A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation

Ngoc-Tu Do; Trung Thanh Tran; Quoc-Hoa Pham

doi:10.1515/cls-2022-0195

Article Open Access

A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation

Ngoc-Tu Do , Trung Thanh Tran and Quoc-Hoa Pham

Published/Copyright: July 26, 2023

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From the journal Curved and Layered Structures Volume 10 Issue 1

Abstract

This article provides a new finite-element procedure based on Reddy’s third-order shear deformation plate theory (TSDT) to establish the motion equation of functionally graded porous (FGP) sandwich plates resting on Kerr foundation (KF). Although Reddy’s TSDT is attractive, it cannot be exploited as expected in finite-element analysis due to the difficulties in satisfying the zero shear stress boundary condition. In this study, the proposed element has four nodes, each with seven degrees of freedom (DOF). The performance of this element is confirmed by conducting various examples, showing its accuracy and range of applications. Then, some studies are performed to present the effects of input parameters on the vibration of FGP sandwich plates resting on KF.

Keywords: sandwich plates; Q4 element; TSDT; Kerr foundation; functionally graded porous

1 Introduction

As is known, research on the mechanical behaviour of functionally graded material (FGM) structures is well understood, and some typical works can be mentioned including beams [1,2,3,4], plates [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], and shells [26,27,28]. Recently, Murin et al. [29] proposed a homogenized beam finite element for modal analysis considering double symmetric cross-section FGM beams. Burzyński et al. [30] analysed the geometrically nonlinear FGM shells using the neutral physical surface approach in the six-parameter shell theory. Brischetto et al. [31] studied the free vibration of FGM plates/cylinders based on 2D and 3D generalized differential quadrature (GDQ) models. Tornabene et al. [32] examined the static bending of anisotropic doubly curved shells with arbitrary geometry and variable thickness resting on a Winkler-Pasternak using higher-order theory. Besides, Viola et al. [33] used the GDQ finite-element method for free vibration analysis of cracked composite structures. Tornabene et al. [34] developed a differential quadrature solution for the free vibrations study of shells and panels of revolution with a free-form meridian. Functionally graded porous (FGP) material is a form of FGMs with the presence of many internal porous. Even so, this material possesses some outstanding mechanical properties such as lightness, excellent energy absorption, and great heat resistance properties. Some typical studies on the mechanical behaviour of FGP structures can be explored in previous studies [35,36,37,38,39,40,41,42,43,44]. Allahkarami et al. [45] analysed the dynamic stability of bi-FGP cylindrical shells resting on an elastic foundation employing a GDQ. Kiarasi et al. [46] conducted a three-dimensional buckling analysis of FGP plates by a combination of GDQ and finite-element method (FEM). Merdaci et al. [47] analysed the higher-order free vibration of FGP plates using Navier’s solution. Sobhani et al. [48] investigated the free vibration of porous graphene oxide powder nano-composites assembled paraboloidal–cylindrical shells using GDQ based on the first shear deformation hypothesis theory.

The sandwich structure basically contains two thin layer skins attached to a light thick core. The advanced properties of this structure are outstanding flexural stiffness, low mass density, noise rejection, and good thermal insulation. However, it is very susceptible to failure due to stress concentration. Some research on the mechanical behaviour of sandwich structures can consist of works as follows: Zenkour et al. [49,50,51,52,53,54,55,56,57,58] used analytical solutions with various plate theories to analyze static bending [49,50,51,52,53,54], vibration and buckling [55,56], and the effect of porosity [57,58]. Thai et al. [59] used a novel first order shear deformation theory to study bending, buckling, and free vibration problems. Houari et al. [60] employed a refined-higher order shear deformation theory to study thermal bending; Nguyen et al. [61,62] extended the smoothed-FEM to examine the bending and free vibration of sandwich plates. Moreover, Li et al. [63,64] employed the Navier approach, Tounsi et al. [65] used a refined plate theory, and Tlidji et al. [66] used an exact solution to examine the mechanical behaviour of FG sandwich plates. Based on quasi-3D theory, Zaoui et al. [67] used Navier solutions to study the vibration of FG sandwich plates supported by the foundation. Neves et al. [68] employed a meshless technique to examine the mechanical characteristics. Farzam-Rad et al. [69] applied an isogeometric analysis to analyze the bending and free vibration. Moreover, Vafakhah and Neya [70] introduced an accurate solution for the static bending problems of FG thick plates.

It can be seen that using the Q4 element to analyze structures yields quick results when done on a computer. However, using only five DOFs per node based on the Lagrange interpolations with a shear correction factor still does not satisfy the stress-free condition at the upper and lower boundary conditions (BCs). Therefore, the main objective of the article is to propose a new finite-element procedure using the Q4 element with 28 DOFs based on the combination of the Lagrange interpolations and the Hermit interpolations to establish the governing equation of FGP sandwich plates. Then, a few examples are conducted to confirm the effectiveness of the proposed element. Moreover, the numerical and graphical results also help present the effect of input parameters on the dynamic response of FGP sandwich plates resting on Kerr foundation (KF).

2 Theoretical formulation

In this section, the FGP sandwich plate model resting on KF is established. The compositional relations of the plate are inferred based on Reddy’s third-order shear deformation plate theory (TSDT). The motion equation of the plate is derived from the Hamiltonian principle, a newly proposed finite-element procedure using the Q4 element based on the combination of Lagrange and Hermite functions.

2.1 FGP sandwich plate and KF models

In this work, the authors consider the FGP sandwich plate resting on KF with geometrical and material parameters, as displayed in Figure 1.

Figure 1

3D view of FGP sandwich plate located on KF.

The sandwich plate comprises a homogeneous metal bottom layer, a completely ceramic top layer, and an FGP core. The ceramic volume fraction V c n and the metal volume fraction V m n ( n = b , c , t ) in the layers are determined by ref. [57]

(1) V c b ( z ) = 0 , z ∈ − h 2 ; h 1 , V c c ( z ) = z − h 1 h 2 − h 1 k , z ∈ [ h 1 ; h 2 ] ; and V m n ( z ) = 1 − V c n ( z ) , V c t ( z ) = 1 , ∈ h 2 ; h 2 ,

where k is the power index.

The FGP material with sinusoidal porosity distribution is expressed as follows [71]:

(2) P c ( z ) = [ P m c + ( P c c − P m c ) V c c ] ( 1 − Φ ( z ) ) ,

in which P denotes the material properties, i.e. elastic modulus E , mass density ρ , and Poisson’s ratio ν ; subscripts below m and c denote the metal and ceramic components, respectively; Φ ( z ) = Ω cos π z h is the sinusoidal porosity distribution function, in which Ω is the porosity factor. Normally, Ω gets values from 0 to 0.5.

In Figure 2, the effective elastic modulus along thickness via different values of k and thickness-ratio h _b-h _e-h _t (namely, scheme) is displayed. Herein, Scheme 0-1-0 corresponds to an isotropic FGP plate.

