Homogenous Asymptotic Methodologies

In this analysis, considering the periodic distribution of viscoelastic material and honeycomb reinforcements in geometry, the homogenous asymptotic theory is applied, and the “rapid” parameter ε is introduced. Asymptotic relationships can be written as

$${\xi }_1\to \frac{{\alpha }_1{A}_1}{\epsilon a},{\xi }_1\to \frac{{\alpha }_2{A}_2}{\epsilon b},z\to \frac{\gamma }{\left(h_1+h_2+h_3\right)}$$

(9)

where a and b are the length and width of the sandwich laminate. The coordinate transformation through asymptotic theory can be expressed as

$$\{\begin{array}{l}\frac{\partial }{\partial {\xi }_1}\to \frac{\partial }{\partial {\alpha }_1}\frac{\partial {\alpha }_1}{\partial {\xi }_1}+\frac{\partial }{\partial {\alpha }_2}\frac{\partial {\alpha }_2}{\partial {\xi }_1}\hfill \\ \frac{\partial }{\partial {\xi }_2}\to \frac{\partial }{\partial {\alpha }_1}\frac{\partial {\alpha }_1}{\partial {\xi }_2}+\frac{\partial }{\partial {\alpha }_2}\frac{\partial {\alpha }_2}{\partial {\xi }_2}\hfill \end{array}$$

(10)

Furthermore, the non-dimensional displacements of the representative volume element at the middle plane can be obtained according to the microscopic asymptotic theory

$$u_i\left(\alpha ,\xi ,z,\epsilon \right)=u_i^{\left(0\right)}\left(\alpha \right)+\epsilon u_i^{\left(1\right)}\left(\alpha ,\xi ,z\right)+{\epsilon }^2{u}_i^{\left(2\right)}\left(\alpha ,\xi ,z\right)+O\left({\epsilon }^3\right)$$

(11a)

$${\sigma }_{ij}\left(\alpha ,\xi ,z,\epsilon \right)={\epsilon }_{ij}^{\left(0\right)}\left(\alpha \right)+\epsilon {\sigma }_{ij}^{\left(2\right)}\left(\alpha ,\xi ,z\right)+{\epsilon }^2{\sigma }_{ij}^{\left(2\right)}\left(\alpha ,\xi ,z\right)+O\left({\epsilon }^3\right)$$

(11b)

where O(ε³) means the infinitesimal of the higher-order term. The relationship between the displacements and stress tensors in the local coordinate for the representative volume element can be defined according to coordinate transform as

$$u_i=v_i\left({\alpha }_1,{\alpha }_2\right)-\epsilon \frac{z}{A_i}w{\left({\alpha }_1,{\alpha }_2\right)}_{,{\alpha }_i}+\epsilon U_i^{\mu v}{\epsilon }_{\mu v}+{\epsilon }^2{V}_i^{\mu v}{\tau }_{\mu v}+O\left({\epsilon }^3\right)$$

(12a)

$$u_3=w\left({\alpha }_1,{\alpha }_2\right)+\epsilon U_3^{\mu v}{\epsilon }_{\mu v}+{\epsilon }^2{V}_3^{\mu v}{\tau }_{\mu v}+O\left({\epsilon }^3\right)$$

(12b)

$${\sigma }_{ij}=b_{ij}^{\mu \nu }{\epsilon }_{\mu \nu }+\epsilon b_{ij}^{*\mu \nu }{\tau }_{\mu \nu }$$

(12c)

where u_i(α₁, α₂), v_i(α₁, α₂) and w_i(α₁, α₂), respectively, represent the displacements in the α_i directions on the α_1-α₂ plane. We can find from Equation (11a,b) that the general global displacements and stresses of the representative volume element can be determined by the general local displacements $U_k^{\mu \nu }$ and $V_k^{*\mu \nu }$, where $U_k^{\mu \nu }=U_k^{\mu \nu }\left({\xi }_1,{\xi }_2,z\right)$ and $V_k^{\mu \nu }=V_k^{\mu \nu }\left({\xi }_1,{\xi }_2,z\right)$, and the local strain ${\epsilon }_{\mu \nu }$ and ${\tau }_{\mu \nu }$ as well as $b_{ij}^{\mu \nu }$ and $b_{ij}^{*\mu \nu }$ denote the general local stress functions. $U_k^{\mu \nu }$ and $V_k^{*\mu \nu }$ can be obtained according to the physical behaviors of the viscoelastic material and elastic reinforcements, and they have three configurations, which can be expressed as

