Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates

Abstract

The free vibration of isotropic gradient elastic thick non-rectangular microplates is analyzed in this paper. To capture the microstructure-dependent effects of microplates, a negative second-order gradient elastic theory with symmetry is utilized. The related equations of motion and boundary conditions are obtained using the energy variational principle. A closed-form solution is presented for simply supported free-vibrational rectangular microplates with four edges. A C¹-type differential quadrature finite element (DQFE) is applied to solve the free vibration of thick microplates. The DQ rule is extended to the straight-sided quadrilateral domain through a coordinate transformation between the natural and Cartesian coordinate systems. The Gauss–Lobato quadrature rule and DQ rule are jointly used to discretize the strain and kinetic energies of a generic straight-sided quadrilateral plate element. Selective numerical examples are validated against those available in the literature. Finally, the impact of various parameters on the free vibration characteristics of annular sectorial and triangular microplates is shown. It indicates that the strain gradient and inertia gradient effects can result in distinct changes in both vibration frequencies and mode shapes.

1. Introduction

The free vibration of plates and plate assemblies is a hot topic that has continually inspired researchers for well over two centuries. From an engineering perspective, the importance of this topic cannot be overemphasized, particularly for its applications in the aeronautical industry, where the top and bottom skins of an aircraft wing are generally idealized as plate assemblies during the structural design. In the context of the first- or third-order shear deformation theory (FSDT or TSDT), researchers have conducted many numerical studies on the vibration characteristics of thick plates. For example, Bui et al. [1] presented new numerical results of the high-frequency modes of Mindlin plates using an effective shear-locking-free meshless method. Based on a modified FSDT, Nam et al. [2] developed a four-node plate element with nine degrees of freedom per node for the static bending and vibration of two-layer composite plates. Tran et al. [3] presented new finite element results of the static bending at high temperatures and the thermal buckling of sandwich FG plates using a modified TSDT. Thai et al. [4] applied the finite element method to simulate the mechanical, electric, and polarization behaviors of TSDT-based piezoelectric nanoplates resting on elastic foundations subjected to static loads. Doan et al. [5] used the TSDT and phase-field approach to simulate the free vibration response of cracked nanoplates while taking into account the flexoelectric effect. Duc et al. [6] established a phase-field fracture model in the context of a new TSDT to study the buckling behavior of multi-cracked FG plates.

Increasing progress in ultra-precision machining techniques has spawned various small-sized beam/plate-like structures in the past decades. Owing to their excellent mechanical, electrical, and thermal performance, such structures have widely served as the major load-bearing objects in Micro-Electro-Mechanical Systems (MEMSs) [7]. However, at the micron or even submicron level, the critical dimensions (e.g., diameter and thickness) of structural members are usually of the same order as the characteristic dimensions of constituent materials (e.g., grain size, void radius, and dislocation spacing), which could induce the microstructure-dependent effects validated by experiments and simulations on the bending of microbeams [8,9,10], the torsion of copper microwires [11,12], and the process of wave propagation in superlattice solids [13,14]. Thus, microstructure-dependent effects should be considered in analyzing static and dynamic problems of small-scale beams and plates for the reliability and design accuracy of MEMS devices. Due to the lack of long-range interactions among adjacent material points, classical continuum mechanics fails to capture size-dependent phenomena.

Classical continuum mechanics needs to be improved by introducing the higher-order spatial derivatives of strain, stress, and inertia terms while preserving its powerful homogenizing characteristic. To realize dimensional homogeneity, one or more material length scale parameters (MLSPs) should be used in non-classical constitutive equations. The original work on gradient-type continuum mechanics can be traced back to Cauchy’s exploratory study on modeling discrete lattices in the 1850s. After that, the Cosserat brothers clearly defined microrotations and couple stress in the early 20th century. The first renaissance of higher-order continuum mechanics was promoted by the representative works of Koiter [15], Mindlin [16,17,18], Toupin [19], and others. Early works focused on the construction of a theoretical framework but lacked experimental validation. Some simplified gradient elasticity theories [8,13,20,21,22,23] were proposed and partly validated in the 1980s and 1990s for engineering applications. Among these theories, the single-parameter gradient elasticity theory (SGET) formulated by Aifantis, Ru, and Altan [20,21] and the modified strain gradient elasticity theory (MSGT) established by Lam et al. [8] are the most attractive. Based on the SGET and MSGT, the size-dependent Bernoulli–Euler beam [24,25], Timoshenko beam [26], Reddy–Levinson beam [27], Kirchhoff plate [28,29], Mindlin plate [30,31], Reddy plate [32], and Kirchhoff–Love cylindrical shell [33,34] models have been developed to predict the static bending, free vibration, and buckling behaviors of microscale devices. Roudbari et al. [35] and Kong [36] reviewed the recent advances in non-classical continuum mechanics models and provided research insights for future studies.

In addition to two types of gradient effects, size-dependent phenomena are also caused by other physical factors (e.g., nonlocal stress and surface energy effects). Thus, multifactorial size-dependent constitutive models have been developed to understand mechanical behavior among microscale members. Recently, the nonlocal strain gradient theory (NSGT) proposed by Lim et al. has attracted the most attention [37]. The NSGT can be regarded as a unification of Eringen’s nonlocal elasticity theory [38] and Aifantis’s strain gradient theory [20]. A nonlocal parameter and a strain gradient parameter are used to weigh the importance of the strain gradient and nonlocal effects. Both the stiffening and softening effects of structural members can be captured by the NSGT. Thus, it has been widely used in modeling small-sized structures with two types of size effects [39,40,41,42,43,44,45,46,47,48]. For instance, Lu et al. [39] proposed a unified nonlocal strain gradient beam model for analyzing the size-dependent bending and buckling behaviors of nanobeams with different slenderness ratios. Ma et al. [44] studied wave propagation in thermo-electro-magneto-mechanical-elastic nanoshells using the nonlocal strain gradient thin and shear-deformable cylindrical shell models. Lu et al. [45] developed a consistent surface-stress-enriched nonlocal strain gradient model for a rectangular buckled plate, by which the critical buckling loads of SSSS, CCSS, and CCCC nanoplates are determined. Lu et al. [48] derived a nonlocal strain gradient model including surface stress effects to analyze the free vibration of moderately thick FG cylindrical nanoshells.

Although the size-dependent continuum modeling of microstructural members has been well studied, the derived governing differential equations can be solved analytically for extremely limited types of boundaries, loadings, and geometric conditions. The reason is that the higher-order gradients introduced by the model can lead to a remarkable rise in the order of equations of motion and boundary conditions. For instance, the deflection of gradient elastic Kirchhoff plates [28,29] and Kirchhoff–Love cylindrical shells [33,34] requires C²-continuity. The deflection and rotations of gradient elastic Mindlin plates [31,49] require C¹-continuity. Moreover, gradient elastic Reddy plates [32] require both the C¹-continuity of rotation and the C²-continuity of deflection. These imply extreme difficulty in solving gradient elastic boundary value problems using both analytical and numerical methods. Although some conventional analytical methods, e.g., the assumed mode method [45,48], Navier method [29,39,40,44,50], extended Kantorovich method [51,52], and p-version Ritz method [31], have been proposed to solve special gradient elastic boundary value problems, so far, few studies have focused on gradient elastic plates with non-rectangular shapes and sudden changes in edge supports and thicknesses.