$Figure 2 Elastic modulus of FGP (Al/Al2O3) sandwich plates through the thickness with Ω = 0.1 \Omega =0.1 . (a) Scheme 0-1-0. (b) Scheme 1-1-1. (c) Scheme 1-1-2. (d) Scheme 2-2-1.$

Figure 2

Elastic modulus of FGP (Al/Al₂O₃) sandwich plates through the thickness with Ω = 0.1 . (a) Scheme 0-1-0. (b) Scheme 1-1-1. (c) Scheme 1-1-2. (d) Scheme 2-2-1.

KF includes an upper layer k _U, a shear layer k _S, and a lower layer k _L can be determined as follows [72]:

(3) R = k L k U k L + k U w ( x , y ) − k S k U k S + k U ∂ w ∂ x 2 + ∂ w ∂ y 2 .

The KF becomes to Pasternak foundation when layer stiffness k _U takes infinity and can be determined by

(4) R = k L w ( x , y ) − k S ∂ w ∂ x 2 + ∂ w ∂ y 2 ,

whereas without an elastic foundation k L = k S = 0 .

2.2 Reddy’s TSDT

In this study, the authors use Reddy’s TSDT because of its simplicity and proven effectiveness in analyzing moderately thick and thin plates. On the other hand, omitting the out-of-plane strains leads to a simpler mathematical procedure with fewer degrees of freedom (DOFs) while maintaining accuracy for the vibration problem.

Applying Reddy’s TSDT [73], the displacement field of sandwich plates is

(5) u ( x , y , z ) = u 0 ( x , y ) − z ( w 0 , x + φ x ) + 4 z 3 3 h 2 w 0 , x , v ( x , y , z ) = v 0 ( x , y ) − z ( w 0 , y + φ y ) + 4 z 3 3 h 2 w 0 , y , w ( x , y , z ) = w 0 ( x , y ) ,

where u 0 , v 0 , w 0 , φ x , and φ y are displacement variables.

Now, the strain field can be inferred by

(6) ε = ε m + z κ 1 + z 3 κ 2 ,

whereThe membrane strain ε m is

(7) ε m = u 0 , x v 0 , y u 0 , y + v 0 , x .

The bending strains κ 1 and κ 2 are

(8) κ 1 = − φ x , x + w x x φ y , y + w y y 2 w x y + φ x , y + φ y , x ,

(9) κ 2 = 4 3 h 2 φ x , x φ y , y φ x , y + φ y , x .

The transverse shear strain γ is

(10) γ = γ 0 + z 2 γ 1 ,

in which

(11) γ 0 = − φ y φ x ; γ 1 = 4 h 2 φ y φ x .

The stress resultants are defined by

(12) N M P T = D m ε m κ 1 κ 2 T ; Q R T = D s γ 0 γ 1 T ,

where

(13) D m = A B B b B F F b B b F b ℍ ; D s = A s B s B s F s ,

in which

(14) ( A , B , F , B b , F b , ℍ ) = ∫ − h 2 h 2 E ( z ) 1 − ν ( z ) 2 1 ν ( z ) 0 ν ( z ) 1 0 0 0 1 2 ( 1 − ν ( z ) ) × ( 1 , z , z 2 , z 3 , z 4 , z 6 ) d z ,

and

(15) ( A s , B s , F s ) = ∫ − h 2 h 2 E ( z ) 2 ( 1 + ν ( z ) ) 1 0 0 1 ( 1 , z 2 , z 4 ) d z .

The constituent material matrices are determined by using the Gaussian quadrature integration method with four interpolating points.

Now, Eq. (11) can be abbreviated in matrix form as follows:

(16) N M P Q R = A B B b 0 0 B F F b 0 0 B b F b ℍ 0 0 0 0 0 A s B s 0 0 0 B s F s ε m κ 1 κ 2 γ 0 γ 1 .

2.3 The motion equations of sandwich plates

The motion equation of sandwich plates following Hamilton’s principle as follows [74]:

(17) ∫ 0 T ( δ U + δ U f + δ W − δ T ) d t = 0 ,

in which the virtual strain energy is

(18) δ U = ∫ S ( ( δ ε m ) T N + ( δ κ 1 ) T M + ( δ κ 2 ) T P + ( δ γ 0 ) T Q + ( δ γ 1 ) T R ) d S .

The virtual deformed elastic foundation is

(19) δ U f = ∫ S R δ w d S .

The virtual work done by the external force is

(20) δ W = − ∫ S p δ w d S ,

where p is the load vector acting on the plate in the z-direction.

The virtual kinetic energy is

(21) δ T = ∫ S ρ ( z ) ( u ̇ δ u ̇ + v ̇ δ v ̇ + w ̇ δ w ̇ ) d Ω = ∫ S δ q ̇ T ℳ q ̇ d S ,

where

(22) q = u 0 v 0 w θ x θ y φ x φ y ,

and

(23) ℳ = ρ ( z ) L m T L m ,

in which ρ ( z ) is the mass density, and

(24) L m = 1 0 0 − z 0 4 z 3 3 h 2 0 0 1 0 0 − z 0 4 z 3 3 h 2 0 0 1 0 0 0 0 .

The weak form for the motion of sandwich plates is obtained by ref. [74]

(25) ∫ S ( ( δ ε ) T D m ε + ( δ γ ) T D s γ ) d S + ∫ S R δ w d S − ∫ S δ q T ℳ q ̈ d S = ∫ S p δ w d S .

2.4 Finite-element procedure

Using the four-node rectangular element with 28 DOFs, as demonstrated in Figure 3a, the nodal displacement vector is defined by

(26) q e = q 1 T q 2 T q 3 T q 4 T T ,

with q i ( i = 1 ÷ 4 ) being the node displacements expressed by

(27) q i = { u 0 i v 0 i w i θ x i θ y i φ x i φ y i } ,

where u 0 , v 0 , φ x , and φ y are interpolated by the Lagrange function (Appendix):

(28a) { u 0 v 0 φ x φ y } = ∑ i = 1 4 N i u 0 i … ∑ i = 1 4 N i v 0 i … ∑ i = 1 4 N i φ x i … ∑ i = 1 4 N i φ y i … T .

Figure 3

The four-node rectangular element: (a) physical coordinates of the element and (b) the natural coordinate.

In finite-element analysis, to satisfy the stress-free BC of the plate, the two derivatives of w in the x- and y-directions are assumed to be two DOF θ x and θ y and are approximated by the Hermite function (Appendix) as follows:

(28b) w = H 1 w 1 + H 2 w 1 , x + H 3 w 1 , y + ⋯ + H 10 w 4 + H 11 w 4 , x + H 12 w 4 , y ,

(28c) θ x = w x + φ x = ∂ ∂ x ( H 1 w 1 + H 2 w 1 , x + H 3 w 1 , y + ⋯ + H 10 w 4 + H 11 w 4 , x + H 12 w 4 , y ) + ∑ i = 1 4 N i φ x i ,

(28d) θ y = w y + φ y = ∂ ∂ y ( H 1 w 1 + H 2 w 1 , x + H 3 w 1 , y + ⋯ + H 10 w 4 + H 11 w 4 , x + H 12 w 4 , y ) + ∑ i = 1 4 N i φ y i .

