Natural Frequencies of Functionally Graded Sandwich Plates Resting on an Elastic Foundation Using Chebyshev Finite Element Method

Nowadays, sandwich plates are common and important components of engineering structures consisting of two skins integrated with a core. These structures stand out for their outstanding bending stiffness, low mass density, efficient noise cancellation, and thermal insulation. They could be widely applied in many fields of engineering and defense technology. However, they also exhibit shortcomings, such as susceptibility to damage due to stress concentration or material discontinuities. Therefore, studying their mechanical behavior is necessary and has practical significance in engineering. Some typical deformation theories proposed and applied to analyze sandwich structures are the first-order shear deformation theory [1–3], the higher-order shear deformation theory [4–7], or the quasi-3D theory [8, 9], and many refined theories [10–17]. Furthermore, readers can find some results on sandwich structure analysis as well as composite structures in available documents [15–24]. For example, in the document [1], the authors presented analytical solutions for free vibration analysis of moderately thick rectangular plates, which were composed of functionally graded materials and supported by either Winkler or Pasternak elastic foundations. Based on the first-order shear deformation plate theory, the analysis procedure solved the exact equations of motion and captured the fundamental frequencies of the functionally graded rectangular plates resting on an elastic foundation. Theoretical formulation, Navier's solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory were presented in [2] for the analysis of through-thickness functionally graded plates. A refined trigonometric higher-order plate theory as in [3] was presented for bending analysis of simply supported functionally graded ceramic-metal sandwich plates. The effects of transverse shear strains as well as the transverse normal strain were taken into account. The number of unknown functions was only four, as opposed to six or more in the case of other shear and normal deformation theories. A general third-order plate theory that accounts for geometric nonlinearity and two constituent material variations through the plate thickness (i.e., functionally graded plates) was presented using the dynamic version of the principle of virtual displacements. The formulation was based on power-law variation of the material through the thickness and the von Kármán nonlinear strains. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity and material gradation through the thickness were specialized to the existing classical and shear deformation plate theories in the literature. Analysis of functionally graded material plates was studied using higher-order shear deformation theory with some special modifications in conjunction with finite element models [4–7]. Furthermore, the vibrational or thermomechanical bending behaviors of functionally graded sandwich plates were also presented in the documents [8–10]. Quasi-3D plate theories considering shear and normal deformations were incorporated to estimate the final results. In the papers [11, 12], the bending and the free flexural vibration behavior of sandwich functionally graded material plates were investigated based on higher-order structural theory. This theory accounted for the realistic variation of the displacements through the thickness. A multi-layered shell formulation was developed in the paper [13] based on a layerwise deformation theory within the framework of isogeometric analysis. The high-order smoothness of non-uniform rational B-splines offered the opportunity to capture the structural deformation efficiently in a rotation-free manner. The derivation also followed a layerwise theory, which assumed a separate displacement field expansion within each layer and considered the transverse displacement component as C0-continuous at layer interfaces, thus resulting in a layerwise continuous transverse strain state. In the paper [14], the authors derived a higher-order shear deformation theory for modeling functionally graded plates accounting for extensibility in the thickness direction. The explicit governing equations and boundary conditions were obtained using the principle of virtual displacements under Carrera’s Unified Formulation. The static and eigenproblems were solved by collocation with radial basis functions.