Advanced numerical methods for gradient elastic beams and plates have come forth through the hard work of researchers. For example, Thai et al. [53] analyzed the size-dependent mechanical behavior of FG microplates by combining the use of MSGT and isogeometric analysis (IGA). Nguyen et al. [54] investigated the vibration behavior of FG microplates with cracks, strain gradient effects, and micro-inertia effects by means of an extended IGA. According to the four-unknown refined plate theory, Nguyen et al. [55] used the IGA to predict the geometrically nonlinear bending responses of small-scale FG plates. Moreover, Nguyen et al. [56] constructed a novel NURBS-based IGA model to study the static bending, free vibration, and buckling of couple-stress-enriched FG microplates with higher-order shear and normal deformation effects. Niiranen et al. [57] performed an IGA on the Galerkin discretization scheme with C²-continuity to address the sixth-order boundary value problems of gradient elastic Kirchhoff plates. Balobanov et al. [58] proposed a single-parameter gradient elastic Kirchhoff–Love shell model of arbitrary geometry and the associated H³-conforming isogeometric Galerkin method. Although the IGA approach can yield arbitrary-order continuous basis functions, there are still inadequacies in the integration of the weak form and the imposition of essential boundary conditions in such a method. In addition, the basis functions of an IGA model often have a larger support domain than those of the related finite element model, implying less sparse system matrices and higher computational expense. According to SGET-based Kirchhoff plates, Babu and Patel [59] established nonconforming C²-continuous rectangular plate finite elements for studying the free vibration and linear buckling of single-walled graphene sheets. However, since the standard FEM is subjected to higher-order continuity conditions, researchers have committed to seeking other alternative methods. Wang [60] developed a weak-form quadrature element method (QEM) to study the free vibration of nonlocal strain gradient Euler–Bernoulli beams. Ishaquddin and Gopalakrishnan [61] presented a weak-form QEM for SGET-based Euler–Bernoulli beams and Kirchhoff plates. To enhance the adaptability of the DQM, combining the advantages of the DQM and FEM may be a good choice. Zhang et al. [62] utilized the advantages of the DQM and FEM for the first time to construct weak-form DQFEs related to isotropic MSGT-based Euler–Bernoulli and Timoshenko beam models, respectively. Soon afterward, they proposed a series of weak-form DQFEs for size-dependent Reddy beams [63,64], Mindlin plates [65,66], and Kirchhoff plates [67,68,69] and showed the efficacy of their developed DQFEM in comparison with the standard FEM.

The aim of this article is to study the free vibration of non-rectangular gradient elastic thick microplates with two types of gradient effects. The remainder of the paper is organized as follows. Section 2 applies the energy variational principle to derive the corresponding equations of motion and boundary conditions. Section 3 develops a quadrilateral differential quadrature finite element to solve the resulting higher-order boundary value problems. In Section 4, we highlight the effectiveness of our theoretical model and solution method by comparing it with other available methods and use it to predict the vibrational behavior of annular sectorial and triangular microplates. Finally, we draw conclusions from our research work in Section 5.

2. Governing Equations of Gradient Elastic Thick Microplates

An originally flat isotropic microplate with moderate thickness h is illustrated in Figure 1, where the plate midplane A coincides with the OXY coordinate plane, and the bold symbols n and s are the unit normal and tangent vectors at a point on the boundary curve $\partial A$, respectively. The material parameters are as follows: Young’s modulus E, shear modulus G, Poisson’s ratio $\nu$, and mass density $\rho$. When the plate receives a transversely distributed load Q on the upper flat surface, there will be a deflection W and two transverse normal rotations ${\mathsf{\Phi }}_X$ and ${\mathsf{\Phi }}_Y$ about the Y- and X-axes, respectively.

For a moderately thick microplate, the displacement field is assumed as

$$U_X=Z{\mathsf{\Phi }}_X\left(X, Y, t\right), U_Y=Z{\mathsf{\Phi }}_Y\left(X, Y, t\right), U_Z=W\left(X, Y, t\right),$$

(1)

where $U_X$, $U_Y$, and $U_Z$ are the displacement components along the X-, Y-, and Z-directions, respectively.

The nonzero components of the Cauchy strain tensor are

$${\epsilon }_{XX}=Z\frac{\partial {\mathsf{\Phi }}_X}{\partial X},{\epsilon }_{YY}=Z\frac{\partial {\mathsf{\Phi }}_Y}{\partial Y},{\epsilon }_{XY}={\epsilon }_{YZ}=\frac{Z}{2}\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial Y}+\frac{\partial {\mathsf{\Phi }}_Y}{\partial X}\right),{\epsilon }_{ZX}={\epsilon }_{XZ}=\frac{1}{2}\left({\mathsf{\Phi }}_X+\frac{\partial W}{\partial X}\right),$$

(2)

$${\epsilon }_{YZ}={\epsilon }_{ZY}=\frac{1}{2}\left({\mathsf{\Phi }}_Y+\frac{\partial W}{\partial Y}\right)$$

Second-order gradient elastic theory [20,21] is initiated from the homogenization of lattice structures by applying the Taylor series to approximate the displacement field of a discrete model. For the negative form, the related constitutive relation is expressed in the following symmetrical form:

$${\sigma }_{ij}=C_{ijkl}\left({\epsilon }_{kl}-l_s^2{\nabla }^2{\epsilon }_{kl}\right)$$

(3)

where $C_{ijkl}$ denotes the elastic constitutive tensor with double symmetry, $l_s$ is the static length scale parameter, ${\nabla }^2$ is the Laplace operator, Latin subscripts run over the symbols $X$, $Y$, and $Z$ unless otherwise indicated, and ${\epsilon }_{kl}$ is the Cauchy strain tensor.