$$U_n^{\lambda \mu }\left({\xi }_1,{\xi }_2,z\right)=\{\begin{array}{l}f\left(m_1{\xi }_1+z\right)+g\left(m_2{\xi }_2+z\right)\mathsf{\Delta }<0\\ f\left(m_1{\xi }_1+z\right)+{\xi }_{1}g\left(m_2{\xi }_2+z\right)\mathsf{\Delta }=0\\ f\left(m_2{\xi }_2+z\right)+g\left(m_1{\xi }_1+z\right)\mathsf{\Delta }>0\end{array}$$

(13a)

$$V_n^{\lambda \mu }\left({\xi }_1,{\xi }_2,z\right)=\{\begin{array}{l}g\left(m_1{\xi }_1+z\right)+f\left(m_2{\xi }_2+z\right)\mathsf{\Delta }<0\\ g\left(m_1{\xi }_1+z\right)+{\xi }_{1}f\left(m_2{\xi }_2+z\right)\mathsf{\Delta }=0\\ g\left(m_2{\xi }_2+z\right)+f\left(m_1{\xi }_1+z\right)\mathsf{\Delta }>0\end{array}$$

(13b)

Applying the asymptotic expansion equations as shown in Equations (9) and (10), two intermediate functions with a double triangular style can be defined, which can be expressed as

$$\begin{array}{l}f\left({\xi }_1,{\xi }_2,z\right)\\ =\{\begin{array}{l}{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^N\left[\sin \left(\frac{n\pi {\xi }_1}{\delta a}+z\right)\sin \left(\frac{m\pi {\xi }_1}{\delta b}+z\right)+\sin \left(\frac{n\pi {\xi }_1}{\delta b}+z\right)\sin \left(\frac{m\pi {\xi }_2}{\delta a}+z\right)\right]when\mathsf{\Delta }<0}}\\ {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^N\left[\sin \left(\frac{n\pi {\xi }_1}{\delta a}+z\right)\sin \left(\frac{m\pi {\xi }_1}{\delta b}+z\right)+{\xi }_1\sin \left(\frac{n\pi {\xi }_1}{\delta b}+z\right)\sin \left(\frac{m\pi {\xi }_2}{\delta a}+z\right)\right]when\mathsf{\Delta }=0}}\\ {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^N\left[\sin \left(\frac{n\pi {\xi }_2}{\delta a}+z\right)\sin \left(\frac{m\pi {\xi }_2}{\delta b}+z\right)+\sin \left(\frac{n\pi {\xi }_1}{\delta b}+z\right)\sin \left(\frac{m\pi {\xi }_2}{\delta a}+z\right)\right]}}when\mathsf{\Delta }<0\end{array}\end{array}$$

(14a)

$$\begin{array}{l}g\left({\xi }_1,{\xi }_2,z\right)\\ =\{\begin{array}{l}{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^N\left[\cos \left(\frac{n\pi {\xi }_2}{\delta a}+z\right)\cos \left(\frac{m\pi {\xi }_2}{\delta b}+z\right)+\cos \left(\frac{n\pi {\xi }_1}{\delta b}+z\right)\cos \left(\frac{m\pi {\xi }_1}{\delta a}+z\right)\right]when\mathsf{\Delta }>0}}\\ {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^N\left[\cos \left(\frac{n\pi {\xi }_2}{\delta a}+z\right)\cos \left(\frac{m\pi {\xi }_2}{\delta b}+z\right)+{\xi }_2\cos \left(\frac{n\pi {\xi }_1}{\delta b}+z\right)\cos \left(\frac{m\pi {\xi }_1}{\delta a}+z\right)\right]when\mathsf{\Delta }=0}}\\ {\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^N\left[\cos \left(\frac{n\pi {\xi }_1}{\delta a}+z\right)\cos \left(\frac{m\pi {\xi }_1}{\delta b}+z\right)+\cos \left(\frac{n\pi {\xi }_2}{\delta b}+z\right)\cos \left(\frac{m\pi {\xi }_2}{\delta a}+z\right)\right]}}when\mathsf{\Delta }>0\end{array}\end{array}$$

(14b)