In recent years, the study of structures embedded in foundations has drawn a lot of attention amongst researchers. To express the interaction between foundation and plate, various hypotheses of foundation models have been introduced [15]. The oldest and simplest hypothesis of elastic medium models, which has only one coefficient substrate reaction, is known as Winkler elastic foundation [16]. Despite its ease of implementation, the Winkler model is defective in providing continuity in the foundation due to separate springs [17]. This hypothesis was improved by the Pasternak model [18] via adding a shear layer over springs. The Pasternak model, including two-parameter substrate (spring and shear layer), is widely used to explain the mechanical interactions of soft plates with different distributions of material properties. In the Kerr foundation [19], nonconcentrated reactions do not occur due to an upper spring layer. It means that, in the Kerr model, a shear layer is surrounded by upper and lower spring layers. From there one can see more related documents. The thermodynamic behavior of functionally graded sandwich plates resting on Winkler/Pasternak/Kerr foundations with various boundary conditions was studied in [20] by using a refined 2D plate theory. The displacement field contained undetermined integral forms and involved only four unknowns to derive. The plate was considered to be subject to a time-harmonic sinusoidal temperature field across its thickness. Three types of foundations were studied. Each was described by a mathematical model. Different boundary conditions were used to study the thermodynamic behavior of sandwich plates on elastic foundations. These models gave an incredible concurrence with the accessible literature. The research [21] investigated the free vibration analysis of advanced composite plates, such as functionally graded plates resting on two-parameter elastic foundations, using a hybrid quasi-3D (trigonometric as well as polynomial) higher-order shear deformation theory. The theory, which did not require a shear correction factor, accounted for shear deformation and thickness stretching effects by a sinusoidal and parabolic variation of all displacements across the thickness. The governing equations of motion for the plates were derived from Hamilton's principle. The closed-form solutions were obtained by using the Navier technique, and natural frequencies were found for simply supported plates by solving the results of eigenvalue problems. The research [22] analyzed the natural frequencies of the imperfect functionally graded sandwich plate comprised of porous face sheets made of functionally graded materials and an isotropic homogeneous core resting on the elastic foundation. To accomplish this, the material characteristics were taken to be changed incessantly along the thickness direction based on the volume fraction of constituents expressed by the modified rule of the mixture, which included porosity volume fraction with three diverse kinds of porosity distribution models. Furthermore, to describe the two-parameter elastic foundation's response on the imperfect plates, the medium was supposed to be linear, homogenous, and isotropic, and it had been modeled using the Winkler-Pasternak model. Moreover, in the kinematic relationship of the imperfect plate resting on the Winkler-Pasternak foundation, third-order shear deformation theory was used, and the motion equations were set up employing Hamilton's principle. A buckling analysis of functionally graded plates of a complex form resting on an elastic foundation and subjected to an in-plane nonuniform loading was performed by the R-functions method as in [23]. The mathematical formulation of the problem was presented within the framework of the classical laminate plate theory. The approach proposed and the software developed consider the heterogeneous subcritical state of the plates. To solve the problems, the Ritz method combined with the R-functions theory was used, etc.

Another aspect: the finite element method was first introduced in 1960 with the basic idea of dividing the entire problem domain into discrete components with simple geometry. Within each element, the representation of dependent variables is accomplished through shape functions. Structural analysis using the finite element method will be more accurate with higher-order shape functions. Nowadays, applying the finite element method to structural analysis has become popular and shows outstanding benefits compared to analytical methods, especially for structures with complex shapes and arbitrary boundary conditions. The study [24] focused on establishing the finite element model based on a new hyperbolic shear deformation theory to investigate the static bending, free vibration, and buckling of the functionally graded sandwich plates with porosity. The novel sandwich plate consisted of one homogenous ceramic core and two different functionally graded face sheets, which could be widely applied in many fields of engineering and defense technology. The discrete governing equations of motion were carried out via Hamilton’s principle and the finite element method. Besides, Chebyshev polynomials play a significant role in approximation theory. They constitute a sequence of orthogonal polynomials defined recursively. Their absolute value in the interval [-1, 1] is bounded by 1. Based on this ideal, the authors [25, 26] proposed a finite element method associated with Chebyshev polynomials of the first kind for the analysis of plate/shell structures. In this approach, Gauss points were employed to formulate the shape functions grounded in Chebyshev polynomials. Consequently, the code was introduced to overcome shear locking and eliminate spurious zero energy modes.

There are different approaches to analyzing this type of structure, and in this article, the finite element method based on Chebyshev polynomials is first used to study the free vibration of FGS plates with HC or SC resting on an elastic foundation. The reliability of this method is verified through numerical examples related to the effects of geometrical parameters, materials, and foundation. Furthermore, increasing the degree of the Chebyshev polynomial will increase the total number of degrees of freedom of the structure, leading to increased computational costs. However, the survey in this paper shows that it is possible to control the degree of the polynomial sufficiently to obtain the required approximate results. Some other advantages of the proposed method, although not explicitly verified in this article, are that the obtained results are not affected by mesh distortion or, although the Chebyshev interpolation polynomial is established in the form of the Lagrange interpolation, Runge’s phenomenon [27] does not occur; therefore, it can be used as an original finite element method for solving PDEs with high accuracy.

This article is organized as follows. The formulation is presented in Sect. 2. To highlight the reliability of the element, some numerical examples are thoroughly studied in Sect. 3. Finally, conclusions are drawn in Sect. 4.