The exploration of the plane stress conditions and Equations (2) and (3) yield the stress–strain equations for gradient elastic Mindlin microplates as follows:

$$\begin{array}{c}{\sigma }_{XX}=\frac{E}{1-{\nu }^2}\left(1-l_s^2{\nabla }^2\right)\left({\epsilon }_{XX}+\nu {\epsilon }_{YY}\right)=\left(1-l_s^2{\nabla }^2\right){\widehat{\sigma }}_{XX}.\hfill \\ {\sigma }_{XY}=\frac{E}{1+\nu }\left(1-l_s^2{\nabla }^2\right){\epsilon }_{XY}=\left(1-l_s^2{\nabla }^2\right){\widehat{\sigma }}_{XY},\hfill \\ {\sigma }_{YY}=\frac{E}{1-{\nu }^2}\left(1-l_s^2{\nabla }^2\right)\left({\epsilon }_{YY}+\nu {\epsilon }_{XX}\right)=\left(1-l_s^2{\nabla }^2\right){\widehat{\sigma }}_{YY}.\hfill \\ {\sigma }_{XZ}=\frac{E}{1+\nu }\left(1-l_s^2{\nabla }^2\right){\epsilon }_{XZ}=\left(1-l_s^2{\nabla }^2\right){\widehat{\sigma }}_{XZ},\hfill \\ {\sigma }_{YZ}=\frac{E}{1+\nu }\left(1-l_s^2{\nabla }^2\right){\epsilon }_{YZ}=\left(1-l_s^2{\nabla }^2\right){\widehat{\sigma }}_{YZ}\hfill \end{array}$$

(4)

where ${\widehat{\sigma }}_{XX}$, ${\widehat{\sigma }}_{XY}$, ${\widehat{\sigma }}_{YY}$, ${\widehat{\sigma }}_{XZ}$, and ${\widehat{\sigma }}_{YZ}$ are classical stresses, and $E$ and $\nu$ are Young’s modulus and Poisson’s ratio, respectively.

Based on Equations (2)–(4) and [28], the strain energy of the present microplate is expressed as

$$\begin{array}{l}{\mathsf{\Pi }}_s=\frac{1}{2}{\displaystyle {\int }_{\mathsf{\Omega }}\left({\widehat{\sigma }}_{XX}{\epsilon }_{XX}+{\widehat{\sigma }}_{YY}{\epsilon }_{YY}+2{\widehat{\sigma }}_{XY}{\epsilon }_{XY}+2K_s{\widehat{\sigma }}_{YZ}{\epsilon }_{YZ}+2K_s{\widehat{\sigma }}_{XZ}{\epsilon }_{XZ}\right)\mathrm{d}\mathsf{\Omega }}+\\ \hspace{1em}\hspace{1em} \frac{l_s^2}{2}{\displaystyle {\int }_{\mathsf{\Omega }}\left(\begin{array}{l}\frac{\partial {\widehat{\sigma }}_{XX}}{\partial X}\frac{\partial {\epsilon }_{XX}}{\partial X}+\frac{\partial {\widehat{\sigma }}_{XX}}{\partial Y}\frac{\partial {\epsilon }_{XX}}{\partial Y}+\frac{\partial {\widehat{\sigma }}_{YY}}{\partial X}\frac{\partial {\epsilon }_{YY}}{\partial X}+\frac{\partial {\widehat{\sigma }}_{YY}}{\partial Y}\frac{\partial {\epsilon }_{YY}}{\partial Y}+2\frac{\partial {\widehat{\sigma }}_{XY}}{\partial X}\frac{\partial {\epsilon }_{XY}}{\partial X}\\ +2\frac{\partial {\widehat{\sigma }}_{XY}}{\partial Y}\frac{\partial {\epsilon }_{XY}}{\partial Y}+2K_s\frac{\partial {\widehat{\sigma }}_{YZ}}{\partial X}\frac{\partial {\epsilon }_{YZ}}{\partial X}+2K_s\frac{\partial {\widehat{\sigma }}_{YZ}}{\partial Y}\frac{\partial {\epsilon }_{YZ}}{\partial Y}+2K_s\frac{\partial {\widehat{\sigma }}_{XZ}}{\partial X}\frac{\partial {\epsilon }_{XZ}}{\partial X}\\ +2K_s\frac{\partial {\widehat{\sigma }}_{XZ}}{\partial Y}\frac{\partial {\epsilon }_{XZ}}{\partial Y}\end{array}\right)\mathrm{d}\mathsf{\Omega }}\\ \hspace{1em}\hspace{1em}=\frac{1}{2}{\displaystyle {\int }_A\{{\mathsf{\Gamma }}_1\left[{\left({\mathsf{\Phi }}_X+\frac{\partial W}{\partial X}\right)}^2+{\left({\mathsf{\Phi }}_Y+\frac{\partial W}{\partial Y}\right)}^2\right]+{\mathsf{\Gamma }}_2{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial Y}+\frac{\partial {\mathsf{\Phi }}_Y}{\partial X}\right)}^2+{\mathsf{\Gamma }}_3\frac{\partial {\mathsf{\Phi }}_X}{\partial X}\frac{\partial {\mathsf{\Phi }}_Y}{\partial Y}}\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\Gamma }}_4\left[{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial X}\right)}^2+{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial Y}\right)}^2\right]+{\mathsf{\Gamma }}_5{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial X}+\frac{{\partial }^{2}W}{\partial X^2}\right)}^2+{\mathsf{\Gamma }}_5{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial X}+\frac{{\partial }^{2}W}{\partial X\partial Y}\right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\Gamma }}_5{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial Y}+\frac{{\partial }^{2}W}{\partial X\partial Y}\right)}^2+{\mathsf{\Gamma }}_5{\left(\frac{{\partial }^{2}W}{\partial Y^2}+\frac{\partial {\mathsf{\Phi }}_Y}{\partial Y}\right)}^2+{\mathsf{\Gamma }}_6{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X\partial Y}+\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X^2}\right)}^2+\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\Gamma }}_6{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial Y^2}+\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X\partial Y}\right)}^2+{\mathsf{\Gamma }}_8\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X\partial Y}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial Y^2}+\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X\partial Y}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X^2}\right)+\\ \hspace{1em}\hspace{1em}\hspace{1em}{\mathsf{\Gamma }}_7\left[{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X\partial Y}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X\partial Y}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial Y^2}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X^2}\right)}^2\right]\}\mathrm{d}A\end{array}$$

(5)

where

$$\begin{array}{c}{\mathsf{\Gamma }}_1=\frac{K_{s}Eh}{4\left(1+ \nu \right)}, {\mathsf{\Gamma }}_2=\frac{E{h}^3}{48\left(1+ \nu \right)}, {\mathsf{\Gamma }}_3=\frac{E\nu h^3}{12\left(1- {\nu }^2\right)}, {\mathsf{\Gamma }}_4=\frac{E{h}^3}{24\left(1- {\nu }^2\right)}, {\mathsf{\Gamma }}_5=\frac{K_{s}Eh{l}_s^2}{4\left(1+ \nu \right)},\hfill \\ \\ {\mathsf{\Gamma }}_6=\frac{E{h}^3{l}_s^2}{48\left(1+ \nu \right)}, {\mathsf{\Gamma }}_7=\frac{E{h}^3{l}_s^2}{24\left(1- {\nu }^2\right)}, {\mathsf{\Gamma }}_8=\frac{E{h}^3\nu l_s^2}{12\left(1- {\nu }^2\right)},\hfill \end{array}$$