The local deformations of each research object in two directions obtained by Equation (13) can be further written as ${\left(U_k^{\mu \nu }\right)}^{\left({\Omega }_n^{\left(L\right)}\right)}$ and ${\left(V_k^{*\mu \nu }\right)}^{\left({\Omega }_n^{\left(L\right)}\right)}$. Once they are determined, their general local stress tensors can be immediately evaluated, which can be written as ${\langle E_{ij}^{\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(\mathrm{L}\right)}\right)}$ and ${\langle z{E}_{ij}^{*\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(\mathrm{L}\right)}\right)}$. Then, the effective equivalent stiffness of the macroscopic composite sandwich structure in global coordinates can be achieved as follows:

$${\langle E_{ij}^{\lambda \mu }\rangle }^{\left(G\right)}={\displaystyle \sum _{n=1}^N{\langle E_{ij}^{\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(L\right)}\right)}}$$

(15a)

$${\langle z{E}_{ij}^{*\lambda \mu }\rangle }^{\left(G\right)}={\displaystyle \sum _{n=1}^N{\langle z{E}_{ij}^{*\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(L\right)}\right)}}$$

(15b)

where ${\langle E_{ij}^{\lambda \mu }\rangle }^{\left(G\right)}$ denotes the global effective modulus, and ${\langle E_{ij}^{\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(L\right)}\right)}$ is the local effective modulus coefficient of the nth research object. In this analysis, a representative volume element consists of nine research objects. Therefore, n = 9 in Equation (16), and its local deformations, stresses and strains can be calculated individually. ${\Omega }_m^{\left(L\right)}$ means the analysis object in the RVE of the composite structure. The superscripts in the brackets, L and G, denote the local coordinate and global coordinate, and these two systems are applied together to evaluate the global dynamic stiffness of the structure. In Equation (15), ijλμ is the fourth-order elastic tensor of the viscoelastic/elastic material. In the present analysis, the fourth-order tensors of the materials are considered to be frequency dependent, and the viscoelastic/elastic materials are assumed to be isotropic. The Kelvin–Voigt single-subscript notation is applied, and this transformation can be written as [39].

$$\begin{array}{l}{(•)}_{11}\to {(•)}_1,{(•)}_{22}\to {(•)}_2\\ {(•)}_{33}\to {(•)}_3,{(•)}_{23}\to {(•)}_4\\ {(•)}_{13}\to {(•)}_5,{(•)}_{12}\to {(•)}_6\end{array}$$

(16)

Finally, the dynamic equivalent effective stiffness of the honeycomb reinforced composite laminated structure can be determined as follows, according to [40]:

$$\langle •\left({\xi }_1,{\xi }_2,z\right)\rangle =\frac{1}{\left|V_{\Omega }\right|}{\displaystyle \underset{V_{\Omega }}{∰}\left[•\left({\xi }_1,{\xi }_2,z\right)\right]\mathrm{d}}{\xi }_1\mathrm{d}{\xi }_2\mathrm{d}z$$

(17)

where $\left|V_{\Omega }\right|$ denotes the RVE’s volume. The mid-coefficient for the bending stiffness can be obtained as

$$\{\begin{array}{l}{\overline{Q}}_{ij}=\frac{{\displaystyle \sum _{n=1}^N{\langle E_{ij}^{\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(\mathrm{L}\right)}\right)}}}{1-\langle v_{12}^{\mathrm{VR}}\rangle \langle v_{21}^{\mathrm{VR}}\rangle },ij=12,21,22\\ {\overline{Q}}_{66}={\displaystyle \sum _{n=1}^N{\langle z{E}_{ij}^{*\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(\mathrm{L}\right)}\right)}}\end{array}$$

(18)

In Equation (18), $\langle v_{21}^{\mathrm{VR}}\rangle$ is the equivalent dynamic Poisson’s ratio of the composite structure, and it can be approximately evaluated according to the composite laminated theory [41] as

$$\langle v_{12}\rangle =\left(t_1+\frac{t_2}{2}\right)v_{12}^{VR}\langle E_{22}^{\mathrm{VR}}\rangle /\left(t_1{E}_{22}^{\mathrm{R}}+\frac{t_2}{2}\langle E_{22}^{\mathrm{VR}}\rangle \right)$$