Considering rectangular FGS plates with HC or SC resting on an elastic foundation as shown in Fig. 1. A group of h₁ – h₂ – h₃ represents the ratio of thicknesses.

Fig. 1.

The model of an FGS plate resting on an elastic foundation

The volume fraction Vc(i) of the ceramic phase of each layer is formulated by [3]

+ For FGS plate with HC (FGS-HC): 1{ Vc(1)=(z−z1z2−z1)nz∈[ z1,z2 ]Vc(2)=1z∈[ z2,z3 ]Vc(3)=(z−z4z3−z4)nz∈[ z3,z4 ] + For FGS plate with SC (FGS-SC): 2{ Vc(1)=1−(z−z1z2−z1)nz∈[ z1,z2 ]Vc(2)=0z∈[ z2,z3 ]Vc(3)=1−(z−z4z3−z4)nz∈[ z3,z4 ]

The effective material properties of FGS plates are determined by 3M(i)(z)=McVc(i)+Mm(1−Vc(i)),i=1,2,3

Here M⁽ⁱ⁾(Z) stands for the material characteristics of each layer.

The mechanical characteristics of the ceramic and metal are used in this article as Al₂O₃ (E = 380 GPa, ρ = 3800 kg/m³, ν = 0.3); Al (E = 70 GPa, ρ = 2707 kg/m³, ν = 0.3); and ZrO₂ (E = 151 GPa, ρ = 3000 kg/m³, ν = 0.3). In addition, Fig. 2 plots the effective elastic modulus versus the thickness of (Al/Al₂O₃)_FGS_(1-2-1).

Fig. 2.

The effective elastic modulus against the thickness of (Al/Al₂O₃)_FGS_(1-2-1) plates

The elastic foundation is modeled with two parameters (α₁, α₂). The foundation reaction is given as a function of the deflection and its Laplacian. The reaction-deflection relation of this foundation is defined by 4ℜ=α1w−α2(∂2w∂x2+∂2w∂y2)

The Chebyshev polynomials as in [25, 26] is given 5Cp(x)=Cp(cosφ)=cospφ,x=cosφ∈[−1,1] 6cos(p+1)φ=2cosφcospφ−cos(p−1)φ or 7Cp+1(x)=2xCp(x)−Cp−1(x),p=0,1,2…

The Chebyshev polynomial C_p(x)with p ≥ 2 will equals 0 at the Gauss points x_i 8xi=−cos[(2i−1)π/2p],i=1,2,3,…

The most important of ideas involves the approximation process of an unknown function f(x) by Lagrangian interpolation polynomial ℵ(x) through these known points (x_k, f(x_k)) based on Chebyshev polynomials as described 9f(x)≈ℵ(x)=∑i=0p−1aiCi(x) and f(xk)=ℵ(xk)

On the interval [-1, 1], Chebyshev polynomials have the orthogonal property so the unknown coefficient a_i may be calculated by 10ai=∑k=1pf(xk)Ci(xk)/∑k=1pCi2(xk)

Hence 11ℵ(x)=∑k=1p∑i=0p−1[ (Ci(xk)Ci(x))/∑j=1pCi2(xj) ]f(xk) with x = cos(π/2p) ξ 12ℵ(ξ)=∑k=1pNk(ξ)f(ξk) where N_k are called the 1D shape functions defined as 13Nk(ξ)=∑i=0p−1[ (Ci(cos(π/2p)ξk)Ci(cos(π/2p)ξ))//∑j=1pCi2(cos(π/2p)ξj) ] 14ξi=−cos[(2i−1)π/2p]/cos(π/2p)i=1,2,…,p

In 2D problems, (ξ, η) ∈ [–1,1] × [–1,1], the shape functions associated with node K(ξ_i, η_j) 15N˜K(ξ,η)=Ni(ξ)Nj(η) with N_i(ξ) or N_j(η) is 1D p-order shape functions related to the set ξ ∈ [–1,1] or the set η ∈ [–1,1] respectively. Fig. 3 gives some third-order shapes functions related to the Chebyshev polynomials in 1D space and 2D space.

Fig. 3.