(6)

where $K_s$ is the shear correction factor. Equation (5) can reduce to its counterpart (see Equation (10) in [66]) when ${\mathsf{\Sigma }}_1={\mathsf{\Gamma }}_5$, ${\mathsf{\Sigma }}_{15}={\mathsf{\Sigma }}_5={\mathsf{\Sigma }}_2=0$, ${\mathsf{\Sigma }}_3+{\mathsf{\Sigma }}_{17}={\mathsf{\Gamma }}_3$, ${\mathsf{\Sigma }}_4+{\mathsf{\Sigma }}_{16}={\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_5$, ${\mathsf{\Sigma }}_{18}={\mathsf{\Gamma }}_2-{\mathsf{\Gamma }}_5$, ${\mathsf{\Sigma }}_{12}=2 {\mathsf{\Gamma }}_6$, ${\mathsf{\Sigma }}_8={\mathsf{\Sigma }}_7={\mathsf{\Sigma }}_6=2 {\mathsf{\Gamma }}_5$, ${\mathsf{\Sigma }}_9={\mathsf{\Gamma }}_1$, ${\mathsf{\Sigma }}_{10}={\mathsf{\Gamma }}_6+{\mathsf{\Gamma }}_7$, ${\mathsf{\Sigma }}_{11}={\mathsf{\Gamma }}_8$, ${\mathsf{\Sigma }}_{13}={\mathsf{\Gamma }}_7$, and ${\mathsf{\Sigma }}_{14}={\mathsf{\Gamma }}_6$.

To capture the inertia gradient effect, the contribution of the velocity gradient should be considered. On the basis of [13,57], the kinetic energy of the present microplate is as follows:

$$\begin{array}{l}{\mathsf{\Pi }}_d={\displaystyle {\int }_{\mathsf{\Omega }}\frac{\rho }{2}\left(\frac{\partial U_X}{\partial t}\frac{\partial U_X}{\partial t}+l_d^2\frac{{\partial }^2{U}_X}{\partial X\partial t}\frac{{\partial }^2{U}_X}{\partial X\partial t}+l_d^2\frac{{\partial }^2{U}_Y}{\partial Y\partial t}\frac{{\partial }^2{U}_Y}{\partial Y\partial t}+l_d^2\frac{{\partial }^2{U}_Z}{\partial Z\partial t}\frac{{\partial }^2{U}_Z}{\partial Z\partial t}\right)\mathrm{d}\mathsf{\Omega }}\\ \hspace{1em}\hspace{1em}=\frac{1}{2}{\displaystyle {\int }_A\left\{\begin{array}{l}\rho h{\left(\frac{\partial W}{\partial t}\right)}^2+\frac{\rho h^3}{12}{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial t}\right)}^2+\frac{\rho h^3}{12}{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial t}\right)}^2+\\ \rho h{l}_d^2\left[{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial t}\right)}^2+{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial t}\right)}^2+{\left(\frac{{\partial }^{2}W}{\partial X\partial t}\right)}^2+{\left(\frac{{\partial }^{2}W}{\partial Y\partial t}\right)}^2\right]+\\ \frac{\rho l_d^2{h}^3}{12}\left[{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial t\partial X}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial t\partial Y}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial t\partial X}\right)}^2+ {\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial t\partial Y}\right)}^2\right]\end{array}\right\}\mathrm{d}A} \end{array}$$

(7)

where $l_d$ is the dynamic length scale parameter.

Similar to the derivation process in [28,30], the virtual work done by external forces is written as the following equation:

$$\begin{array}{cc}\hfill \delta {\mathsf{\Pi }}_e& ={\displaystyle {\int }_{A}Q\delta W\mathrm{d}X\mathrm{d}Y}+{\displaystyle {\int }_{\partial A}{\overline{V}}_{\left(W\right)}\delta W\mathrm{d}s}+{\displaystyle {\int }_{\partial A}{\overline{M}}_{\left(W\right)}\frac{\partial \delta W}{\partial n}\mathrm{d}s}+{\displaystyle {\int }_{\partial A}{\overline{V}}_{\left({\mathsf{\Phi }}_s\right)}\delta {\mathsf{\Phi }}_s\mathrm{d}s}\hfill \\ \hfill & +{\displaystyle {\int }_{\partial A}{\overline{M}}_{\left({\mathsf{\Phi }}_s\right)}\frac{\partial \delta {\mathsf{\Phi }}_s}{\partial n}\mathrm{d}s}+{\displaystyle {\int }_{\partial A}{\overline{V}}_{\left({\mathsf{\Phi }}_n\right)}\delta {\mathsf{\Phi }}_n\mathrm{d}s}+{\displaystyle {\int }_{\partial A}{\overline{M}}_{\left({\mathsf{\Phi }}_n\right)}\frac{\partial \delta {\mathsf{\Phi }}_n}{\partial n}\mathrm{d}s}\hfill \end{array}$$

(8)

where $Q$ is the distributed transverse load, ${\overline{V}}_{\left(W\right)}$, ${\overline{V}}_{\left({\mathsf{\Phi }}_s\right)}$, and ${\overline{V}}_{\left({\mathsf{\Phi }}_n\right)}$ are generalized shear forces, and ${\overline{M}}_{\left(W\right)}$, ${\overline{M}}_{\left({\mathsf{\Phi }}_s\right)}$, and ${\overline{M}}_{\left({\mathsf{\Phi }}_n\right)}$ are generalized bending moments.

The displacement-based equations of motion and boundary conditions of gradient elastic thick microplates can be obtained using the variational formulations provided in [69].

For any $\left(X, Y\right)\in A$ and $t\in \left(t_1, t_2\right)$:

$$\begin{array}{c}Q+2{\mathsf{\Gamma }}_1\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial X}+\frac{\partial {\mathsf{\Phi }}_Y}{\partial Y}\right)-2{\mathsf{\Gamma }}_5\left(\frac{{\partial }^{4}W}{\partial X^4}+2\frac{{\partial }^{4}W}{\partial Y^2\partial X^2}+\frac{{\partial }^{4}W}{\partial Y^4}\right)+2{\mathsf{\Gamma }}_1\left(\frac{{\partial }^{2}W}{\partial X^2}+\frac{{\partial }^{2}W}{\partial Y^2}\right)\hfill \\ -2{\mathsf{\Gamma }}_5\left(\frac{{\partial }^3{\mathsf{\Phi }}_X}{\partial X^3}+\frac{{\partial }^3{\mathsf{\Phi }}_Y}{\partial X^2\partial Y}+\frac{{\partial }^3{\mathsf{\Phi }}_X}{\partial X\partial Y^2}+\frac{{\partial }^3{\mathsf{\Phi }}_Y}{\partial Y^3}\right)-\rho h\frac{{\partial }^{2}W}{\partial t^2}+\rho h{l}_d^2\left(\frac{{\partial }^{4}W}{\partial X^2\partial t^2}+\frac{{\partial }^{4}W}{\partial Y^2\partial t^2}\right)=0,\hfill \end{array}$$

