A novel finite element formulation for static bending analysis of functionally graded porous sandwich plates

Abstract

This article aims to propose a finite element formulation based on Quasi-3D theory for the static bending analysis of functionally graded porous (FGP) sandwich plates. The FGP sandwich plates consist of three layers including the bottom skin of homogeneous metal, the top skin of fully ceramic and the FGP core layer with uneven porosity distribution. A quadrilateral (Q4) element with nine degrees of freedom (DOFs) per node is derived and employed in analyzing the static bending response of the plate under uniform and/or sinusoidally distributed loads. The accuracy of the present finite element formulation is verified by comparing the obtained numerical results with the published results in the literature. Then, some numerical examples are performed to examine the effects of the parameters including power-law index k and porosity coefficient \xi on the static bending response of rectangular FGP sandwich plates. In addition, a problem with a complicated L-shape model is conducted to illustrate the superiority of the proposed finite element method.

Graphical abstract

Keywords

sandwich plates / functionally graded porous / static bending / Quasi-3D theory / Q4 element

1 Introduction

Functionally graded materials (FGMs) were first studied by Japanese scientists in 1984 at Sendai Institute. These materials are made of two or more component materials, usually ceramic and metal in which material properties vary smoothly and continuously from one surface to the other one. They have many outstanding features compared to traditional materials, such as high-temperature resistance, good corrosion resistance, etc. Therefore, FGMs have been applied in high-tech fields such as aerospace engineering, nuclear power, biology and so on. The outstanding characteristics of FGMs can eliminate stress concentration and delamination phenomena which often occur in laminated composite materials. The initial studies on behavior analysis of structures using FGMs can be summarized as those involved with beams [1–4], plates [5–25] and shells [26–28]. Functionally graded porous (FGP) material is a form of FGMs including many pores that develop in the process of making the material. This material possesses several outstanding mechanical properties such as low density, excellent energy-absorbing capability, great thermal resistant properties, etc. They have been widely applied in various fields including aerospace, automotive industry, and civil engineering. Some works on the mechanical behavior of structures using FGP are as follows. Rezaei and Saidi [29,30] used the exact solutions to analyze the vibration of porous-cellular plates. Zhao et al. [31,32] employed an improved Fourier method to examine the bending and dynamical response of FGP doubly-curved panels and shells. Li et al. [33] analyzed the free vibration and dynamic buckling behavior of sandwich FGP plate reinforced GPL for nonlinear problems, and Sahmani et al. [34] studied large-amplitude vibrations of FGP micro/nano-plates. Gou et al. [35,36] used the deep collocation method based on neuron architecture search-and-transfer learning for heterogeneous porous materials. Readers can see the application of artificial intelligence (AI) in scientific and mechanical problems for plate and shell structures in [37–39]. In general, almost the above works are based on analytical solutions, so they are limited in terms of models, loads and boundary conditions.

Sandwich structures are basically fabricated by attaching two thin skins to a thick lightweight core. Recently, sandwich structures have been used popularly in aerospace vehicles due to their outstanding bending rigidity, low mass density, good noise cancellation and insulation. However, they are highly susceptible to failure due to stress concentration at load areas, and due to geometrical and material discontinuities of sandwich plates. Some researchers have studied the static bending, buckling and free vibration behavior of the functionally graded material (FGM) plates, employing refined plate theories. Specifically, Zenkour et al. [40–49] uses analytical solutions with many improvements in plate theory to obtain accurate results in the problems of analyzing bending and stress [40,41], mechanical and thermal bending [42–45], free vibration and buckling [46,47], and the effect of porosity [48,49]. Thai et al. [50] used a new first-order shear deformation theory (FSDT), while Sid Ahmed Houari et al. [51] refined a higher-order shear deformation theory (HSDT). Nguyen et al. [52,53] developed a smoothed finite element method for the mixed interpolation of tensorial components technique of triangular elements (MITC3). Li et al. [54,55] employed the Navier approach, while Tounsi et al. [56] developed a refined trigonometric shear deformation theory. In addition, Tlidji et al. [57] used an analytical method to study behavior of FGM sandwich plates. Using Quasi-3D theory, some typical works can be mentioned as follows. Zaoui et al. [58] used two different shear deformation theories (2D, Quasi-3D) and Navier solutions to analyze the vibrations response of FGM plate on elastic foundation. Neves et al. [59] used a meshless technique based on sinusoidal Quasi-3D deformation theory to study the static bending, free vibration, and stability of FGM sandwich plate. Farzam-Rad et al. [60] used Quasi-3D shear deformation theory and isogeometric analysis for the static and free vibration analysis of FGM sandwich plate with attention to the physical neutral surface position. Vafakhah and Navayi Neya [61] proposed an accurate solution for the static bending problems of simple support rectangular FGM thick plates with 3D model.

Use of the quadrilateral (Q4) element is straightforward and achieves initial results in analyzing the mechanical behavior of structures. However, using only five degrees of freedom (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 boundaries of plates. In addition, the above-mentioned literature reveals that there is still a research gap for proposing new effective finite element formulations for analysis of FGP sandwich plates. The work is therefore conducted to fill in this research gap by proposing a new finite element formulation using the Q4 element with nine DOFs per node 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 examined to explore the accuracy and reliability of the proposed method. Moreover, the numerical and graphical results help to demonstrate the effect of geometrical parameters and material properties on the static bending of FGP sandwich plates.