Replacing Eq. (26) into Eq. (6), the strain vectors are re-written by

(29a) ε = B 1 B 2 B 3 q e ,

(29b) γ = B 4 B 5 q e ,

with

(30a) B 1 = ∑ i = 1 4 N i , x 0 0 0 0 0 0 0 N i , y 0 0 0 0 0 N i , y N i , x 0 0 0 0 0 ,

(30b) B 2 = − ∑ i = 1 4 0 0 H ( 3 i − 2 ) , x x H ( 3 i − 1 ) , x x H ( 3 i ) , x x 0 N i , x 0 0 H ( 3 i − 2 ) , y y H ( 3 i − 1 ) , y y H ( 3 i ) , y y N i , y 0 0 0 2 H ( 3 i − 2 ) , x y 2 H ( 3 i − 1 ) , x y 2 H ( 3 i ) , x y N i , x N i , y ,

(30c) B 3 = 4 3 h 2 ∑ i = 1 4 0 0 0 0 0 N i , x 0 0 0 0 0 0 0 N i , y 0 0 0 0 0 N i , x N i , y ,

and

(31a) B 4 = − ∑ i = 1 4 0 0 0 0 0 0 N i 0 0 0 0 0 N i 0 ,

(31b) B 5 = 4 h 2 ∑ i = 1 4 0 0 0 0 0 0 N i 0 0 0 0 0 N i 0 .

Next, substituting Eq. (29) into Eq. (25), the governing equation of the sandwich plate element is defined by

(32) M e q e ̈ + ( K e + K e f ) q e = F e ,

where the element stiffness matrix K e is

(33) K e = ∫ S B 1 B 2 B 3 B 4 B 5 T A B B b 0 B F F b 0 B b F b ℍ 0 0 0 A s B s 0 0 B s F s B 1 B 2 B 3 B 4 B 5 d S .

The foundation element stiffness matrix K e f is

(34) K e f = k 1 ∫ S N w T N w d S + k 2 ∫ S ∂ N w ∂ x T ∂ N w ∂ x + ∂ N w ∂ y T ∂ N w ∂ y d S ,

where N w is defined by

(35) N w = ∑ i = 1 4 0 0 H 3 i − 2 0 0 0 0 .

The element mass matrix M e is

(36) M e = ∫ S N T ℳ N d S ,

where N is the shape function matrix determined by

(37) N = ∑ i = 1 4 N i 0 0 0 0 0 0 0 N i 0 0 0 0 0 0 0 H ( 3 i − 2 ) H ( 3 i − 1 ) H 3 i 0 0 0 0 H ( 3 i − 2 ) , x H ( 3 i − 1 ) , x H 3 i , x 0 N i 0 0 H ( 3 i − 2 ) , y H ( 3 i − 1 ) , y H 3 i , y N i 0 0 0 0 0 0 0 N i 0 0 0 0 0 N i 0 .

The element load vector F e is

(38) F e = ∫ S N T p dS ,

where p = 0 0 q 0 0 0 0 0 T , with q 0 being the uniform distribution load.

Now, the motion equation of the sandwich plate is

(39) M q ̈ + K q = F ,

where K = ∑ nel K e , M = ∑ nel M e , and F = ∑ nel F e is the global stiffness matrix, the global mass matrix, and the global load vector of the sandwich plate, respectively, and symbol “nel” represents the number of discretized elements of the sandwich plate.

When the force vector is a time function F = F ( t ) , the motion equation of sandwich plates is now

(40) M q ̈ + K q = F ( t ) .

To solve Eq. (40), we employ the Newmark direct integral [78,79].

In this study, the types of BCs are given by

Clamped (C): u 0 = v 0 = w = θ x = θ y = φ x = φ y = 0 for all edges;
Simply supported (S): u 0 = w = θ x = φ x = 0 at y = 0 , y = b or v 0 = w = θ y = φ y = 0 at x = 0 , x = a ;
Free (F): All DOFs are different from zero.

3 Numerical results and discussions

This section performs the following main tasks: (i) verify the accuracy of the proposed element; (ii) provide new results in the vibration of FGP sandwich plates resting on KF.

In order to facilitate numerical exploration, the dimensionless parameters are defined by

(41) ω ⁎ = a π 2 ρ c h D c ; D c = E c h 3 12 ( 1 − ν c 2 ) ; K L = k L a 4 D c ; K S = k S a 2 D c ; K U = k U a 4 D c .

3.1 Verification studies

First, the first dimensionless frequencies ω ¯ = ω h ρ m / E m of SSSS FGM (Al/Al₂O₃) thick square plate ( a / h = 5 ) with different mesh sizes are listed in Table 1. FGM with the rule of mixture and material properties is provided in Table 2. From obtained results, it can be found that the obtained frequencies change little with a mesh size larger than 16 × 16 and agree well with those of existing work using the quadrature element [77]. Therefore, the authors use mesh size 16 × 16 for the next examples.

Table 1

First dimensionless frequencies of SSSS FGM (Al/Al₂O₃) square plates

Mesh size	K
Mesh size	0	0.2	0.5	1	2	5	10
8 × 8	0.38414	0.36016	0.33199	0.30132	0.27282	0.25072	0.23936
10 × 10	0.38267	0.35886	0.33085	0.30032	0.27186	0.24959	0.23816
12 × 12	0.38154	0.35784	0.32996	0.29955	0.27113	0.24874	0.23726
14 × 14	0.38064	0.35704	0.32926	0.29893	0.27055	0.24808	0.23657
16 × 16	0.37993	0.35640	0.32870	0.29844	0.27009	0.24755	0.23601
18 × 18	0.37935	0.35587	0.32823	0.29804	0.26972	0.24713	0.23556
20 × 20	0.37886	0.35544	0.32785	0.29770	0.26940	0.24677	0.23519
Wang et al. [77]	0.38323	0.3543	0.33046	0.29949	0.27070	0.24934	0.23920

Table 2

Mechanical properties of FGP sandwich plates

Materials component	Elastic modulus (GPa)	Mass densities (kg/m³)	Poisson’s ratio
Al₂O₃ (ceramic)	E c = 380	ρ c = 3 , 800	ν c = 0.3
Al (metal)	E m = 70	ρ m = 2 , 707	ν m = 0.3
Si₃N₄ (ceramic)	E c = 348.43	ρ c = 2 , 370	ν c = 0.3
SUS304 (metal)	E m = 201.04	ρ m = 8 , 166	ν m = 0.3

Second, the first dimensionless frequencies ω ⁎ = ω h ρ / E of isotropic square plates with various elastic parameters are presented in Table 3. The plate is made by Al with material properties, as listed in Table 2. It can be seen that the obtained results are in good agreement with those of Li et al. [75] (using simple quasi-3D theory) and Shahsavari et al. [76] (employing quasi-3D hyperbolic theory).