(9)

$$\begin{array}{c}{\mathsf{\Gamma }}_3\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X\partial Y}+2{\mathsf{\Gamma }}_4\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X^2}-2{\mathsf{\Gamma }}_1\left({\mathsf{\Phi }}_X+\frac{\partial W}{\partial X}\right)+2{\mathsf{\Gamma }}_2\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial Y^2}+\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X\partial Y}\right)-{\mathsf{\Gamma }}_8\left(\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X^3\partial Y}+\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X\partial Y^3}\right)\hfill \\ +2{\mathsf{\Gamma }}_5\left(\frac{{\partial }^{3}W}{\partial X^3}+\frac{{\partial }^{3}W}{\partial X\partial Y^2}+\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X^2}+\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial Y^2}\right)-2{\mathsf{\Gamma }}_6\left(\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X^2\partial Y^2}+\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X^3\partial Y}+\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial Y^4}+\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X\partial Y^3}\right)\hfill \\ -\rho h\left(\frac{h^2}{12}+l_d^2\right)\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial t^2}-2{\mathsf{\Gamma }}_7\left(\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X^4}+\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X^2\partial Y^2}\right)+\frac{\rho h^3{l}_d^2}{12}\left(\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X^2\partial t^2}+\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial Y^2\partial t^2}\right)=0,\hfill \end{array}$$

(10)

$$\begin{array}{c}{\mathsf{\Gamma }}_3\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X\partial Y}+2{\mathsf{\Gamma }}_4\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial Y^2}-2{\mathsf{\Gamma }}_1\left({\mathsf{\Phi }}_Y+\frac{\partial W}{\partial Y}\right)+2{\mathsf{\Gamma }}_2\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X^2}+\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial X\partial Y}\right)-{\mathsf{\Gamma }}_8\left(\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X\partial Y^3}+\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X^3\partial Y}\right)\hfill \\ +2{\mathsf{\Gamma }}_5\left(\frac{{\partial }^{3}W}{\partial Y^3}+\frac{{\partial }^{3}W}{\partial X^2\partial Y}+\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial Y^2}+\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial X^2}\right)-2{\mathsf{\Gamma }}_6\left(\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X^2\partial Y^2}+\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X^3\partial Y}+\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X^4}+\frac{{\partial }^4{\mathsf{\Phi }}_X}{\partial X\partial Y^3}\right)\hfill \\ -\rho h\left(\frac{h^2}{12}+l_d^2\right)\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial t^2}-2{\mathsf{\Gamma }}_7\left(\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X^2\partial Y^2}+\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial Y^4}\right)+\frac{\rho h^3{l}_d^2}{12}\left(\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial X^2\partial t^2}+\frac{{\partial }^4{\mathsf{\Phi }}_Y}{\partial Y^2\partial t^2}\right)=0.\hfill \end{array}$$

(11)

Because of the introduction of higher-order partial derivatives and boundary conditions, the present model is difficult to solve using an analytical or semi-analytical method. The available works focus on seeking analytical/numerical solutions for gradient elastic beams and plates with simple loading and boundary conditions.

3. Solution Procedure

3.1. Navier Method

For a simply supported gradient elastic rectangular microplate, the Navier method can be used to derive the analytical free vibration solution. In this case, $W$, ${\mathsf{\Phi }}_X$, and ${\mathsf{\Phi }}_Y$ can be written as

$$\begin{array}{c}W={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}{e}^{j\omega t}\sin \left({\alpha }_{m}X\right)\sin \left({\beta }_{n}Y\right)}},\hfill \\ {\mathsf{\Phi }}_X={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{B}_{mn}{e}^{j\omega t}\cos \left({\alpha }_{m}X\right)\sin \left({\beta }_{n}Y\right)}},\hfill \\ {\mathsf{\Phi }}_Y={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{C}_{mn}{e}^{j\omega t}\sin \left({\alpha }_{m}X\right)\cos \left({\beta }_{n}Y\right)}},\hfill \end{array}$$

(12)

where $A_{mn}$, $B_{mn}$, and $C_{mn}$ are Fourier coefficients, $\omega$ is the vibration frequency, $j$ is the imaginary unit, $L_X$ and $L_Y$ are the length and width of a rectangular microplate, respectively, and ${\alpha }_m=m\pi /L_X$${\beta }_n=n\pi /L_Y$. The following expression is obtained by substituting Equation (12) into Equations (9)–(11).

$$\left\{\left[\begin{array}{ccc}{K}_{11}^{\left(mn\right)}& K_{12}^{\left(mn\right)}& K_{13}^{\left(mn\right)}\\ K_{21}^{\left(mn\right)}& K_{22}^{\left(mn\right)}& K_{23}^{\left(mn\right)}\\ K_{31}^{\left(mn\right)}& K_{32}^{\left(mn\right)}& K_{33}^{\left(mn\right)}\end{array}\right]-{\omega }^2\left[\begin{array}{ccc}{M}_{11}^{\left(mn\right)}& M_{12}^{\left(mn\right)}& M_{13}^{\left(mn\right)}\\ M_{21}^{\left(mn\right)}& M_{22}^{\left(mn\right)}& M_{23}^{\left(mn\right)}\\ M_{31}^{\left(mn\right)}& M_{32}^{\left(mn\right)}& M_{33}^{\left(mn\right)}\end{array}\right]\right\}\left[\begin{array}{c}{A}_{mn}\\ B_{mn}\\ C_{mn}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],$$

(13)

where

$$\begin{array}{c}{K}_{21}^{\left(mn\right)}=K_{12}^{\left(mn\right)}=2 {\mathsf{\Gamma }}_1{\alpha }_m+2 {\mathsf{\Gamma }}_5{\alpha }_m\left({\alpha }_m^2+{\beta }_n^2\right),\hfill \\ K_{31}^{\left(mn\right)}=K_{13}^{\left(mn\right)}=2{\mathsf{\Gamma }}_1{\beta }_n+2{\mathsf{\Gamma }}_5 {\beta }_n\left({\alpha }_m^2+{\beta }_n^2\right),\hfill \\ K_{22}^{\left(mn\right)}=2 {\mathsf{\Gamma }}_1+2\left({\mathsf{\Gamma }}_4 +{\mathsf{\Gamma }}_5 \right){\alpha }_m^2+2 \left({\mathsf{\Gamma }}_2+{\mathsf{\Gamma }}_5\right){\beta }_n^2+2 \left({\alpha }_m^2+{\beta }_n^2\right)\left({\beta }_n^2{\mathsf{\Gamma }}_6+{\alpha }_m^2{\mathsf{\Gamma }}_7\right),\hfill \\ K_{23}^{\left(mn\right)}=\left({\mathsf{\Gamma }}_3+2 {\mathsf{\Gamma }}_2\right){\alpha }_m{\beta }_n+\left({\mathsf{\Gamma }}_8+2{\mathsf{\Gamma }}_6\right){\alpha }_m^3{\beta }_n+\left({\mathsf{\Gamma }}_8+2{\mathsf{\Gamma }}_6\right){\alpha }_m{\beta }_n^3,\hfill \end{array}$$