This article is organized as follows. Section 1 presents a general introduction and literature review. Sections 2 and 3 provide the finite element formulations based on Quasi-3D theory for static bending analysis of FGP sandwich plates. Section 4 demonstrates the numerical results and discussion. Finally, Section 5 gives major conclusions.

2 The functionally graded porous sandwich plate

In this paper, the FGP sandwich plate as shown in Fig.1 is considered.

In Fig.1, a, b, and h are the length, width, and thickness dimensions of sandwich plates, respectively. The sandwich plate structure consists of the FGP core layer, a bottom layer of homogeneous metal, and a fully ceramic top layer. The ceramic volume fraction V_{\text{c}}^n(n=1,2,3) in the layers, i.e., the bottom layer, the core layer and the top layer are determined according to power-law function [48]:

(1)\{\begin{array}{ll}{V}_{\text{c}}^1(\mathit{\text{z}})=0,& \mathit{\text{z}}\in \left[{\displaystyle \frac{-h}{2}};h_1\right],\\ V_{\text{c}}^2(\mathit{\text{z}})={\left({\displaystyle \frac{\mathit{\text{z}}-h_1}{h_2-h_1}}\right)}^k,& \mathit{\text{z}}\in \left[h_1;h_2\right],\\ V_{\text{c}}^3(\mathit{\text{z}})=1,& \mathit{\text{z}}\in \left[h_2;{\displaystyle \frac{h}{2}}\right],\end{array}

where k represents the power-law index; z is the integral variable according to the plate thickness.

The metal volume fraction in the layers is inferred by the formula:

(2)V_{\text{m}}^n(\mathit{\text{z}})=1-V_{\text{c}}^n(\mathit{\text{z}}).

The core layer is made by FGP material with uneven porosity distribution expressed by the rule of the mixture as follows:

(3)P^2(\mathit{\text{z}})=P_{\text{m}}^2+\left(P_{\text{c}}^2-P_{\text{m}}^2\right)V_{\text{c}}^2-\frac{\xi }{2}\left(1-\frac{2|\mathit{\text{z}}|}{h}\right)\left(P_{\text{m}}^2+P_{\text{c}}^2\right) ,

where P represents the effective material properties such as elastic modulus E, mass density ρ, and Poisson’s ratio \nu; subscripts ‘m’ and ‘c’ denote the metal and ceramic constituents, respectively; \xi is the porosity coefficient.

Besides, Fig.2 plots the effective elastic modulus through thickness with different values of k and thickness ratio between the bottom layer, the core layer and the top layer (so-called scheme h_{\text{b}}\text{-} h_{\text{c}} \text{-}{h}_{\text{t}}) with mechanical properties as shown in Tab.1. Herein, scheme 0-1-0 corresponds to the case of isotropic FGP plates.

3 Theoretical formulation

3.1 The Quasi-3D theory for functionally graded porous sandwich plates

According to Quasi-3D theory, the displacement field of FGP sandwich plates can be expressed as follows [62]:

(4)\{\begin{array}{c}u\left(x,y,\mathit{\text{z}}\right)=u_0\left(x,y,0\right)-\mathit{\text{z}}{w}_{\text{b,}x}-f\left(\mathit{\text{z}}\right)w_{\text{s,}x},\\ v\left(x,y,\mathit{\text{z}}\right)=v_0\left(x,y,0\right)-\mathit{\text{z}}{w}_{\text{b,}y}-f\left(\mathit{\text{z}}\right)w_{\text{s,}y},\\ w\left(x,y,\mathit{\text{z}}\right)=w_{\text{b}}+w_{\text{s}}+g\left(\mathit{\text{z}}\right)w_{\mathit{\text{z}}},\end{array}

where u_0, v_0, w_{\mathrm{b}}, w_{\mathrm{s}}, and w_{\mathit{\text{z}}} are displacements variables in the neutral plane at the point \left(x,y,\mathit{\text{z}}\right). f(\mathit{\text{z}})={\displaystyle \frac{4{\mathit{\text{z}}}^3}{3h^2}}; g(\mathit{\text{z}})=1-{\displaystyle \frac{4{\mathit{\text{z}}}^2}{h^2}} are the continuous functions [63]. Note that f\left(\mathit{\text{z}}\right) is the continuous function that is established basing on three following rules: 1) the function must be continuous; 2) the deformed surface is curved; and 3) the stress-free condition at the upper and lower boundaries of the plate is satisfied.

Recently, many continuum functions f\left(\mathit{\text{z}}\right) have been created by scientists all over the world [64–68]. In this work, the Reddy’s continuous function is chosen due to its proven efficiency and simple polynomial. Besides, the current authors’ previous studies regarding the performance of these functions [69] illustrated that using different functions can affect the numerical performance. It should be noted that the new component g\left(\mathit{\text{z}}\right)w_{\mathit{\text{z}}} in Eq. (4) is added to investigate the thickness stretching effect on the static bending of FGP sandwich plates.