Table 3

First dimensionless frequencies of SSSS isotropic square plates resting on KF with K L = 100 is fixed

K U	K S	h / a	Method	ω ⁎
100	0	0.05	Li et al. [75]	0.0158
			Shahsavari et al. [76]	0.0157
			Present	0.0155
		0.1	Li et al. [75]	0.0616
			Shahsavari et al. [76]	0.0615
			Present	0.0595
		0.15	Li et al. [75]	0.1337
			Shahsavari et al. [76]	0.1337
			Present	0.1273
		0.2	Li et al. [75]	0.2276
			Shahsavari et al. [76]	0.2278
			Present	0.2143
100	100	0.05	Li et al. [75]	0.0284
			Shahsavari et al. [76]	0.0285
			Present	0.0282
		0.1	Li et al. [75]	0.1121
			Shahsavari et al. [76]	0.1125
			Present	0.1112
		0.15	Li et al. [75]	0.2468
			Shahsavari et al. [76]	0.2487
			Present	0.2454
		0.2	Li et al. [75]	0.4268
			Shahsavari et al. [76]	0.4332
			Present	0.3903

Finally, the present element with a mesh size of 16 × 16 is deployed to analyze the dynamic response of the SSSS FGM (SUS304/Si₃N₄) square plate. In this time history analysis, the FGM plate under a suddenly uniform load with q 0 ( x , y ) = 50 MPa . The geometrical dimensions: a = b = 0.2 m , h = 0.025 m , and the time step is taken as Δ t = 0.2 μs . Observing Figure 4, the obtained displacement response is in good agreement with those of Abuteir et al. [80] using a curved eight-node degenerated shell element in both value and shape. From the above examples, it can be concluded that our formula and program are correct and reliable.

Figure 4

The deflection response of SSSS FGM (SUS304/Si₃N₄) square plate center.

3.2 Free vibration problem

First, Figure 5 shows the first six mode shapes of the SSSS FGP square sandwich plate (Scheme 1-2-1) with a thickness of h = a / 50 . Accordingly, the material properties are provided in Table 2 with a power-law index of k = 1 , a porosity factor of Ω = 0.1 , and foundation parameters K L = K U = 100 , K S = 10 . From this figure, it can be concluded that the second and the third mode shapes, the fifth and the sixth mode shapes are the same as each other, and differ only in the view direction. It is suitable for square sandwich plates under completely simple support.

Figure 5

Mode shapes of the SSSS FGP sandwich plate.

Second, the simultaneous influence of two material parameters ( k , Ω ) on dimensionless frequencies of FGP square sandwich plates (Scheme 1-8-1) with a thickness of h = a / 25 is demonstrated in Figure 6. In this study, foundation parameters get values K L = K U = 50 , K S = 10 . Observing this figure, we see that the first dimensionless frequency decreases rapidly as k rises from 0 to 2, little changes with k bigger than 2, and peaks in the case of ceramic plates (k = 0). This is perfectly reasonable since the increase of k reduces the sandwich plate stiffness. More interestingly, the first dimensionless frequency of FGP square sandwich plates increases when the porosity factor Ω increases for all BCs. This can be explained by the fact that although Ω reduces the stiffness but also reduces the mass of the plate, this simultaneous impact is the cause of the increased frequency. Moreover, it is easy to see that the dimensionless frequency of CCCC FGP sandwich plates is the largest because the CCCC FGP sandwich plate is the stiffest of all BCs.

$Figure 6 Effect of material parameters ( k , Ω ) (k,\text{Ω})\hspace{.25em} on the dimensionless frequency of FGP square sandwich plates. (a) The SSSS FGP square sandwich plates. (b) The SCCS FGP square sandwich plates. (c) The SCSC FGP square sandwich plates. (d) The CCCC FGP square sandwich plates.$

Figure 6

Effect of material parameters ( k , Ω ) on the dimensionless frequency of FGP square sandwich plates. (a) The SSSS FGP square sandwich plates. (b) The SCCS FGP square sandwich plates. (c) The SCSC FGP square sandwich plates. (d) The CCCC FGP square sandwich plates.

Next, Table 4 provides more than the first dimensionless frequencies of CSSC FGP square sandwich plates with the thickness h = a / 75 and foundation parameters get values K L = K U = 75 , K S = 20 via the material parameters and schemes. Note that Schemes 2-2-1 and 1-4-2 are the case of an asymmetrical sandwich plate. Furthermore, Table 5 presents the first six dimensionless frequencies of FGP square sandwich plates via different schemes with material properties k = 5 ; Ω = 0.4 . From these results, it can be concluded that increasing the FGP core size results in an increase in the dimensionless frequencies.

Table 4

Dimensionless frequency of the CSSC FGP square sandwich plates via the material parameters and schemes

Schemes	Ω	k
Schemes	Ω	0	0.5	1.5	2.5	4.5	10.5
0-1-0	0	3.3709	3.1098	2.9262	2.8901	2.8871	2.8749
	0.1	3.4417	3.1783	2.9949	2.9602	2.9592	2.9502
	0.2	3.5211	3.2549	3.0712	3.0381	3.0392	3.0337
	0.3	3.6107	3.3412	3.1567	3.1253	3.1286	3.1269
	0.4	3.7129	3.4393	3.2533	3.2235	3.2293	3.2317
	0.5	3.8304	3.5518	3.3633	3.3352	3.3436	3.3509
1-1-1	0	2.8541	2.8401	2.8292	2.8293	2.8343	2.8434
	0.1	2.8886	2.874	2.8629	2.8628	2.8674	2.8758
	0.2	2.9242	2.9091	2.8979	2.8976	2.9017	2.9095
	0.3	2.9609	2.9455	2.9343	2.9338	2.9374	2.9445
	0.4	2.9987	2.9831	2.9721	2.9714	2.9746	2.9809
	0.5	3.0375	3.022	3.0114	3.0107	3.0134	3.0189
1-4-1	0	3.0771	2.9486	2.8771	2.87	2.8816	2.9054
	0.1	3.1392	3.0113	2.9411	2.9345	2.9462	2.9703
	0.2	3.2076	3.08	3.0109	3.0048	3.0167	3.0409
	0.3	3.2834	3.1556	3.0875	3.0819	3.0938	3.1181
	0.4	3.3678	3.2392	3.1721	3.167	3.1788	3.2029
	0.5	3.4624	3.3324	3.2661	3.2613	3.273	3.2968
2-2-1	0	2.8178	2.8341	2.8547	2.869	2.8859	2.9036
	0.1	2.8632	2.8788	2.8988	2.9125	2.9288	2.946
	0.2	2.9111	2.9259	2.9451	2.9583	2.9739	2.9904
	0.3	2.9619	2.9757	2.9939	3.0065	3.0213	3.0371
	0.4	3.0157	3.0284	3.0455	3.0572	3.0711	3.0861
	0.5	3.0729	3.0843	3.1	3.1108	3.1237	3.1377