(14)

$$M_{11}^{\left(mn\right)}=\rho h+\rho h{l}_d^2\left({\alpha }_m^2+{\beta }_n^2\right), M_{33}^{\left(mn\right)}=M_{22}^{\left(mn\right)}=\frac{\rho h^3}{12}+\rho h{l}_d^2+\frac{\rho l_d^2{h}^3}{12}\left({\alpha }_m^2+{\beta }_n^2\right).$$

(15)

For a non-trivial solution of $A_{mn}$, $B_{mn}$, and $C_{mn}$, it is required that the determinant of the coefficient matrix of Equation (13) vanish. The determinant of the coefficient matrix in Equation (13) is a cubic equation in ${\omega }^2$, the smallest (positive) root of which gives the mnth natural frequency, ${\omega }_{mn}$, for the free vibration of the plate.

3.2. Differential Quadrature Finite Element Method (DQFEM)

A quadrilateral DQFE is derived to address the general free vibration problem of the present gradient elastic model. Figure 2 illustrates a 2D DQ-based geometric mapping scheme to satisfy the C¹-continuity conditions of $W$, ${\mathsf{\Phi }}_X$, and ${\mathsf{\Phi }}_Y$ and a natural-to-Cartesian coordinate transformation to make the DQ rule feasible for a straight-sided quadrilateral domain.

From Figure 2, we can derive the partial derivative relationship between the global and local coordinate systems as follows:

$$\left[\begin{array}{c}\frac{\partial }{\partial X}\\ \frac{\partial }{\partial Y}\end{array}\right]={\mathit{J}}_{\overline{X}\overline{Y}}^{-1}\left[\begin{array}{c}\frac{\partial }{\partial \overline{X}}\\ \frac{\partial }{\partial \overline{Y}}\end{array}\right], \left[\begin{array}{c}\frac{{\partial }^2}{\partial X^2}\\ \frac{{\partial }^2}{\partial Y^2}\\ \frac{{\partial }^2}{\partial X\partial Y}\end{array}\right]=\left[\begin{array}{ccc}{\left(\frac{\partial \overline{X}}{\partial X}\right)}^2& {\left(\frac{\partial \overline{Y}}{\partial X}\right)}^2& 2\frac{\partial \overline{X}}{\partial X}\frac{\partial \overline{Y}}{\partial X}\\ {\left(\frac{\partial \overline{X}}{\partial Y}\right)}^2& {\left(\frac{\partial \overline{Y}}{\partial Y}\right)}^2& 2\frac{\partial \overline{X}}{\partial Y}\frac{\partial \overline{Y}}{\partial Y}\\ \frac{\partial \overline{X}}{\partial X}\frac{\partial \overline{X}}{\partial Y}& \frac{\partial \overline{Y}}{\partial X}\frac{\partial \overline{Y}}{\partial Y}& \frac{\partial \overline{Y}}{\partial X}\frac{\partial \overline{X}}{\partial Y}+\frac{\partial \overline{X}}{\partial X}\frac{\partial \overline{Y}}{\partial Y}\end{array}\right]\left[\begin{array}{c}\frac{{\partial }^2}{\partial {\overline{X}}^2}\\ \frac{{\partial }^2}{\partial {\overline{Y}}^2}\\ \frac{{\partial }^2}{\partial \overline{X}\partial \overline{Y}}\end{array}\right],$$

(16)

where

$${\mathit{J}}_{\overline{X}\overline{Y}}=\left[\begin{array}{cc}\frac{\partial X}{\partial \overline{X}}& \frac{\partial Y}{\partial \overline{X}}\\ \frac{\partial X}{\partial \overline{Y}}& \frac{\partial Y}{\partial \overline{Y}}\end{array}\right], {\mathit{J}}_{\overline{X}\overline{Y}}^{-1}=\left[\begin{array}{cc}\frac{\partial \overline{X}}{\partial X}& \frac{\partial \overline{Y}}{\partial X}\\ \frac{\partial \overline{X}}{\partial Y}& \frac{\partial \overline{Y}}{\partial Y}\end{array}\right].$$

(17)

Using Equations (16) and (17), Equation (5) can be transformed into a natural coordinate system:

$$\begin{array}{l}{\mathsf{\Pi }}_s={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1[{\beta }_1{\left(\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\right)}^2+{\beta }_2{\left(\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}\right)}^2+{\beta }_3{\left(\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}\right)}^2+{\beta }_4\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}+{\beta }_5\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}+}}\\ \hspace{1em}\hspace{1em}{\beta }_6\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}+{\beta }_7{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{X}}^2}\right)}^2+{\beta }_8{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{Y}}^2}\right)}^2+{\beta }_9{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial \overline{Y}}\right)}^2+{\beta }_{10}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{Y}}^2}+\\ \hspace{1em}\hspace{1em}{\beta }_{11}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{12}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{Y}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{13}{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{X}}^2}\right)}^2+{\beta }_{14}{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{Y}}^2}\right)}^2+{\beta }_{15}{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial \overline{Y}}\right)}^2\\ \hspace{1em}\hspace{1em}+{\beta }_{16}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{Y}}^2}+{\beta }_{17}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{18}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{Y}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{19}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{X}}^2}+\\ \hspace{1em}\hspace{1em}{\beta }_{20}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{Y}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{Y}}^2}+{\beta }_{21}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{Y}}^2}+{\beta }_{22}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{Y}}^2}+{\beta }_{23}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial \overline{Y}}+\\ \hspace{1em}\hspace{1em}{\beta }_{24}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial {\overline{Y}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{25}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{Y}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{26}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial {\overline{X}}^2}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial \overline{Y}}+{\beta }_{27}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial \overline{Y}}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial \overline{Y}}+\\ \hspace{1em}\hspace{1em}{\beta }_{28}\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}+{\beta }_{29}\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}+{\beta }_{30}\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}+{\beta }_{31}\frac{{\partial }^{2}W}{\partial {\overline{Y}}^2}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}+{\beta }_{32}\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}\\ \hspace{1em}\hspace{1em}+{\beta }_{33}\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}+{\beta }_{34}\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}+{\beta }_{35}\frac{{\partial }^{2}W}{\partial {\overline{X}}^2}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}+{\beta }_{36}\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}+{\beta }_{37}\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}\\ \hspace{1em}\hspace{1em}+{\beta }_{38}\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}+{\beta }_{39}\frac{{\partial }^{2}W}{\partial \overline{X}\partial \overline{Y}}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}+{\beta }_{40}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}+{\beta }_{41}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}+{\beta }_{42}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}\\ \hspace{1em}\hspace{1em}+{\beta }_{43}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}+{\beta }_{44}\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}+{\beta }_{45}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}+{\beta }_{46}{\left(\frac{\partial W}{\partial \overline{X}}\right)}^2+{\beta }_{47}{\left(\frac{\partial W}{\partial \overline{Y}}\right)}^2+\\ \hspace{1em}\hspace{1em}{\beta }_{48}{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{X}}\right)}^2+{\beta }_{49}{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial \overline{Y}}\right)}^2+{\beta }_{50}{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{X}}\right)}^2+{\beta }_{51}{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial \overline{Y}}\right)}^2+{\beta }_{52}\frac{\partial W}{\partial \overline{X}}\frac{\partial W}{\partial \overline{Y}}+{\beta }_{53}{\mathsf{\Phi }}_X \frac{\partial W}{\partial \overline{X}}\\ \hspace{1em}\hspace{1em}+{\beta }_{54}{\mathsf{\Phi }}_X \frac{\partial W}{\partial \overline{Y}}+{\beta }_{55}{\mathsf{\Phi }}_Y \frac{\partial W}{\partial \overline{X}}+{\beta }_{56}{\mathsf{\Phi }}_Y \frac{\partial W}{\partial \overline{Y}}+{\beta }_{57}\left({\mathsf{\Phi }}_X^2+{\mathsf{\Phi }}_Y^2\right)]\left|{\mathit{J}}_{\overline{X}\overline{Y}}\right|\mathrm{d}\overline{X}\mathrm{d}\overline{Y},\end{array}$$