Then, the strain components are inferred from the displacement field as follows:

\begin{array}{rl}& \mathit{\epsilon }=\left\{\begin{array}{c}{\mathit{\epsilon }}_1\\ 0\end{array}\right\}+\left\{\begin{array}{c}0\\ {\mathit{\epsilon }}_2\end{array}\right\},\\ & {\mathit{\epsilon }}_1={\left\{\begin{array}{cccc}{\epsilon }_x& {\epsilon }_y& {\epsilon }_{\mathit{\text{z}}}& {\gamma }_{xy}\end{array}\right\}}^{\text{T}},\\ & {\mathit{\epsilon }}_2={\left\{\begin{array}{cc}{\gamma }_{y\mathit{\text{z}}}& {\gamma }_{x\mathit{\text{z}}}\end{array}\right\}}^{\text{T}},\end{array}

where

\begin{array}{rl}& {\mathit{\epsilon }}_1={\mathit{\epsilon }}_{\text{m}}+\mathit{\text{z}}{\mathit{\kappa }}_{\text{b}}+f(\mathit{\text{z}}){\mathit{\kappa }}_{\text{s}}+g_{,\mathit{\text{z}}}{\mathit{\epsilon }}_{\mathit{\text{z}}},\\ & {\mathit{\epsilon }}_2=g(\mathit{\text{z}}){\mathit{\epsilon }}_{\text{s1}}+g(\mathit{\text{z}}){\mathit{\epsilon }}_{\text{s2}},\\ & {\mathit{\epsilon }}_{\text{m}}={\left\{\begin{array}{cccc}{u}_{0,x}& v_{0,y}& 0& \left(u_{0,y}+v_{0,x}\right)\end{array}\right\}}^{\text{T}},\\ & {\mathit{\kappa }}_{\text{b}}=-{\left\{\begin{array}{cccc}{w}_{\text{b,}xx}& w_{\text{b,}yy}& 0& 2w_{\text{b,}xy}\end{array}\right\}}^{\text{T}},\\ & {\mathit{\kappa }}_{\text{s}}=-{\left\{\begin{array}{cccc}{w}_{\text{s,}xx}& w_{\text{s,}yy}& 0& 2w_{\text{s,}xy}\end{array}\right\}}^{\text{T}},\\ & {\mathit{\epsilon }}_{\mathit{\text{z}}}={\left\{\begin{array}{cccc}0& 0& w_{\mathit{\text{z}}}& 0\end{array}\right\}}^{\text{T}},\\ & {\mathit{\epsilon }}_{\text{s1}}={\left\{\begin{array}{cc}{w}_{\text{s,}y}& w_{\text{s,}x}\end{array}\right\}}^{\text{T}},\\ & {\mathit{\epsilon }}_{\text{s2}}={\left\{\begin{array}{cc}{w}_{\mathit{\text{z}},y}& w_{\mathit{\text{z}},x}\end{array}\right\}}^{\text{T}}.\end{array}

From Hooke’s law, the stress−strain relation is:

\begin{array}{rl}& \mathit{\sigma }=\left\{\begin{array}{c}{\mathit{\sigma }}_1\\ 0\end{array}\right\}+\left\{\begin{array}{c}0\\ {\mathit{\sigma }}_2\end{array}\right\},\\ & {\mathit{\sigma }}_1={\left\{\begin{array}{cccc}{\sigma }_x& {\sigma }_y& {\sigma }_{\mathit{\text{z}}}& {\tau }_{xy}\end{array}\right\}}^{\text{T}}={\mathit{D}}_1{\mathit{\epsilon }}_1,\\ & {\mathit{\sigma }}_2={\left\{\begin{array}{cc}{\tau }_{y\mathit{\text{z}}}& {\tau }_{x\mathit{\text{z}}}\end{array}\right\}}^{\text{T}}={\mathit{D}}_2{\mathit{\epsilon }}_2,\end{array}

in which

\begin{array}{rl}& {\mathit{D}}_1=E\left(\mathit{\text{z}}\right)\left[\begin{array}{cccc}{a}_1& a_2& a_2& 0\\ a_2& a_1& a_2& 0\\ a_2& a_2& a_1& 0\\ 0& 0& 0& a_3\end{array}\right],\\ & {\mathit{D}}_2=E\left(\mathit{\text{z}}\right)\left[\begin{array}{cc}{a}_3& 0\\ 0& a_3\end{array}\right],\end{array}

\begin{array}{rl}& a_1=\frac{1-v{(\mathit{\text{z}})}^2}{1-3v{(\mathit{\text{z}})}^2-2v{(\mathit{\text{z}})}^3},\\ & a_2=\frac{v(\mathit{\text{z}})\left(1+v(\mathit{\text{z}})\right)}{1-3v{(\mathit{\text{z}})}^2-2v{(\mathit{\text{z}})}^3},\\ & a_3=\frac{1}{2\left(1+v(\mathit{\text{z}})\right)}.\end{array}

The strain energy is calculated by:

(7)U=\frac{1}{2}{\int }_V{\mathit{\epsilon }}^{\text{T}}\mathit{\sigma }\text{d}V= \frac{1}{2}{\int }_V\left({\mathit{\epsilon }}_1^{\text{T}}{\mathit{\sigma }}_1+{\mathit{\epsilon }}_2^{\text{T}}{\mathit{\sigma }}_2\right)\text{d}V,

where V is the volume of the sandwich plate. When the sandwich plate is affected by the distribution load q\left(x,y\right), which is perpendicular to the neutral plane of the sandwich plate, the work done by the external force is determined by

(8)W={\int }_{\mathrm{\Omega }}\left(w_{\text{b}}+w_{\text{s}}+w_{\text{z}}\right)q\left(x,y\right)\text{d}\mathrm{\Omega },

in which Ω is the neutral plane area of the sandwich plate.