(19)

where $\langle E_{22}\rangle$ is the equivalent modulus of the composite honeycomb reinforced layer, and it is composed of viscoelastic material and elastic material. The reinforcements are made of elastic material, and the filler is built of viscoelastic material. Then, it can be evaluated according to Hill’s generalized self-consistent model based on the Halpin–Tsai model [42,43,44] and macroscopically uniform theory [45]

$$E_{22}^{VR}=\frac{1}{\frac{f_R^{}}{E_{22}^R}+\frac{f_{VEM}^{}}{E_{22}^{VEM}}-f_{VEM}^{}{f}_R^{}{\alpha }_{tt}}$$

(20)

$${\alpha }_{tt}=\frac{\frac{{\left(v_R\right)}^2{E}_{22}^{VEM}}{E_{22}^R}+\frac{{\left(v_{VEM}\right)}^2{E}_{22}^R}{E_{22}^{VEM}}-2v_R{v}_{VEM}}{f_R^{}{E}_{22}^R+f_{VEM}^{}{E}_{22}^{VEM}}$$

(21)

$$v_{12}^{VR}=f_R^{}{v}_R+f_{VEM}^{}{v}_{VEM}$$

(22a)

$${\rho }_{}^{VR}=f_R^{}{\rho }_R+f_{VEM}^{}{\rho }_{VEM}$$

(22b)

where f_VEM and f_R are the volume fractions of the viscoelastic material and the honeycomb reinforcement in a representative volume element, and v_R and v_VEM denote the Poisson’s ratio of reinforcements and viscoelastic fillers. The density of the honeycomb reinforced layer can be obtained through Equation (22a,b), and ρ_R and ρ_VEM denote the density of the viscoelastic and elastic materials.

By substituting the above equations into Equation (10), the dynamic equivalent Poisson’s ratio can be achieved. In the present analysis, the hexagonal honeycomb is considered; thus, $\langle v_{12}\rangle =\langle v_{21}\rangle$ can be assumed.

Accordingly, the dynamic equivalent bending stiffness of the honeycomb reinforced composite laminated with viscoelastic material can be obtained by drawing back the above equation into Equation (8) as

$$\langle D_{ij}^{\mathrm{G}}\rangle =\{\begin{array}{l}{\displaystyle {\int }_{-\frac{h}{2}}^{-\frac{h}{2}}\frac{{\displaystyle \sum _{n=1}^N{\langle E_{ij}^{\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(\mathrm{L}\right)}\right)}}}{1-\langle v_{12}^{\mathrm{VR}}\rangle \langle v_{21}^{\mathrm{VR}}\rangle }{z}^2\mathrm{d}z,}ij=12,21,22\\ {\displaystyle {\int }_{-\frac{h}{2}}^{-\frac{h}{2}}{\displaystyle \sum _{n=1}^N{\langle z{E}_{ij}^{*\lambda \mu }\rangle }^{\left({\Omega }_n^{\left(\mathrm{L}\right)}\right)}}{z}^2\mathrm{d}z,}ij=66\end{array}$$

(23)

Finally, the dynamic critical buckling loads of the composite honeycomb reinforced laminated can be determined as

$$\langle \widetilde{N_b}\rangle =-\frac{{\pi }^2\left[\langle D_{11}^{\mathrm{G}}\rangle m^4+2\left(\langle D_{12}^{\mathrm{G}}\rangle +2\langle D_{66}^{\mathrm{G}}\rangle \right)m^2{n}^2{k}_a{}^2+\langle D_{22}^{\mathrm{G}}\rangle n^4{k}_a{}^4\right]}{a^2\left(m^2+k_n{n}^2{k}_a{}^2\right)}$$

(24)

4. Numerical Analysis

In this section, a numerical analysis is performed. To visualize the regularity of the parameters’ influence on the critical buckling loads, the axis load parameter k_n, the geometry parameter k_a, the thickness ratio of the core layer, the honeycomb reinforcement’s width d_t and the dynamic impulse load’s frequency f are qualitatively and quantitatively compared and discussed. Geometrically, the external loads and physical parameters in the numerical section are defined as follows:

Geometrical parameters: face layers and honeycomb reinforcement.

t₁ = 2.5 mm, t₂ = At₁, d_t = Bt₁, where A = {2,4,8,12,16}, B= {1,2,3,4,5}.

a = L_a = {0.1:0.1:1}, b = a/k_a, k_a = {1:1:10}, L_honey = a/10. The fraction of the VEM filler and the hexagonal honeycomb reinforcement in the core layer are determined by L_honey and d_t.