Table 5

Dimensionless frequencies of FGP square sandwich plates via different schemes

Dimensionless frequency	ω 1 ⁎	ω 2 ⁎	ω 3 ⁎	ω 4 ⁎	ω 5 ⁎	ω 6 ⁎
Scheme	SSSS ( h = a /10)
1-1-1	2.5289	4.4657	4.4949	4.9819	4.9865	6.1504
1-2-1	2.6037	4.5552	4.5885	4.8291	4.8358	6.2374
1-4-1	2.7087	4.6283	4.6331	4.6786	4.7299	6.3591
1-6-1	2.7398	4.5421	4.5424	4.7108	4.7588	6.3863
1-4-2	2.6234	4.5921	4.6229	4.9739	4.9798	6.2950
Scheme	CCCC ( h = a /100)
1-1-1	3.4543	6.2351	6.2359	8.6876	10.4161	10.448
1-2-1	3.5525	6.3983	6.3993	8.9051	10.6691	10.7014
1-4-1	3.6733	6.5839	6.5852	9.1412	10.9343	10.9666
1-6-1	3.7007	6.6167	6.6181	9.1755	10.9661	10.9981
1-4-2	3.5551	6.3876	6.3885	8.88	10.6347	10.6662

Finally, to confirm the effectiveness of the finite-element method compared with the analytical method, especially when the geometrical model becomes complex (un-symmetry), the authors consider the FGP L-shape sandwich plate with geometrical parameters, as presented in Figure 7. Figure 8 plots the first six mode shapes of the fully clamped FGP L-shape sandwich plates (Scheme 1-2-1) with the thickness h = a / 50 and the rest of the parameters are k = 1 ; Ω = 0.5 , K L = K U = K S = 50 .

Figure 7

The schematic diagram for the FGP L-shape sandwich plate.

Figure 8

Mode shapes of the fully clamped FGP L-shape sandwich plate.

3.3 Forced vibration problem

In this section, the authors examine the dynamic response of FGP sandwich plates under the uniformly distributed transverse load that vary with time as follows:

(42) q = q 0 F ( t ) ,

where

(43) F ( t ) = Step loading 1 , 0 ≤ t ≤ t s 0 , t ≥ t s Triangular loading 1 − t t s , 0 ≤ t ≤ t s 0 , t ≥ t s Sine loading sin π t t s , 0 ≤ t ≤ t s 0 , t ≥ t s Blast loading { e − γ t ,

in which t s = 0.005 s , γ = 330 s − 1 , q 0 = 3.448 MPa .

First, Figure 9 demonstrates the influence of the parameter k on the forced vibration of the SSSS FGP square sandwich plate with the rest of the parameters: h = a / 20 ; Ω = 0.3 , and foundation parameters get values K L = K U = 50 , K S = 25 . We can see that the deflection response of sandwich plates increases in accordance with the increase of the parameter k, because power-law index k makes the sandwich plate stiffness decrease.

Figure 9

The dynamic response of SSSS FGP square sandwich plates (Scheme 1-4-1) via the power-law index k. (a) The deflection response of the sandwich plate center under the step loading. (b) The deflection response of the sandwich plate center under the triangular. (c) The deflection response of the sandwich plate center under the sine loading. (d) The deflection response of the sandwich plate center under the blast loading.

Second, Figure 10 displays the effect of the parameter Ω on the dynamic response of CCCC FGP square sandwich plates with the rest of the parameters: h = a / 40 , k = 2 , and foundation parameters get values K L = K U = 80 , K S = 30 . It can be observed that the increase of the parameter Ω leads to decrease in sandwich plate stiffness; therefore, the deflection response of sandwich plates increases.

$Figure 10 The dynamic response of FGP square sandwich plates (Scheme 1-6-1) via porosity factor Ω {\Omega } . (a) The deflection response of the sandwich plate center under the step loading. (b) The deflection response of the sandwich plate center under the triangular. (c) The deflection response of the sandwich plate center under the sine loading. (d) The deflection response of the sandwich plate center under the blast loading.$

Figure 10

The dynamic response of FGP square sandwich plates (Scheme 1-6-1) via porosity factor Ω . (a) The deflection response of the sandwich plate center under the step loading. (b) The deflection response of the sandwich plate center under the triangular. (c) The deflection response of the sandwich plate center under the sine loading. (d) The deflection response of the sandwich plate center under the blast loading.

Finally, Figure 11 shows the effect of schemes on the dynamic response of FGP square sandwich plates with parameters such as the thickness h = a / 15 , the material parameters k = 0.5 ; Ω = 0.4 , and foundation parameters get values K L = K U = 60 , K S = 15 . It can be concluded that the scheme significantly affects the dynamic behaviour of the sandwich plate. In particular, the sandwich plate with a larger size FGP core (corresponding to Schemes 0-1-0, 1-8-1, 1-4-1, 1-2-1, 1-1-1) will make the sandwich plate “stiffer” thus reducing the deflection response.

Figure 11

The dynamic response of FGP square sandwich plates via schemes. (a) The deflection response of the sandwich plate center under the step loading. (b) The deflection response of the sandwich plate center under the triangular. (c) The deflection response of the sandwich plate center under the sine loading. (d) The deflection response of the sandwich plate center under the blast loading.

Note that, in Figures 9–11, the responses of the plate are separated into two phases. In the first phase, the plate is forced vibration due to being subjected to transverse loads, the deflection response of plates depends on the type of loading, i.e. step, triangular, sine, and blast loading. And in the second phase, the plate is free vibration, i.e. the vibration of the plate is harmonic.

4 Conclusions

This article has successfully developed a Q4 element with seven DOFs per node to investigate the dynamic response of FGP sandwich plates. Then, the authors also analyze and discuss the effects of geometrical parameters and material properties on the dynamic response of FGP sandwich plates. From the formula and obtained numerical results, it can be seen that using the Q4 element based on Lagrange interpolations and Hermit interpolations easily mesh the element even with complex geometric domains and avoids “the shear-locking” phenomenon as well as satisfying the zero shear stress boundary condition that appears in the classical Q4 element. Besides, using sandwich structures with an FGP core not only helps reduce the overall mass but also increases the flexural strength of the structure. The material properties such as power-law index k, the porosity factor Ω , and the thickness ratio between layers significantly affect the dynamic response of FGP sandwich plates. In general, the power-law index k and the porosity factor Ω reduce the sandwich plate stiffness, while the FGP core helps reduce the weight of the structure and improves the flexural strength of sandwich plates. Finally, the numerical results are expected to be very useful for the calculation and design of FGP sandwich panels in engineering practice.

Funding information: Authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.