3.2 Finite element formulation

Using the Q4 element with 36 DOFs (each node with nine DOFs) to discretize the model of the FGP sandwich plate (Fig.3). The node displacement vector of the element is defined as follows:

(9)\underset{36\times 1}{{\mathit{d}}_{\mathrm{e}}}={\left[\begin{array}{cccc}{\mathit{d}}_{\text{m}}^{\text{T}}& {\mathit{d}}_{\text{b}}^{\text{T}}& {\mathit{d}}_{\text{s}}^{\text{T}}& {\mathit{d}}_{\mathit{\text{z}}}^{\text{T}}\end{array}\right]}^{\text{T}},

herein

(10)\begin{array}{rl}& \underset{8\times 1}{{\mathit{d}}_{\text{m}}}={\left\{\begin{array}{cc}{\mathit{d}}_{\text{u}}^{\text{T}}& {\mathit{d}}_{\text{v}}^{\text{T}}\end{array}\right\}}^{\text{T}},\underset{4\times 1}{{\mathit{d}}_{\text{u}}}={\left\{\begin{array}{cccc}{u}_{01}& u_{02}& u_{03}& u_{04}\end{array}\right\}}^{\text{T}},\\ & \underset{4\times 1}{{\mathit{d}}_{\text{v}}}={\left\{\begin{array}{cccc}{v}_{01}& v_{02}& v_{03}& v_{04}\end{array}\right\}}^{\text{T}},\\ & \underset{12\times 1}{{\mathit{d}}_{\text{b}}}={\left\{\begin{array}{cccc}{\mathbf{\Delta }}_{\text{b1}}^{\text{T}}& {\mathbf{\Delta }}_{\text{b2}}^{\text{T}}& {\mathbf{\Delta }}_{\text{b3}}^{\text{T}}& {\mathbf{\Delta }}_{\text{b4}}^{\text{T}}\end{array}\right\}}^{\text{T}},\\ & \underset{3\times 1}{{\mathit{\Delta }}_{\text{b}i}}={\left\{\begin{array}{ccc}{w}_{\text{b}i}& w_{\text{b},xi}& w_{\text{b},yi}\end{array}\right\}}^{\text{T}},\\ & \underset{3\times 1}{{\mathit{\Delta }}_{\text{s}i}}={\left\{\begin{array}{ccc}{w}_{\text{s}i}& w_{\text{s},xi}& w_{\text{s},yi}\end{array}\right\}}^{\text{T}},\\ & \underset{12\times 1}{{\mathit{d}}_{\text{s}}}={\left\{\begin{array}{cccc}{\mathbf{\Delta }}_{\text{b1}}^{\text{T}}& {\mathbf{\Delta }}_{\text{b2}}^{\text{T}}& {\mathbf{\Delta }}_{\text{b3}}^{\text{T}}& {\mathbf{\Delta }}_{\text{b4}}^{\text{T}}\end{array}\right\}}^{\text{T}},\\ & \underset{3\times 1}{{\mathit{\Delta }}_{\text{s}i}}={\left\{\begin{array}{ccc}{w}_{\text{s}i}& w_{\text{s,}xi}& w_{\text{s},yi}\end{array}\right\}}^{\text{T}},\left(i=1,...,4\right)\\ & \underset{4\times 1}{{\mathit{d}}_{\mathit{\text{z}}}}={\left\{\begin{array}{cccc}{w}_{\mathit{\text{z}}1}& w_{\mathit{\text{z}}2}& w_{\mathit{\text{z}}3}& w_{\mathit{\text{z}}4}\end{array}\right\}}^{\text{T}}.\end{array}

The node displacement components of the element are defined by

(11)u_0=\mathit{N} {\mathit{d}}_{\text{u}}, v_0=\mathit{N} {\mathit{d}}_{\text{v}}, w_{\text{b}} =\mathit{H}{\mathit{d}}_{\text{b}},w_{\text{s}} =\mathit{H}{\mathit{d}}_{\text{s}},w_{\mathit{\text{z}}}=\mathit{N}{\mathit{d}}_{\mathit{\text{z}}} ,

with \mathit{N}=\left[\begin{array}{cccc}{N}_1& N_2& N_3& N_4\end{array}\right], \mathit{H} =\left[\begin{array}{cccc}{H}_1& H_2& ...& H_{12}\end{array}\right] are Lagrange and the Hermit interpolations given by

\{\begin{array}{c}{N}_1=\frac{1}{4}\left(1-r\right)\left(1-s\right),N_2=\frac{1}{4}\left(1+r\right)\left(1-s\right),\\ N_3=\frac{1}{4}\left(1+r\right)\left(1+s\right),N_4=\frac{1}{4}\left(1-r\right)\left(1+s\right),\end{array}

and

\{\begin{array}{rl}& H_1=\frac{1}{8}\left(1-r\right)\left(1-s\right)\left(2-r-s-r^2-s^2\right),\\ & H_2=\frac{1}{8}\left(1-r\right)\left(1-s\right)\left(1-r^2\right),\\ & H_3=\frac{1}{8}\left(1-r\right)\left(1-s\right)\left(1-s^2\right),\\ & H_4=\frac{1}{8}\left(1+r\right)\left(1-s\right)\left(2+r-s-r^2-s^2\right),\\ & H_5=-\frac{1}{8}\left(1+r\right)\left(1-s\right)\left(1-r^2\right),\\ & H_6=\frac{1}{8}\left(1+r\right)\left(1-s\right)\left(1-r^2\right),\\ & H_7=\frac{1}{8}\left(1+r\right)\left(1+s\right)\left(2+r+s-r^2-s^2\right),\\ & H_8=-\frac{1}{8}\left(1+r\right)\left(1+s\right)\left(1-r^2\right),\\ & H_9=-\frac{1}{8}\left(1+r\right)\left(1+s\right)\left(1-s^2\right),\\ & H_{10}=\frac{1}{8}\left(1-r\right)\left(1+s\right)\left(2-r+s-r^2-s^2\right),\\ & H_{11}=\frac{1}{8}\left(1-r\right)\left(1+s\right)\left(1-r^2\right),\\ & H_{12}=-\frac{1}{8}\left(1-r\right)\left(1+s\right)\left(1-s^2\right),\end{array}

Substituting Eq. (11) into Eq. (5b), we get the strain vector as follows.