The third-order shape functions in 1D & 2D

The finite element formulation related to Chebyshev polynomials is established as in [25, 26]. The plate is divided by the quadrilateral elements with five degrees of freedom in each node of an element. These components can be approximated through the displacement components at node K 16u={ uovowoβxβy }T==∑K=1(p+1)(q+1)N˜K(ξ,η){ uoKvoKwoKβxKβyK }T==NKdK with NK=N˜Kdiag(1,1,1,1,1)dK={ uoKvoKwoKβxKβyK }T are nodal displacement vector associated node K. Based on the first-order shear deformation theory, the kinematics of the plate is calculated through five displacement components u_o, v_o, w_o, β_x and β_y in the mid-surface 17u(x,y,z)=u0(x,y)+zβxv(x,y,z)=v0(x,y)+zβyw(x,y,z)=w0(x,y) with u_o, v_o, w_o are three displacement components, and α_x β_y are the rotations related to the x- and y-axes. The in-plane strain vector and the transverse shear strain vector are shown 18ε=[ εxxεyyγxy ]=[ u0,xv0,yu0,y+v0,x ]︸εm+z[ βx,xβy,yβx,y+βy,x ]︸εb=εm+zεb 19εs=[ γxzγyz ]=[ βx+w0,xβy+w0,y ]

The constitutive relation for functionally graded plate can be defined as below 20{ σxxσyyτxyτxzτyz }=[ Q11Q12000Q21Q2200000Q6600000Q5500000Q44 ]{ εxxεyyγxyγxzγyz } with the material constants are given by 21Q11=Q22=E(z)/[ 1−v2(z) ],Q12=Q21=v(z)E(z)/[ 1−v2(z) ],Q44=Q55=Q66=E(z)/2[1+v(z)]

By applying the Hamilton’s principle, the weak-form for free vibration analysis can be presented by 22∫Ωδε¯TDb*ε¯dΩ+∫Ωδγ¯TDs*γ¯dΩ+∫ΩℜwdΩ=∫ΩδuTmu¨dΩ where 23ε¯={ εm,εb }T,γ¯={ εs }T and m=[ I1I2I2I3 ] with the mass inertia terms I_i (i = 1,2,3) are given by 24(I1,I2,I3)=∫−h/2h/2ρ(z)(1,z,z2)[ 100010001 ]dz

Besides, Db* and Ds* , the material matrices, are written by 25Db*=[ ABBD ],Ds*=As with 26(A,B,D)=∫−h/2h/2(1,z,z2)[ Q11Q120Q21Q22000Q66 ]dz,As=56∫−h/2h/2[ Q5500Q44 ]dz

The discretized systems for free vibration analysis may be given as 27(K−ω2M)d=0 where K is the global stiffness matrix including the plate and elastic foundation, M is the global mass matrix, respectively. Furthermore, the Gauss quadrature rule is taken on the integration through a set of (p+1)(q+1) Gauss points, in which p and q are the orders of N_i(ξ) and N_j(η). The total numbers of Gauss points and degrees of freedom per each element are (p+1)(q+1) and 5(p+1)(q+1). The values of p and q are set to 3 for all numerical examples by following [16]. The BCs are given by

Clamped (C): u₀ = v₀ = w₀ = β_x = β_y = 0 at all edges.

Simply supported (S): u₀ = w₀ = β_x = 0 at y = 0 & y = b and v₀ = w₀ = β_y = 0 at x = 0 & x = a.

In order to clarify how the proposed technique is incorporated into a finite element code, a numerical implementation is briefly pre-sented as follows. –

Discretize the domain into quadrilateral elements and form the matrices of node coordinates (coord) and element connections (nodes).

The normalized parameters in this study are expressed by 28k1=α1a4/Dm,k2=α2a2/Dm,Dm=Emh3/12/(1−vm2),ω¯=(ωa2h)ρoEo,ρo=1 kg/m3,Eo=1GPa

Firstly, the free vibration analysis of thin (h = a/200) and thick (h = a/10) square plates with two boundary conditions, SSSS and CCCC, is considered. The Young’s modulus E = 200 GPa, Poisson ratio ν = 0.3, and mass density ρ = 8000 kg/m³ are the material properties. The first four normalized frequencies of these structures ω* = [12ωρa⁴(1 – v²)/(Eh²)]^1/4 are presented in Table 1 and Fig. 4. These results are also compared with other results of MITC4 element [28] and exact solutions [29].

Fig. 4.

The comparison of the first four normalized frequencies of square plate

Secondly, consider the SSSS square (Al/Al₂O₃) FGM plate re-sting on an elastic foundation. The dimensionless frequency is nor-malized by ω**=(ωh)ρb/Eb.