Appendix

Lagrange function:

N 1 = 1 4 ( 1 − χ ) ( 1 − ξ ) ; N 2 = 1 4 ( 1 + χ ) ( 1 − ξ ) N 3 = 1 4 ( 1 + χ ) ( 1 + ξ ) ; N 4 = 1 4 ( 1 − χ ) ( 1 + ξ ) .

Hermit function:

H 1 = 1 8 ( 1 − χ ) ( 1 − ξ ) ( 2 − χ − ξ − χ 2 − ξ 2 ) ; H 2 = 1 8 ( 1 − χ ) ( 1 − ξ ) ( 1 − χ 2 ) H 3 = 1 8 ( 1 − χ ) ( 1 − ξ ) ( 1 − ξ 2 ) ; H 4 = 1 8 ( 1 + χ ) ( 1 − ξ ) ( 2 + χ − ξ − χ 2 − ξ 2 ) H 5 = − 1 8 ( 1 + χ ) ( 1 − ξ ) ( 1 − χ 2 ) ; H 6 = 1 8 ( 1 + χ ) ( 1 − ξ ) ( 1 − χ 2 ) H 7 = 1 8 ( 1 + χ ) ( 1 + ξ ) ( 2 + χ + ξ − χ 2 − ξ 2 ) ; H 8 = − 1 8 ( 1 + χ ) ( 1 + ξ ) ( 1 − χ 2 ) H 9 = − 1 8 ( 1 + χ ) ( 1 + ξ ) ( 1 − ξ 2 ) ; H 10 = 1 8 ( 1 − χ ) ( 1 + ξ ) ( 2 − χ + ξ − χ 2 − ξ 2 ) H 11 = 1 8 ( 1 − χ ) ( 1 + ξ ) ( 1 − χ 2 ) ; H 12 = − 1 8 ( 1 − χ ) ( 1 + ξ ) ( 1 − ξ 2 ) .

References

[1] Vo TP, Thai H-T, Nguyen T-K, Maheri A, Lee J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct. 2014;64:12–22.10.1016/j.engstruct.2014.01.029Search in Google Scholar

[2] Asghari M, Ahmadian M, Kahrobaiyan M, Rahaeifard M. On the size-dependent behavior of functionally graded micro-beams. Mater Des (1980-2015). 2010;31:2324–9.10.1016/j.matdes.2009.12.006Search in Google Scholar

[3] Trinh LC, Vo TP, Osofero AI, Lee J. Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach. Compos Struct. 2016;156:263–75.10.1016/j.compstruct.2015.11.010Search in Google Scholar

[4] Vo TP, Thai H-T, Nguyen T-K, Inam F. Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica. 2014;49:155–68.10.1007/s11012-013-9780-1Search in Google Scholar

[5] Reddy J. Analysis of functionally graded plates. Int J Numer Methods Eng. 2000;47:663–84.10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO;2-8Search in Google Scholar

[6] Luat DT, Van Thom D, Thanh TT, Van Minh P, Van Ke T, Van Vinh P. Mechanical analysis of bi-functionally graded sandwich nanobeams. Adv Nano Res. 2021;11:55–71.Search in Google Scholar

[7] Nguyen V-H, Nguyen T-K, Thai H-T, Vo TP. A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos B Eng. 2014;66:233–46.10.1016/j.compositesb.2014.05.012Search in Google Scholar

[8] Zenkour AM. Generalized shear deformation theory for bending analysis of functionally graded plates. Appl Math Modell. 2006;30:67–84.10.1016/j.apm.2005.03.009Search in Google Scholar

[9] Li S, Zheng S, Chen D. Porosity-dependent isogeometric analysis of bi-directional functionally graded plates. Thin-Walled Struct. 2020;156:106999.10.1016/j.tws.2020.106999Search in Google Scholar

[10] Tran TT, Le PB. Nonlocal dynamic response analysis of functionally graded porous L-shape nanoplates resting on elastic foundation using finite element formulation. Eng Comput. 2023;39:809–25.10.1007/s00366-022-01679-6Search in Google Scholar

[11] Zhao X, Lee Y, Liew KM. Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J Sound Vib. 2009;319:918–39.10.1016/j.jsv.2008.06.025Search in Google Scholar

[12] Nguyen-Xuan H, Tran LV, Nguyen-Thoi T, Vu-Do H. Analysis of functionally graded plates using an edge-based smoothed finite element method. Compos Struct. 2011;93:3019–39.10.1016/j.compstruct.2011.04.028Search in Google Scholar

[13] Nguyen H-N, Canh TN, Thanh TT, Ke TV, Phan V-D, Thom DV. Finite element modelling of a composite shell with shear connectors. Symmetry. 2019;11:527.10.3390/sym11040527Search in Google Scholar

[14] Karamanlı A, Vo TP. Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method. Compos B Eng. 2018;144:171–83.10.1016/j.compositesb.2018.02.030Search in Google Scholar

[15] Thai CH, Zenkour A, Wahab MA, Nguyen-Xuan H. A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis. Compos Struct. 2016;139:77–95.10.1016/j.compstruct.2015.11.066Search in Google Scholar

[16] Tran TT, Pham Q-H, Nguyen-Thoi T. Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method. Def Technol. 2021;17:971–86.10.1016/j.dt.2020.06.001Search in Google Scholar

[17] Tran TT, Nguyen P-C, Pham Q-H. Vibration analysis of FGM plates in thermal environment resting on elastic foundation using ES-MITC3 element and prediction of ANN. Case Stud. Therm Eng. 2021;24:100852.10.1016/j.csite.2021.100852Search in Google Scholar

[18] Tran TT, Pham Q-H, Nguyen-Thoi T. Dynamic analysis of functionally graded porous plates resting on elastic foundation taking into mass subjected to moving loads using an edge-based smoothed finite element method. Shock Vib. 2020;2020. 10.1155/2020/8853920.Search in Google Scholar

[19] Pham Q-H, Tran VK, Tran TT, Nguyen-Thoi T, Nguyen P-C. A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation. Case Stud Therm Eng. 2021;26:101170.10.1016/j.csite.2021.101170Search in Google Scholar

[20] Pham Q-H, Tran TT, Tran VK, Nguyen P-C, Nguyen-Thoi T. Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element. Alex Eng J. 2022;61:1788–802.10.1016/j.aej.2021.06.082Search in Google Scholar

[21] Pham Q-H, Tran TT, Tran VK, Nguyen P-C, Nguyen-Thoi T, Zenkour AM. Bending and hygro-thermo-mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation. Mech Adv Mater Struct. 2022;29:5885–905.10.1080/15376494.2021.1968549Search in Google Scholar

[22] Thanh TT, Van KeT, Hoa PQ, Trung NT. An edge-based smoothed finite element for buckling analysis of functionally graded material variable-thickness plates. Vietnam J Mech. 2021;43:221–35.10.15625/0866-7136/15503Search in Google Scholar