(13)\begin{array}{rl}& {\mathit{\epsilon }}_{\text{m}}={\mathit{B}}_{\text{m}}{\mathit{d}}_{\text{m}},{\mathit{\kappa }}_{\text{b}}={\mathit{C}}_{\text{b}}{\mathit{d}}_{\text{b}},{\mathit{\kappa }}_{\text{s}}={\mathit{C}}_{\text{s}}{\mathit{d}}_{\text{s}},\\ & {\mathit{\epsilon }}_{\mathit{\text{z}}}={\mathit{B}}_{\mathit{\text{z}}}{\mathit{d}}_{\mathit{\text{z}}},{\mathit{\epsilon }}_{\text{s1}}={\mathit{B}}_{\text{s1}}{\mathit{d}}_{\text{s}},{\mathit{\epsilon }}_{\text{s2}}={\mathit{B}}_{\text{s2}}{\mathit{d}}_{\mathit{\text{z}}},\end{array}

where

(14)\begin{array}{rl}& \underset{4\times 8}{{\mathit{B}}_{\text{m}}}=\left[\begin{array}{cc}{\mathit{N}}_{,x}& 0\\ 0& {\mathit{N}}_{,y}\\ 0& 0\\ {\mathit{N}}_{,y}& {\mathit{N}}_{,x}\end{array}\right],\underset{4\times 12}{{\mathit{C}}_{\text{b}}}=\underset{4\times 12}{{\mathit{C}}_{\text{s}}}=-\left[\begin{array}{c}{\mathit{H}}_{,xx}\\ {\mathit{H}}_{,yy}\\ 0\\ 2{\mathit{H}}_{,xy}\end{array}\right],\\ & \underset{4\times 4}{{\mathit{B}}_{\mathit{\text{z}}}}=\left[\begin{array}{c}0\\ 0\\ \mathit{N}\\ 0\end{array}\right],\underset{2\times 12}{{\mathit{B}}_{\text{s1}}}=\left[\begin{array}{c}{\mathit{H}}_{,y}\\ {\mathit{H}}_{,x}\end{array}\right],\underset{2\times 4}{{\mathit{B}}_{\text{s2}}}=\left[\begin{array}{c}{\mathit{N}}_{,y}\\ {\mathit{N}}_{,x}\end{array}\right].\end{array}

Substituting Eq. (13) into Eq. (5b) and combining with Eq. (7), the strain energy of element sandwich plates is obtained as follows.

(15)U_{\text{e}} =\frac{1}{2}{\mathit{d}}_{\text{e}}^{\text{T}}{\mathit{K}}_{\text{e}} {\mathit{d}}_{\text{e}}=\frac{1}{2}{\mathit{d}}_{\text{e}}^{\text{T}} \left[\begin{array}{cccc}{\mathit{k}}_{\text{mm}}& {\mathit{k}}_{\text{mb}}& {\mathit{k}}_{\text{ms}}& {\mathit{k}}_{\text{m}\mathit{\text{z}}}\\ {\mathit{k}}_{\text{mb}}^{\text{T}}& {\mathit{k}}_{\text{bb}}& {\mathit{k}}_{\text{bs}}& {\mathit{k}}_{\text{b}\mathit{\text{z}}}\\ {\mathit{k}}_{\text{ms}}^{\text{T}}& {\mathit{k}}_{\text{bs}}^{\text{T}}& {\mathit{k}}_{\text{ss}}& {\mathit{k}}_{\text{s}\mathit{\text{z}}}\\ {\mathit{k}}_{\text{m}\mathit{\text{z}}}^{\text{T}}& {\mathit{k}}_{\text{b}\mathit{\text{z}}}^{\text{T}}& {\mathit{k}}_{\text{s}\mathit{\text{z}}}^{\text{T}}& {\mathit{k}}_{\mathit{\text{z}}\mathit{\text{z}}}\end{array}\right]{\mathit{d}}_{\text{e}},

where {\mathit{K}}_{\text{e}} is the element stiffness matrix with the submatrices determined by

\begin{array}{rl}& \underset{8\times 8}{{\mathit{k}}_{\mathrm{m}\mathrm{m}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}{\mathit{B}}_{\text{m}}^{\text{T}}{\mathit{D}}_1{\mathit{B}}_{\text{m}}\text{d}\mathrm{\Omega },\\ & \underset{12\times 12}{{\mathbf{k}}_{\text{bb}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}{\mathit{\text{z}}}^2{\mathit{C}}_{\text{b}}^{\text{T}}{\mathit{D}}_1{\mathit{C}}_{\text{b}}\text{d}\mathrm{\Omega },\\ & \underset{12\times 12}{{\mathit{k}}_{\text{ss}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}{f}^2{\mathit{C}}_{\text{s}}^{\text{T}}{\mathit{D}}_1{\mathit{C}}_{\text{s}}\text{d}\mathrm{\Omega }+{\int }_{{\mathrm{\Omega }}_{\text{e}}}{g}^2{\mathit{B}}_{\text{s}}^{\text{T}}{\mathit{D}}_2{\mathit{B}}_{\text{s}}\text{d}\mathrm{\Omega },\\ & \underset{4\times 4}{{\mathit{k}}_{\mathit{\text{z}}\mathit{\text{z}}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}{g}_{,\mathit{\text{z}}}^2{\mathit{B}}_{\mathit{\text{z}}}^{\text{T}}{\mathit{D}}_1{\mathit{B}}_{\mathit{\text{z}}}\text{d}\mathrm{\Omega }+{\int }_{{\mathrm{\Omega }}_{\text{e}}}{g}^2{\mathit{B}}_{\text{s2}}^{\text{T}}{\mathit{D}}_2{\mathit{B}}_{\text{s2}}\text{d}\mathrm{\Omega },\\ & \underset{8\times 12}{{\mathit{k}}_{\text{mb}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}\mathit{\text{z}}{\mathit{B}}_{\text{m}}^{\text{T}}{\mathit{D}}_1{\mathit{C}}_{\text{b}}\text{d}\mathrm{\Omega },\end{array}