Observing that the obtained results converge at a mesh size of 4 × 4 with the order of shape functions p = q = 3 and are in good agreement with those of solutions using quasi-3D theory [30] and third-order shear deformation theory [31], as given in Table 2. An error of about 2% compared to the results from quasi-3D theory is acceptable when using only coarse meshes.

Next, the SSSS square (Al/ZrO₂) FGS-HC plate is considered. The first normalized frequencies ω¯ are provided in Table 3. It can be observed that the obtained results are completely consistent with the results of [32], employing a closed-form solution based on the hyperbolic shear deformation theory. Moreover, it is clear from the table that the natural frequencies are increasing with the existence of the elastic foundation. Clearly, the natural frequencies are increasing with the increasing values of two parameters: spring stiffness and shear layer stiffness. Table 3 also shows that, as the volume ratio of ceramic in the sandwich plate increases, the natural frequencies of the plate increase.

Continuously, the dimensionless frequencies of the square (Al/ZrO2) FGS plates with different values of the power-law index, ratio of thicknesses, and BCs are listed in Table 4 and Table 5. From these tables, it can be seen that when the power-law index n increases, the frequencies of the FGS-HC plates decrease while the frequencies of the FGS-SC increase. The reason is that as n increases, the volume fraction of ceramic decreases for the FGS-HC plate, leading to a decrease in the FGS-HC plate stiffness; so the frequency of the FGS plate decreases. The frequency of FGS-HC plates with n = 0 is highest, because when n = 0, FGS-HC plates become homogeneous ceramic plates. On the other hand, when n increases, the frequencies of the FGS-SC plates increase, because when n increases, the volume fraction of the ceramic components increases, and the FGS plate stiffness increases. Besides, when n = 0, the FGS-SC plates become homogeneous metal plates, so the frequency of FGS plates is the smallest. In addition, the thicker ceramic core results in a stiffer FGS-HC plate, causing the frequency of the FGS plate to increase. This is in contrast to the FGS-SC plate, when the thicker metal core leads to a softer FGS-SC plate, so the frequency of the FGS plates reduces.

Figs. 5 and 6 present the first four modeshapes of FGS plates. Observing that the 2nd and 3rd eigenmodes are the same as each other, owing to the FGS plates under the same “S” or “C” boundary conditions at the edges.

Fig. 5.

The first four mode shapes of (SSSS) FGS-HC plate with a/b = 1, a/h = 20, k₁ = 10, k₂ = 5, n = 1, ratio of thicknesses (1-1-1)

Fig. 6.

The first four mode shapes of (CCCC) FGS-SC plate with a/b = 1, a/h = 25, k₁ = 15, k₂ = 10, n = 2, ratio of thicknesses (1-2-1)

The article aims to conduct the frequency analysis of FGS plates resting on an elastic foundation by using a finite element method based on Chebyshev polynomials. The idea of this element is to use the high-order shape functions that satisfy the interpolation condition at the points based on Chebyshev polynomials as well as to use the full Gauss quadrature rule for the establishment of the stiffness matrix and mass matrix. Numerical examples and comments are made to illustrate in detail the influence of geometric parameters and material properties on the free vibrations of the FGS plates. According to the numerical outcomes, some important observations are summarized as follows: i. The effectiveness of using the finite element method based on Chebyshev polynomials in the analysis of free vibrations of FGS plates on the elastic foundation. ii. Elastic foundation increases the frequency of FGS plates, as expected. More specifically, the natural frequencies are increasing with the existence of an elastic foundation. The natural frequencies are increasing with the increasing values of two parameters: spring stiffness and shear layer stiffness. iii. While increasing the value of the power-law index leads to the decreasing value of the natural frequency for hardcore sandwich plates, the natural frequency increases with increasing value of the power-law index for softcore sandwich plates. In conclusion, it is evident that the elastic foundation has a significant effect on the mechanical behavior of sandwich plates.

The obtained results can be useful for extended calculation, design, and fabrication of FGS plates working under various conditions. In the near future, the proposed method will continue to demonstrate the stability of the results when meshing distortion. In addition, although the Chebyshev interpolation polynomial is established in the form of Lagrange interpolation, the Runge phenomenon does not occur, so it can be used as an original finite element method to solve PDEs with high accuracy. Besides, the task of non-linear analysis of plate/shell structures will be developed. Last but not least, the computational cost for complex structures will also be considered to help readers have a specific view of the advantages and disadvantages of this proposed method.