Geometrical parameters: N_y = k_nN_x = k_nN₀, and k_n = {−2:0.5:2}. N_x and N_y denote the compression forces on the simply supported composite laminated structure, and the negative kn means the tensile force.

Physical parameters: face layers and honeycomb reinforcement made of aluminum.

Poisson’s ratio v_R = 0.33, Young’s modulus E_R = 68 GPa, shear modulus G_R12 = 25.6 Gpa, density ρ_R = 2700 kgm⁻³.

Physical parameters: VEM fillers.

Poisson’s ratio v_VEM = 0.49, density ρ_VEM = 1200 kgm⁻³_. Young’s modulus and the shear modulus G_R12 of the VEM are frequency dependent.

4.1. Critical Dynamic Buckling Load Affected by the Load Parameter k_n

As is known, the dynamic buckling of the composite structure is affected by two axial loads. The buckling load is changed if the tensional load is applied in one direction and if the compressional load is performed in the other direction. Moreover, the buckling load is variable when two compressional axial forces are loaded. To analyze the regularity and mechanism of the buckling loads affected by two axial forces, the load parameters of the honeycomb reinforced laminated composite structure are defined as k_n, where N_y = N₀ and N_x = k_nN_y = k_nN₀. The data are provided in

Table 1. Buckling loads vs. k_n.

a		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
kn	−2	921.69	230.42	102.41	57.61	36.87	25.60	18.81	14.4	11.38	9.22
	−1.5	582.62	145.66	64.74	36.41	23.30	16.18	11.89	9.10	7.19	5.83
	−1	395.41	98.85	43.93	24.71	15.82	10.98	8.07	6.18	4.88	3.95
	0	307.70	76.93	34.19	19.23	12.31	8.55	6.28	4.81	3.80	3.08
	1	198.55	49.64	22.06	12.41	7.94	5.52	4.05	3.10	2.45	1.99
	1.5	153.49	38.37	17.05	9.59	6.14	4.26	3.13	2.40	1.89	1.53
	2	49.170	12.29	5.46	3.07	1.97	1.37	1.00	0.77	0.61	0.49

. The buckling loads affected by k_n are plotted in Figure 2, where k_a = 1, and the length and width of the composite laminated plate are equal and denoted as a. t₁ = t₃, t₂ = 4t₁, d_t = 1.5t₁ and the length of the composite laminate ranges from 0.1 to 1 m. As shown in this figure, by increasing the load parameter, the buckling load is decreased significantly. Figure 3a shows that it is affected by the dimensions of the composite honeycomb reinforced laminate at the defined load parameters, and the buckling loads are decreased by increasing the laminate’s geometry dimensions. To visualize the buckling load of the composite laminate quantitatively, the tensile and compression loads are numerically analyzed and plotted in Figure 3b. The reference load parameter is defined as k_n = 1, and the reference buckling load is N₀ (k_n = 1). The buckling load of the honeycomb reinforced laminate is increased by raising the tensile loads, whereas it is decreased by increasing the compression loads. It can be seen in the figure that the buckling load can be dramatically increased by the tensile forces, whereas it is decreased slowly by the compression forces.

4.2. Critical Buckling Load Affected by the Length/Width Geometry Parameter

Except for the dynamic equivalent effective stiffness of the honeycomb reinforced composite laminate, the critical dynamic buckling is directly influenced by the length and width of the honeycomb reinforced composite sandwich structure at the same time. The geometry shape of the composite structure is determined by the length–width ratio k_a. Accordingly, the critical dynamic buckling is affected by the geometrical parameter k_a. In this section, the critical dynamic buckling of the honeycomb reinforced laminated composite structure is analyzed numerically. In this section, t₁ = t₃ = 2.5 mm, t₂ = 4t₁, d_t = 1.5t₁, k_n = 1 and the buckling loads affected by k_a are shown in

Table 2. Buckling loads vs. k_a.