(16)\begin{array}{rl}& \underset{8\times 12}{{\mathit{k}}_{\text{ms}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}f{\mathit{B}}_{\text{m}}^{\text{T}}{\mathit{D}}_1{\mathit{C}}_{\text{s}}\text{d}\mathrm{\Omega },\\ & \underset{12\times 12}{{\mathit{k}}_{\text{bs}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}\mathit{\text{z}}f{\mathit{C}}_{\text{b}}^{\text{T}}{\mathit{D}}_1{\mathit{C}}_{\text{s}}\text{d}\mathrm{\Omega },\\ & \underset{12\times 4}{{\mathit{k}}_{\text{b}\mathit{\text{z}}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}\mathit{\text{z}}{g}_{,\mathit{\text{z}}}{\mathit{C}}_{\text{b}}^{\text{T}}{\mathit{D}}_1{\mathit{B}}_{\mathit{\text{z}}}\mathrm{d}\mathrm{\Omega },\\ & \underset{12\times 4}{{\mathit{k}}_{\text{s}\mathit{\text{z}}}}={\int }_{{\mathrm{\Omega }}_{\text{e}}}f{g}_{,\mathit{\text{z}}}{\mathit{C}}_{\text{s}}^{\text{T}}{\mathit{D}}_1{\mathit{B}}_{\mathit{\text{z}}}\text{d}\mathrm{\Omega }+{\int }_{{\mathrm{\Omega }}_{\text{e}}}{g}^2{\mathit{B}}_{\text{s1}}^{\text{T}}{\mathit{D}}_2{\mathit{B}}_{\text{s2}}\text{d}\mathrm{\Omega }.\end{array}

Substituting Eq. (11) into Eq. (8), the work done by the external force acting on the element is:

(17)W_{\text{e}} ={\int }_{{\mathrm{\Omega }}_{\text{e}}}\left(w_{\text{b}}+w_{\text{s}}+w_{\mathit{\text{z}}}\right)q\left(x,y\right)\text{d}\mathrm{\Omega }= {\mathit{F}}_{\text{e}}{\mathit{d}}_{\text{e}},

in which, {\mathit{F}}_{\text{e}}={\int }_{\mathrm{\Omega }}{\mathit{N}}_{\text{w}}^{\text{T}}q\left(x,y\right)\text{d}\mathrm{\Omega } is the node load vector and \underset{1\times 36}{{\mathit{N}}_{\text{w}}}=\left[\begin{array}{ccccc}0& 0& \mathit{H}& \mathit{H}& \mathit{N}\end{array}\right] is the shape function matrix of the displacement w which is determined according to the node displacement of the element.

By using the virtual work principle \delta \left({\displaystyle \sum _{nel}^{}{U}_{\text{e}}-{\displaystyle \sum _{nel}^{}{W}_{\text{e}}}}\right)= 0, the governing equations of the sandwich plate are derived by the following formulation:

(18)\mathit{K}\mathit{D}=\mathit{F},

where D is the global node displacement vector; \mathit{K}= {\displaystyle \sum _{nel}{\mathit{K}}_{\text{e}},\mathit{F}={\displaystyle \sum _{nel}{\mathit{F}}_{\text{e}}}} are the global stiffness matrix and the global load vector of the sandwich plate, respectively, and symbol “nel” represents the number of discretized elements of the sandwich plate. Note that, {\mathit{K}}_{\text{e}} and {\mathit{F}}_{\text{e}} are determined by using the Gaussian quadrature integration method with four interpolating points. After imposing the boundary condition and solving Eq. (18), the displacement and strain field can be determined through their relations.

For the finite element analysis, boundary conditions (BCs) are taken according to the geometric constraints at the edges. In this paper, BCs are given as follows.

Clamped support (C):

u_0=v_0=w_{\text{b}} =w_{\text{s}}=w_{\mathit{\text{z}}}=w_{\text{b,}x}=w_{\text{b,}y}=w_{\text{s,}x}=w_{\text{s,}y}=0 at all edges.

Simple support (S):

u_0=w_{\text{b}}=w_{\text{s}}=w_{\mathit{\text{z}}} =w_{\text{b,}x} =w_{\text{s,}x} =0 at or x = a/2,

v_0=w_{\text{b}}=w_{\text{s}}=w_{\mathit{\text{z}}} =w_{\text{b,}y} =w_{\text{s,}y} =0 at or y = b/2.

Free support (F): all DOFs at the boundary edge are non-zero.