[23] Nguyen P-C, Pham QH, Tran TT, Nguyen-Thoi T. Effects of partially supported elastic foundation on free vibration of FGP plates using ES-MITC3 elements. Ain Shams Eng J. 2022;13:101615.10.1016/j.asej.2021.10.010Search in Google Scholar

[24] Tran TT, Pham Q-H, Nguyen-Thoi T. An edge-based smoothed finite element for free vibration analysis of functionally graded porous (FGP) plates on elastic foundation taking into mass (EFTIM). Math Probl Eng. 2020;2020:1–17.10.1155/2020/8278743Search in Google Scholar

[25] Sobhy M. A comprehensive study on FGM nanoplates embedded in an elastic medium. Compos Struct. 2015;134:966–80.10.1016/j.compstruct.2015.08.102Search in Google Scholar

[26] Tornabene F. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng. 2009;198:2911–35.10.1016/j.cma.2009.04.011Search in Google Scholar

[27] Mantari J. Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells. Compos B Eng. 2015;83:142–52.10.1016/j.compositesb.2015.08.048Search in Google Scholar

[28] Torabi J, Kiani Y, Eslami M. Linear thermal buckling analysis of truncated hybrid FGM conical shells. Compos B Eng. 2013;50:265–72.10.1016/j.compositesb.2013.02.025Search in Google Scholar

[29] Murin J, Aminbaghai M, Hrabovsky J, Gogola R, Kugler S. Beam finite element for modal analysis of FGM structures. Eng Struct. 2016;121:1–18.10.1016/j.engstruct.2016.04.042Search in Google Scholar

[30] Burzyński S, Chróścielewski J, Daszkiewicz K, Witkowski W. Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory. Compos B Eng. 2016;107:203–13.10.1016/j.compositesb.2016.09.015Search in Google Scholar

[31] Brischetto S, Tornabene F, Fantuzzi N, Viola E. 3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders. Meccanica. 2016;51:2059–98.10.1007/s11012-016-0361-ySearch in Google Scholar

[32] Tornabene F, Viscoti M, Dimitri R. Static analysis of anisotropic doubly-curved shells with arbitrary geometry and variable thickness resting on a Winkler-Pasternak support and subjected to general loads. Eng Anal Bound Elem. 2022;140:618–73.10.1016/j.enganabound.2022.02.021Search in Google Scholar

[33] Viola E, Tornabene F, Fantuzzi N. Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape. Compos Struct. 2013;106:815–34.10.1016/j.compstruct.2013.07.034Search in Google Scholar

[34] Tornabene F, Liverani A, Caligiana G. General anisotropic doubly-curved shell theory: a differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian. J Sound Vib. 2012;331:4848–69.10.1016/j.jsv.2012.05.036Search in Google Scholar

[35] Rezaei A, Saidi A. Application of carrera unified formulation to study the effect of porosity on natural frequencies of thick porous–cellular plates. Compos B Eng. 2016;91:361–70.10.1016/j.compositesb.2015.12.050Search in Google Scholar

[36] Rezaei A, Saidi A. Exact solution for free vibration of thick rectangular plates made of porous materials. Compos Struct. 2015;134:1051–60.10.1016/j.compstruct.2015.08.125Search in Google Scholar

[37] Zhao J, Xie F, Wang A, Shuai C, Tang J, Wang Q. A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions. Compos B Eng. 2019;156:406–24.10.1016/j.compositesb.2018.08.115Search in Google Scholar

[38] Zhao J, Xie F, Wang A, Shuai C, Tang J, Wang Q. Vibration behavior of the functionally graded porous (FGP) doubly-curved panels and shells of revolution by using a semi-analytical method. Compos B Eng. 2019;157:219–38.10.1016/j.compositesb.2018.08.087Search in Google Scholar

[39] Li Q, Wu D, Chen X, Liu L, Yu Y, Gao W. Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler–Pasternak elastic foundation. Int J Mech Sci. 2018;148:596–610.10.1016/j.ijmecsci.2018.09.020Search in Google Scholar

[40] Sahmani S, Aghdam MM, Rabczuk T. Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Compos Struct. 2018;198:51–62.10.1016/j.compstruct.2018.05.031Search in Google Scholar

[41] Guo H, Zhuang X, Chen P, Alajlan N, Rabczuk T. Stochastic deep collocation method based on neural architecture search and transfer learning for heterogeneous porous media. Eng Comput. 2022;38:5173–98.10.1007/s00366-021-01586-2Search in Google Scholar

[42] Guo H, Zhuang X, Chen P, Alajlan N, Rabczuk T. Analysis of three-dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis. Eng Comput. 2022;38:5423–44.10.1007/s00366-022-01633-6Search in Google Scholar

[43] Guo H, Zheng H, Zhuang X. Numerical manifold method for vibration analysis of Kirchhoff’s plates of arbitrary geometry. Appl Math Modell. 2019;66:695–727.10.1016/j.apm.2018.10.006Search in Google Scholar

[44] Guo H, Zheng H. The linear analysis of thin shell problems using the numerical manifold method. Thin-Walled Struct. 2018;124:366–83.10.1016/j.tws.2017.12.027Search in Google Scholar

[45] Allahkarami F, Tohidi H, Dimitri R, Tornabene F. Dynamic stability of bi-directional functionally graded porous cylindrical shells embedded in an elastic foundation. Appl Sci. 2020;10:1345.10.3390/app10041345Search in Google Scholar

[46] Kiarasi F, Babaei M, Asemi K, Dimitri R, Tornabene F. Three-dimensional buckling analysis of functionally graded saturated porous rectangular plates under combined loading conditions. Appl Sci. 2021;11:10434.10.3390/app112110434Search in Google Scholar

[47] Merdaci S, Adda HM, Hakima B, Dimitri R, Tornabene F. Higher-order free vibration analysis of porous functionally graded plates. J Compos Sci. 2021;5:305.10.3390/jcs5110305Search in Google Scholar

[48] Sobhani E, Masoodi AR, Dimitri R, Tornabene F. Free vibration of porous graphene oxide powder nano-composites assembled paraboloidal-cylindrical shells. Compos Struct. 2023;304:116431.10.1016/j.compstruct.2022.116431Search in Google Scholar

[49] Zenkour A. A comprehensive analysis of functionally graded sandwich plates: Part 1-Deflection and stresses. Int J Solids Struct. 2005;42:5224–42.10.1016/j.ijsolstr.2005.02.015Search in Google Scholar

[50] Zenkour AM. The effect of transverse shear and normal deformations on the thermomechanical bending of functionally graded sandwich plates. Int J Appl Mech. 2009;1:667–707.10.1142/S1758825109000368Search in Google Scholar

[51] Zenkour A, Alghamdi N. Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads. Mech Adv Mater Struct. 2010;17:419–32.10.1080/15376494.2010.483323Search in Google Scholar

[52] Zenkour A, Sobhy M. Thermal buckling of various types of FGM sandwich plates. Compos Struct. 2010;93:93–102.10.1016/j.compstruct.2010.06.012Search in Google Scholar