(18)

where ${\beta }_{\mathrm{m}}$ is the coordinate transformation coefficient, as shown in Appendix A.

Based on Equations (7), (16) and (17), the kinetic energy for the gradient elastic thick plate element is rewritten as

$${\mathsf{\Pi }}_d={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1\left\{\begin{array}{l}{\alpha }_1{\left(\frac{\partial W}{\partial t}\right)}^2+{\alpha }_2\left[{\left(\frac{\partial {\mathsf{\Phi }}_X}{\partial t}\right)}^2+{\left(\frac{\partial {\mathsf{\Phi }}_Y}{\partial t}\right)}^2\right]+{\alpha }_3{\left(\frac{{\partial }^{2}W}{\partial \overline{X}\partial t}\right)}^2+\\ {\alpha }_4{\left(\frac{{\partial }^{2}W}{\partial \overline{Y}\partial t}\right)}^2+{\alpha }_5\frac{{\partial }^{2}W}{\partial \overline{X}\partial t}\frac{{\partial }^{2}W}{\partial \overline{Y}\partial t}+{\alpha }_6\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial t}\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{Y}\partial t}+\\ {\alpha }_6\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial t}\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{Y}\partial t}+{\alpha }_7\left[{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{X}\partial t}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{X}\partial t}\right)}^2\right]+\\ {\alpha }_8\left[{\left(\frac{{\partial }^2{\mathsf{\Phi }}_Y}{\partial \overline{Y}\partial t}\right)}^2+{\left(\frac{{\partial }^2{\mathsf{\Phi }}_X}{\partial \overline{Y}\partial t}\right)}^2\right]\end{array}\right\}\left|{\mathit{J}}_{\overline{X}\overline{Y}}\right|\mathrm{d}\overline{X}\mathrm{d}\overline{Y}}},$$

(19)

where

$$\begin{array}{c}{\alpha }_1=\frac{\rho h}{2}, {\alpha }_2=\frac{\rho h\left(h^2+12l_d^2 \right)}{24}, {\alpha }_3=\frac{\rho h{l}_d^2}{2} \left[{\left(\frac{\partial \overline{X}}{\partial X}\right)}^2+{\left(\frac{\partial \overline{X}}{\partial Y}\right)}^2\right],\hfill \\ {\alpha }_4=\frac{\rho h{l}_d^2}{2}\left[{\left(\frac{\partial \overline{Y}}{\partial X}\right)}^2+{\left(\frac{\partial \overline{Y}}{\partial Y}\right)}^2\right], {\alpha }_5=\rho h{l}_d^2\left(\frac{\partial \overline{X}}{\partial X}\frac{\partial \overline{Y}}{\partial X}+\frac{\partial \overline{X}}{\partial Y}\frac{\partial \overline{Y}}{\partial Y}\right),\hfill \\ {\alpha }_6=\frac{\rho h^3{l}_d^2}{12}\left(\frac{\partial \overline{X}}{\partial X}\frac{\partial \overline{Y}}{\partial X}+\frac{\partial \overline{X}}{\partial Y}\frac{\partial \overline{Y}}{\partial Y}\right), {\alpha }_7=\frac{\rho h^3{l}_d^2}{24}\left[{\left(\frac{\partial \overline{X}}{\partial X}\right)}^2+{\left(\frac{\partial \overline{X}}{\partial Y}\right)}^2\right],\hfill \\ {\alpha }_8=\frac{\rho h^3{l}_d^2}{24}\left[{\left(\frac{\partial \overline{Y}}{\partial X}\right)}^2+{\left(\frac{\partial \overline{Y}}{\partial Y}\right)}^2\right].\hfill \end{array}$$

(20)

Gauss–Lobatto (GL) quadrature points and weight coefficients in Figure 2 are given by

$${\overline{Y}}_1={\overline{X}}_1=-1, {\overline{Y}}_2={\overline{X}}_2=-1/\sqrt{5}, {\overline{X}}_3={\overline{Y}}_3=1/\sqrt{5}, {\overline{X}}_4={\overline{Y}}_4=1,$$

(21)

$$C_1^{\left(\overline{X}\right)}=C_1^{\left(\overline{Y}\right)}=C_4^{\left(\overline{X}\right)}=C_4^{\left(\overline{Y}\right)}=1/6, C_2^{\left(\overline{X}\right)}=C_2^{\left(\overline{Y}\right)}=C_3^{\left(\overline{X}\right)}=C_3^{\left(\overline{Y}\right)}=5/6.$$

(22)

Next, the Lagrange interpolation technique is used to obtain the trial functions of $W$,${\mathsf{\Phi }}_X$, and ${\mathsf{\Phi }}_Y$:

$$\Delta ={\displaystyle \sum _{i=1}^4{\displaystyle \sum _{j=1}^4{l}_{\overline{X}\left(i\right)}\left(\overline{X}\right)l_{\overline{Y}\left(j\right)}\left(\overline{Y}\right){\Delta }_{ij},}}$$

(23)

where ${\Delta }_{ij}$ represents the function value of $W$, ${\mathsf{\Phi }}_X$, or ${\mathsf{\Phi }}_Y$ at the ijth GL quadrature point; $l_{\overline{X}\left(i\right)}\left(\overline{X}\right)$ and $l_{\overline{Y}\left(j\right)}\left(\overline{Y}\right)$ are the Lagrange interpolation polynomials along the $\overline{X}$- and $\overline{Y}$-directions, respectively.