4 Numerical results and discussion

This section aims to perform the numerical examples to illustrate novel contributions including: 1) verification of the reliability of the proposed method; 2) presentation of new results in the bending displacement and stress of FGP sandwich plates including rectangular and L-shape models.

4.1 Verification examples

Firstly, consider a rectangular FG sandwich plate as shown in Fig.1. The uniformly distributed load q_0 or the sinusoidal distributed load q\left(x,y\right)=q_0\sin \left({\displaystyle \frac{\text{π}x}{a}}\right)\sin \left({\displaystyle \frac{\text{π}y}{b}}\right) is applied. The material properties are provided in Tab.1. The dimensionless parameters are introduced by the following formulations:

For the rectangular sandwich plate:

\begin{array}{rl}& w^{\ast }=\frac{10E_{0}h}{q_0{a}^2}w\left(\frac{a}{2},\frac{b}{2}\right),\\ & {\sigma }_x^{\ast }\left(\mathit{\text{z}}\right)=\frac{10h^2}{q_0{a}^2}{\sigma }_x\left(\frac{a}{2},\frac{b}{2},\mathit{\text{z}}\right),\\ & \\ & {\sigma }_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right)=\frac{10h^2}{q_0{a}^2}{\sigma }_{\mathit{\text{z}}}\left(\frac{a}{2},\frac{b}{2},\mathit{\text{z}}\right),\\ & {\tau }_{x\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right)=\frac{h}{q_{0}a}{\tau }_{x\mathit{\text{z}}}\left(0,\frac{b}{2},\mathit{\text{z}}\right),\\ & E_0=1\text{GPa}.\end{array}

For the L-shape sandwich plate:

\begin{array}{rl}& w^{\ast }=\frac{10E_{0}h}{q_0{a}^2}w\left(\frac{a}{4},\frac{3b}{4}\right),\\ & {\sigma }_x^{\ast }\left(\mathit{\text{z}}\right)=\frac{10h^2}{q_0{a}^2}{\sigma }_x\left(\frac{a}{4},\frac{3b}{4},\mathit{\text{z}}\right),\\ & {\sigma }_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right)=\frac{10h^2}{q_0{a}^2}{\sigma }_{\mathit{\text{z}}}\left(\frac{a}{4},\frac{3b}{4},\mathit{\text{z}}\right),\\ & {\tau }_{x\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right)=\frac{h}{q_{0}a}{\tau }_{x\mathit{\text{z}}}\left(a,\frac{3b}{4},\mathit{\text{z}}\right).\end{array}

The displacement w^{\ast } and stress {\sigma }_x^{\ast }\left(\mathit{\text{z}}\right) of the completely simply supported square sandwich (SSSS) plate with FGM skins and ceramic core under the action of the sinusoidally distributed force are shown in Tab.2. Moreover, some comparison results can be seen in Fig.4. In these table and figure, numerical results of the present work are compared with published work of Zenkour [40] using third-order shear deformation theory (TSDT) and with work of Thai et al. [50] employing FSDT. The thickness of plate is h=a/10, the material properties are shown in Tab.1 with the ceramic being replaced by ZrO₂ with E_{\text{c}}=151 \text{GPa},v_{\text{c}} =\mathrm{0.3.} It can be seen that the displacement and stress converge at mesh size 16\times 16. Moreover, the obtained results are close to those of TSDT [40] and better than those of FSDT [50]. It can be asserted that the results from our work are more reliable than those of Thai et al. [50] because the FSDT has omitted the higher-order displacement components. The computed results are in good agreement with those of other published, specifically, less than 1% for the displacement and are less than 1.8% for the stress. So, it can be confirmed that the current formula guarantees accuracy and reliability. From here, the mesh size 16\times 16 is used for the next examples.

Next, to ensure the performance of the proposed element, we compare the displacement and stress results of the FGM plate with those of Vasiraja and Nagaraj [70] using ANSYS^®15.0 software. In that study, they used a Shell181 element with a mesh size 50 × 50. The results shown in Tab.3 and Tab.4 once again confirm the reliability and accuracy of the present element.

Finally, the ability to overcome shear locking of this element for the cases of FGM (Al₂O₃/Al) square plates with different BCs is demonstrated in Fig.5. It is observed that the dimensionless deflection of the center-point of FGM plates is almost independent of the change of the length-to-thickness ratio {\displaystyle \frac{a}{h}} when the plates become very thin. It can be concluded that the shear locking problem is corrected for the plates with four BCs, and also for five different power-law indexes, k = 0, 0.5, 1, 2, 4.

4.2 The rectangular functionally graded porous sandwich plate

Firstly, the effect of power-law index k on the static response of square FGP sandwich plates is studied. Fig.6 and Tab.5 show displacement and stress of SSSS square FGP sandwich plate (scheme 1-2-1) with the thickness a/h=25 (a is fixed), and with the porosity coefficient \xi =0.1. Fig.7 and Tab.6 illustrate displacement and stress of SCSC square FGP sandwich plate (scheme 1-1-1) with the thickness a/h=30 and with the porosity coefficient \xi =0.15. These figure and table show that increase in power-law index k leads to increase of the displacement and stress of FGP sandwich plates. This is as expected because the increase of the power-law index k results in increasing the volume of metal in the structure and thus it reduces the stiffness of FGP sandwich plates.