[53] Zenkour A, Alghamdi N. Thermomechanical bending response of functionally graded nonsymmetric sandwich plates. J Sandw Struct Mater. 2010;12:7–46.10.1177/1099636209102264Search in Google Scholar

[54] Zenkour AM. Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory. J Sandw Struct Mater. 2013;15:629–56.10.1177/1099636213498886Search in Google Scholar

[55] Daikh AA, Zenkour AM. Free vibration and buckling of porous power-law and sigmoid functionally graded sandwich plates using a simple higher-order shear deformation theory. Mater Res Express. 2019;6:115707.10.1088/2053-1591/ab48a9Search in Google Scholar

[56] Zenkour A. A comprehensive analysis of functionally graded sandwich plates: Part 2—Buckling and free vibration. Int J Solids Struct. 2005;42:5243–58.10.1016/j.ijsolstr.2005.02.016Search in Google Scholar

[57] Daikh AA, Zenkour AM. Effect of porosity on the bending analysis of various functionally graded sandwich plates. Mater Res Express. 2019;6:065703.10.1088/2053-1591/ab0971Search in Google Scholar

[58] Zenkour AM. A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities. Compos Struct. 2018;201:38–48.10.1016/j.compstruct.2018.05.147Search in Google Scholar

[59] Thai H-T, Nguyen T-K, Vo TP, Lee J. Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. Eur J Mech-A/Solids. 2014;45:211–25.10.1016/j.euromechsol.2013.12.008Search in Google Scholar

[60] Houari MSA, Tounsi A, Bég OA. Thermoelastic bending analysis of functionally graded sandwich plates using a new higher order shear and normal deformation theory. Int J Mech Sci. 2013;76:102–11.10.1016/j.ijmecsci.2013.09.004Search in Google Scholar

[61] Nguyen T-K, Nguyen V-H, Chau-Dinh T, Vo TP, Nguyen-Xuan H. Static and vibration analysis of isotropic and functionally graded sandwich plates using an edge-based MITC3 finite elements. Compos B Eng. 2016;107:162–73.10.1016/j.compositesb.2016.09.058Search in Google Scholar

[62] Pham Q-H, Nguyen P-C, Tran TT, Nguyen-Thoi T. Free vibration analysis of nanoplates with auxetic honeycomb core using a new third-order finite element method and nonlocal elasticity theory. Eng Comput. 2023;39:233–51.10.1007/s00366-021-01531-3Search in Google Scholar

[63] Li D, Deng Z, Xiao H, Zhu L. Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores. Mech Adv Mater Struct. 2018;25:179–91.10.1080/15376494.2016.1255814Search in Google Scholar

[64] Li D, Deng Z, Xiao H, Jin P. Bending analysis of sandwich plates with different face sheet materials and functionally graded soft core. Thin-Walled Struct. 2018;122:8–16.10.1016/j.tws.2017.09.033Search in Google Scholar

[65] Tounsi A, Houari MSA, Benyoucef S. A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerosp Sci Technol. 2013;24:209–20.10.1016/j.ast.2011.11.009Search in Google Scholar

[66] Tlidji Y, Daouadji TH, Hadji L, Tounsi A, Bedia EAA. Elasticity solution for bending response of functionally graded sandwich plates under thermomechanical loading. J Therm Stresses. 2014;37:852–69.10.1080/01495739.2014.912917Search in Google Scholar

[67] Zaoui FZ, Ouinas D, Tounsi A. New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Compos B Eng. 2019;159:231–47.10.1016/j.compositesb.2018.09.051Search in Google Scholar

[68] Neves A, Ferreira A, Carrera E, Cinefra M, Roque C, Jorge R, et al. Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos B Eng. 2013;44:657–74.10.1016/j.compositesb.2012.01.089Search in Google Scholar

[69] Farzam-Rad SA, Hassani B, Karamodin A. Isogeometric analysis of functionally graded plates using a new quasi-3D shear deformation theory based on physical neutral surface. Compos B Eng. 2017;108:174–89.10.1016/j.compositesb.2016.09.029Search in Google Scholar

[70] Vafakhah Z, Neya BN. An exact three dimensional solution for bending of thick rectangular FGM plate. Compos B Eng. 2019;156:72–87.10.1016/j.compositesb.2018.08.036Search in Google Scholar

[71] Kim J, Żur KK. Reddy J. Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates. Compos Struct. 2019;209:879–88.10.1016/j.compstruct.2018.11.023Search in Google Scholar

[72] Keshtegar B, Motezaker M, Kolahchi R, Trung N-T. Wave propagation and vibration responses in porous smart nanocomposite sandwich beam resting on Kerr foundation considering structural damping. Thin-Walled Struct. 2020;154:106820.10.1016/j.tws.2020.106820Search in Google Scholar

[73] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech. 1984;51(4):745–52.10.1115/1.3167719Search in Google Scholar

[74] Reddy JN. Mechanics of laminated composite plates and shells. Theory and Analysis. Boca Raton (FL), USA: CRC Press; 2003.10.1201/b12409Search in Google Scholar

[75] Li M, Soares CG, Yan R. Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple quasi-3D HSDT. Compos Struct. 2021;264:113643.10.1016/j.compstruct.2021.113643Search in Google Scholar

[76] Shahsavari D, Shahsavari M, Li L, Karami B. A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp Sci Technol. 2018;72:134–49.10.1016/j.ast.2017.11.004Search in Google Scholar

[77] Wang X, Yuan Z, Jin C. 3D free vibration analysis of multi-directional FGM parallelepipeds using the quadrature element method. Appl Math Modell. 2019;68:383–404.10.1016/j.apm.2018.11.030Search in Google Scholar

[78] Pham Q-H, Nguyen P-C, Tran TT. Dynamic response of porous functionally graded sandwich nanoplates using nonlocal higher-order isogeometric analysis. Compos Struct. 2022;290:115565.10.1016/j.compstruct.2022.115565Search in Google Scholar

[79] Vasiraja N, Nagaraj P. The effect of material gradient on the static and dynamic response of layered functionally graded material plate using finite element method. Bull Pol Acad Sci: Tech Sci. 2019;67(4):827–38.10.24425/bpasts.2019.130191Search in Google Scholar

[80] Abuteir B, Harkati E, Boutagouga D, Mamouri S, Djeghaba K Thermo-mechanical nonlinear transient dynamic and Dynamic-Buckling analysis of functionally graded material shell structures using an implicit conservative/decaying time integration scheme. Mech Adv Mater Struct. 2022;29:5773–92.10.1080/15376494.2021.1964115Search in Google Scholar

Received: 2022-11-27

Revised: 2023-03-22

Accepted: 2023-04-26

Published Online: 2023-07-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cls-2022-0195

Keywords for this article

sandwich plates; Q4 element; TSDT; Kerr foundation; functionally graded porous

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