a		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
ka	2	553.83	138.46	61.540	34.61	22.15	15.38	11.30	8.65	6.84	5.54
	4	1198.17	299.54	133.13	74.89	47.93	33.28	24.45	18.72	14.79	11.98
	6	2193.74	548.44	243.75	137.11	87.75	60.94	44.77	34.28	27.08	21.94
	8	3574.88	893.72	397.21	223.43	143.00	99.30	72.96	55.86	44.13	35.75
	10	5346.98	1336.75	594.11	334.19	213.88	148.53	109.12	83.55	66.01	53.47

. The results are the same as those shown in Figure 4; the geometry dimensions of the composite laminate affect its buckling sharply by increasing the length and width of the composite structures, and the structure’s buckling is decreased significantly. However, it can be observed from Figure 5 that the structure’s buckling loads are increased by raising the geometry parameters, and a larger k_a can always provide a bigger one when the other parameters are defined. To analyze the rate changes in the buckling load on the geometrical parameters, Figure 6 is provided. It can be seen in this figure that the honeycomb reinforced composite structure’s buckling load sharply increases when k_a is smaller than five, but it slowly increases when k_a is larger than five in the considered ranges.

4.3. Critical Buckling Load Affected by the Load Frequency

The load frequency affects the critical buckling load of the composite structures, which is the same as the results of the analysis performed by Xu et al. [46]. By increasing the loading frequency, the critical buckling loads are regularly increased. Considering the frequency-dependent properties of the viscoelastic material and by analyzing the load frequency influence on the structure’s buckling load, the loading frequency ranging from 1 to 1000 Hz is numerically analyzed in this section. t₁ = t₃ = 2.5 mm, t₂ = 4t₁, d_t = 1.5t₁, k_n = [−2, −1, 0, 1, 2], a = b = 0.1 m and k_a = 1. As shown in Figure 7, by increasing the loading frequency, the buckling load of the honeycomb reinforced laminated structure is generally increased in the considered frequency, ranging from 1 Hz to 1000 Hz. However, it should be noted that, in the present analysis, the structure’s buckling load, which is influenced by the load’s frequency, can be divided into three sections according to its change rate by red dotted lines, as shown in Figure 7. It is separated into three sections in terms of the load frequency, which are: 1~100.3 Hz, 100.3~700.9 Hz and 700.9~1000 Hz. In the considered frequency ranges, the buckling load of the structure is sharply increased and is then kept stable in a defined range. Then, it goes up significantly, and the critical frequencies are 100.3 Hz and 700.9 Hz. As a key factor that affects the structure’s buckling, the trend of the structure buckling is influenced by the load frequency, which is the same as the normalized effective equivalent stiffness.

4.4. Critical Buckling Load Affected by the Core Layer’s Thickness

The core layer’s thickness of the honeycomb reinforced composite laminated structure directly influences the structure’s dynamic effective equivalent stiffness, and the stiffness further affects the buckling load of the composite structures. In this section, the thickness of the honeycomb reinforcement and viscoelastic material composed layer is analyzed numerically. The geometrical, physical and load parameters are: t₁ = t₃ = 2.5 mm, d_t = 1.5t₁, k_n = k_a =1 and a = b, and the numerical data are provided in

Table 3. Buckling loads vs. core layer’s thickness.

a		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\frac{t_2}{t_1}$	2	85.38	21.35	9.49	5.34	3.42	2.37	1.74	1.33	1.05	0.85
	4	290.83	72.71	32.31	18.18	11.63	8.08	5.94	4.54	3.59	2.91
	8	1368.22	342.06	152.02	85.51	54.73	38.01	27.92	21.38	16.89	13.68
	12	3785.82	946.45	420.65	236.61	151.43	105.16	77.26	59.15	46.74	37.86
	16	8053.88	2013.47	894.88	503.37	322.16	223.72	164.36	125.84	99.43	80.54

. Similar to that which has been researched, the thickness of the honeycomb and viscoelastic material composed layer are defined as t₂ = t_Rt₁. t_R = {2, 4, 6, 8, 12, 16}. The buckling load of is sharply decreased by increasing the composite laminate’s geometrical dimensions, which is the same as the trend displayed in Figure 8. Moreover, it can be observed from Figure 8 and Figure 9 that the buckling load of the structure is increased by increasing the honeycomb reinforced layer’s thickness. To display the buckling load’s increasing rate on the honeycomb layer’s thickness, Figure 9 is provided. In the figure, the buckling load of the composite honeycomb laminate at t_R = 2 is considered as the reference value.