Secondly, Fig.8 and Tab.7 show displacement and stress response of fully clamped (CCCC) square FGP sandwich plates (scheme 1-1-1) with the thickness a/h=50 versus porosity coefficient \xi with power-law index k=2. In addition, Fig.9 and Tab.8 present displacement and stress response of SFSF square FGP sandwich plates (scheme 1-4-1) with the thickness a/h=15 versus porosity coefficient \xi with power-law index k=1. It can be observed that the increase of the porosity coefficient \xi results in the increase of the displacement and stress response of FGP sandwich plates. This is reasonable since the increase in volume makes reduce the stiffness of FGP sandwich plates.

Next, the effect of the thickness ratio between layers (schemes) on the static response of FGP sandwich plates are plotted in Fig.10 and Fig.11 and provided in Tab.9 and Tab.10 with porosity coefficient \xi =0.4 and power-law index k=4. Specifically, Fig.10 and Tab.9 present the displacement and stress response of SSFS FGP sandwich plates with the thickness a/h=20 while Fig.11 and Tab.10 show the static response of SSCS FGP sandwich plates with the thickness a/h=40. From these figures and tables, it can be found that when the FGP sandwich plate thickness is constant, the thickness ratio between layers significantly affects the static bending of FGP sandwich plates. Observing the graphs, we can see that the scheme 1-1-1 gives the smallest displacement and stress response, whereas the scheme 0-1-0 (the isotropic FGP plate) gives the largest displacement and stress response. This confirms that the FGP core has a positive effect on the static response of sandwich plates and is more efficient than the isotropic FGP plates. Herein, C-point has coordinates \left(0,{\displaystyle \frac{b}{2}},\mathit{\text{z}}\right).

Besides, Tab.11 provides more displacement and stress results of SSSS FGP sandwich plates versus power-law index k corresponding to specific values of a/h ratios and schemes while Tab.12 presents the displacement and stress results of CCCC FGP sandwich plates versus porosity coefficient \xi. Note that, the order of BCs is denoted sequentially from edge 1 to edge 4.

4.3 The L-shape functionally graded porous sandwich plate

This part investigates the static bending of FGP L-shape sandwich plates with geometrical parameters as shown in Fig.12, so as to illustrate the effectiveness of the finite element method compared with the analytical method in analyzing the mechanical behavior of the structures, especially when the geometrical model becomes more complex (un-symmetry).

Firstly, some displacement and stress results of SSSSSS FGP L-shape sandwich plates (scheme 1-2-1) with h/a=20;\xi =0.3 versus variations with power-law index k and different mesh sizes are provided in Tab.13, which indicates that the numerical results converge at mesh size 16 × 16. Then, Fig.13 presents the displacement and stress response of SSSSSS FGP L-shape sandwich plates, and illustrates that the maximum displacement of FGP L-shape sandwich plates shifts to the left and the elastic line is no longer as symmetric as in the case of rectangular sandwich plates. This is due to the asymmetry of L-shape sandwich plates.

Similarly, the displacement and stress response of CCCCCC FGP L-shape sandwich plates (scheme 1-4-1) with a/h=40,k= 10 versus porosity coefficient \xi and different mesh sizes are listed in Tab.14, which confirms that the obtained results also converge at a mesh size 16\times 16. Next, the displacement and stress response of CCCCCC FGP L-shape sandwich plates is plotted in Fig.14. Note that, A-point has coordinates \left({\displaystyle \frac{a}{4}},{\displaystyle \frac{3a}{4}},\mathit{\text{z}}\right), B-point has coordinates \left(a,{\displaystyle \frac{3a}{4}},\mathit{\text{z}}\right) and AB-line has the equation as y={\displaystyle \frac{3a}{4}} (see Fig.12), the order of BCs is denoted sequentially from edge 1 to edge 6 and mesh size N_x\times N_y with N_x,N_y are respectively the number of elements divided along the x- and y-axis.

Furthermore, Tab.15 and Tab.16 present some new results on the static response of FGP L-shape sandwich plates with variation of the geometrical and material parameters.

Finally, the variation of deformation field of FGP L-shape sandwich plates versus with different BCs is shown visually in Fig.15. It can be concluded that with the same geometry parameters and material properties, the displacement of the CCCCCC L-shape sandwich plate is smaller than those of L-shape sandwich plates with other BCs because the clamped boundary condition is less flexible than the simply supported and free BCs.

5 Conclusions

This paper proposes a Q4 finite element with nine DOFs per node for the investigation of the static response of FGP sandwich plates including L-shape models. The finite element formulation based on Quasi-3D theory helps to describe more accurately the stress−strain field of plates, especially with thick plates. Besides, the use of Quasi-3D theory satisfies the stress-free condition at the upper and lower boundaries of plates condition and does not require use of any anti “shear-locking” techniques. The paper also analyses and discusses the effects of geometrical parameters, material properties on the static response of FGP sandwich plates consisting of L-shape models. From the formulas and obtained numerical results, some main conclusions can be highlighted as follows.

1) Using the Q4 element combined with Quasi-3D theory easily discretizes the problem domain even with complex geometric domains and avoids “the shear-locking” phenomenon that appears in the classical Q4 element.

2) Using sandwich structures with an FGP core not only reduces the overall mass but also increases the flexural strength of the structure in some specific cases.

3) The material properties such as power-law index k, porosity coefficient \xi and geometric parameters such as length-thickness ratios, thickness ratio between the bottom layer significantly affect the static bending of FGP sandwich plates. In general, power-law index k and porosity coefficient \xi make reduce the plate’s stiffness. Moreover, adjusting these parameters can help control the stiffness of sandwich plates as desired.

4) The numerical results provided are useful for calculations for design of FGP sandwich plates in engineering practice.

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