4.5. Critical Buckling Load Affected by the Honeycomb Reinforcement’s Thickness

The composite structure’s dynamic equivalent effective stiffness is directly affected by the honeycomb reinforcement’s width. By increasing the reinforcement’s width, the dynamic equivalent effective stiffness of the composite structure is increased, and furthermore, the structure’s buckling loads in the considered research are significantly influenced. In the present analysis, to illustrate the trend of the buckling load affected by the honeycomb reinforcement’s width, a qualitative investigation and quantitative comparison are performed, as plotted in Figure 10 and Figure 11, and the data is showed in

Table 4. Buckling loads vs. wall thickness of honeycomb.

a		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$\frac{d_t}{t_1}$	1	299.73	74.93	33.3	18.73	11.99	8.33	6.12	4.68	3.7	3
	1.75	441.04	110.26	49	27.56	17.64	12.25	9	6.89	5.44	4.41
	2.5	737.25	184.31	81.92	46.08	29.49	20.48	15.05	11.52	9.1	7.37
	3.25	1060.38	265.1	117.82	66.27	42.42	29.46	21.64	16.57	13.09	10.6
	4	1445.42	361.36	160.6	90.34	57.82	40.15	29.5	22.58	17.84	14.45

. The physical and geometrical parameters in this section are: t₁ = t₃ = 2.5 mm, t₂ = 5t₁ =, d_t = 1~4 t₁, k_n = k_a =1 and a = b. It can be seen in Figure 10 that, with the same geometry parameters, the buckling load of the honeycomb reinforced composite laminated structure is sharply increased by raising the reinforcement’s width. Moreover, the buckling loads at the defined width of the honeycomb reinforcement are compared in Figure 12. It can be observed that the buckling loads of the composite structure are decreased by increasing the dimensions.

5. Validation

In the present analysis, the buckling load of a honeycomb reinforced composite laminated structure is examined by the asymptotic method. To validate the method performed and the results obtained in the present analysis, the structure, which is the same as that which is referred in [37], is established. By submitting the equivalent stiffness of the composite laminate in the method applied in [37], and by defining the load parameter k_n = 1 and k_n = −1, the buckling loads obtained by the two methods can be compared. A comparison of the two methods is plotted in Figure 12, and it can be seen that the buckling loads obtained by the asymptotic methods are well consistent with [37]. The calculation error decreases significantly by increasing the structure’s geometry dimensions.

6. Conclusions and Discussion

In the present analysis, the buckling load of a honeycomb reinforced laminated structure filled with viscoelastic material is examined by employing the equivalent stiffness parameters obtained through asymptotic methods and the Halpin–Tsai model. The geometrical and physical parameters that influence the composite laminate’s buckling loads are qualitatively and quantitatively discussed and compared through an analysis. The following conclusions can be obtained from this research:

(1)

The physical parameters, such as the elastic modulus and shear modulus of the viscoelastic material, are affected by the load’s frequencies; accordingly, the buckling loads of the composite structure are influenced by the load’s frequency. The trend of the buckling loads affected by the loading frequency is the same as the composite structure’s dynamic equivalent effective stiffness, and the original phenomenon can be traced back to the behaviors of viscoelastic materials.

(2)

Buckling loads are affected by the geometry parameters of the composite sandwich structure directly. The buckling load is significantly increased by increasing by the width of the composite structure, but it is decreased sharply by increasing the length of the sandwich structure. This phenomenon shows the same results as those of the buckling analysis of classical beams and plates.

(3)

The composite structure’s buckling load is influenced by the honeycomb reinforcement layer’s thickness and the honeycomb’s width. The main reason is that the equivalent stiffness of the composite sandwich structure is changed by the honeycomb reinforcement’s width and height. It should be noted that, by comparing it to the honeycomb reinforcement layer’s thickness, the honeycomb’s width affects the structure’s buckling loads to a relatively smaller degree.

The moisture and temperature of the environment as well as the loading amplitude influence the viscoelastic material’s physical parameters as well, which is the same as the frequency-dependent properties. The background temperature in the research is considered to be 20 °C, the loading frequency ranges from 1 to 1000 Hz and the moisture effect on the viscoelastic material is neglected. Moreover, the loading amplitude properties on the viscoelastic material are not taken into consideration. In further research, those factors will be introduced.