For a standard parent domain $\left[-1, 1\right]\times \left[-1, 1\right]$, the 1st and 2nd partial derivatives of $W$, ${\mathsf{\Phi }}_X$, and ${\mathsf{\Phi }}_Y$ at all GL quadrature points are expressed as follows:

$$\begin{array}{c}\left[\begin{array}{c}{\mathit{D}}_{\overline{X}}^{\left(1\right)}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{X}}^{\left(1\right)}{\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{X}}^{\left(1\right)}{\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right]={\mathit{A}}_{\overline{X}}^{\left(1\right)}\left[\begin{array}{c}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right], \left[\begin{array}{c}{\mathit{D}}_{\overline{Y}}^{\left(1\right)}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{Y}}^{\left(1\right)}{\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{Y}}^{\left(1\right)}{\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right]={\mathit{A}}_{\overline{Y}}^{\left(1\right)}\left[\begin{array}{c}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right], \left[\begin{array}{c}{\mathit{D}}_{\overline{X}}^{\left(2\right)}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{X}}^{\left(2\right)}{\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{X}}^{\left(2\right)}{\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right]={\mathit{A}}_{\overline{X}}^{\left(2\right)}\left[\begin{array}{c}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right],\\ \left[\begin{array}{c}{\mathit{D}}_{\overline{X}\overline{Y}}^{\left(1\oplus 1\right)}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{X}\overline{Y}}^{\left(1\oplus 1\right)}{\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{X}\overline{Y}}^{\left(1\oplus 1\right)}{\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right]={\mathit{A}}_{\overline{X}\overline{Y}}^{\left(1\oplus 1\right)}\left[\begin{array}{c}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right], \left[\begin{array}{c}{\mathit{D}}_{\overline{Y}}^{\left(2\right)}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{Y}}^{\left(2\right)}{\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathit{D}}_{\overline{Y}}^{\left(2\right)}{\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right]={\mathit{A}}_{\overline{Y}}^{\left(2\right)}\left[\begin{array}{c}{\mathit{W}}_{\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}\\ {\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}\end{array}\right],\end{array}$$

(24)

where ${\mathit{D}}_{\overline{X}\overline{Y}}^{\left(p\oplus q\right)}{\mathsf{\Delta }}_{\left(\mathrm{GL}\right)}$ is the following partial derivative matrix:

$$\begin{array}{cc}\hfill {\mathit{D}}_{\overline{X}\overline{Y}}^{\left(p\oplus q\right)}{\mathsf{\Delta }}_{\left(\mathrm{GL}\right)}=& [{\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{11}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{21}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{31}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{41},\hfill \\ \hfill & {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{12}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{22}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{32}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{42},\hfill \\ \hfill & {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{13}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{23}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{33}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{43},\hfill \\ \hfill & {{\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{14}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{24},{\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{34}, {\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{44}]}^{\mathrm{T}}\hfill \end{array}$$

(25)

with ${\left(\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}\right)}_{ij}$ denoting the function value of $\frac{{\partial }^{p+q}\Delta }{\partial {\overline{X}}^p\partial {\overline{Y}}^q}$ at the ijth GL quadrature point. ${\mathit{W}}_{\left(\mathrm{GL}\right)}$, ${\mathsf{\Phi }}_{X\left(\mathrm{GL}\right)}$, and ${\mathsf{\Phi }}_{Y\left(\mathrm{GL}\right)}$ are defined in the following vectors:

$${\Delta }_{\left(\mathrm{GL}\right)}={\left[{\Delta }_{11}, {\Delta }_{21}, {\Delta }_{31}, {\Delta }_{41}, {\Delta }_{12}, {\Delta }_{22}, {\Delta }_{32}, {\Delta }_{42}, {\Delta }_{13}, {\Delta }_{23}, {\Delta }_{33}, {\Delta }_{43}, {\Delta }_{14}, {\Delta }_{24}, {\Delta }_{34}, {\Delta }_{44}\right]}^{\mathrm{T}},$$

(26)

and ${\mathit{A}}_{\overline{X}}^{\left(1\right)}$, ${\mathit{A}}_{\overline{Y}}^{\left(1\right)}$, ${\mathit{A}}_{\overline{X}}^{\left(2\right)}$, ${\mathit{A}}_{\overline{X}\overline{Y}}^{\left(1\oplus 1\right)}$, and ${\mathit{A}}_{\overline{Y}}^{\left(2\right)}$ are 16 × 16 weight coefficient matrices, as detailed in [65].

Based on Equation (22), the following weigh coefficient matrix ${\mathit{C}}_{\left(\mathrm{GL}\right)}$ formed at all quadrature points is defined:

$${\mathit{C}}_{\left(\mathrm{GL}\right)}=diag\left(\left[1, 5, 5, 1, 5, 25, 25, 5, 5, 25, 25, 5, 1, 5, 5, 1\right]\right)/36,$$

(27)

The element stiffness and mass matrices and load vector are determined using the same discretization procedure as depicted in [66,70]. Based on the previous work in [57], we divide the clamped and simply supported boundaries into two types according to the normal curvature, i.e., single and double attributes.

Simply supported (S):

$$\mathrm{Sin}\mathrm{gle} \mathrm{attribute}: W={\mathsf{\Phi }}_s=0$$

(28)

$$\mathrm{Double} \mathrm{attribute}: W={\mathsf{\Phi }}_s=\frac{\partial {\mathsf{\Phi }}_n}{\partial n}=0$$

(29)

Clamped (C):

$$\mathrm{Sin}\mathrm{gle} \mathrm{attribute}: W={\mathsf{\Phi }}_s={\mathsf{\Phi }}_n=0$$

(30)

$$\mathrm{Double} \mathrm{attribute}: W={\mathsf{\Phi }}_s={\mathsf{\Phi }}_n=\frac{\partial {\mathsf{\Phi }}_n}{\partial n}=0$$

(31)

Free (F): No constraint.

4. Numerical Results and Discussion

4.1. Model Validation

The convergence and accuracy of our model are verified by some selective examples. The vibration frequencies predicted by the pb-2 Ritz method for thick macroplates and by the Navier method for thick microplates are used as benchmarks.

Figure 3 and Figure 4 illustrate three mesh densities for an annular sectorial plate and an equilateral triangular plate, respectively. For the annular sectorial case, ${\overline{\omega }}_n=\omega R_{\mathrm{out}}^2\sqrt{\rho h/D}$, inner radius $R_{\mathrm{in}}=0.5$, outer radius $R_{\mathrm{out}}=1.0$, $h=0.1$, sectorial angle $\alpha {=45}^{°}$, $\nu =0.3$, and $K_s=5/6$; for the equilateral triangular plate, ${\overline{\omega }}_n=\omega L^2\sqrt{\rho h/D}$, side length $L=1.0$, $h=0.1$, $\nu =0.3$, and $K_s={\pi }^2/12$. Note that $D=E{h}^3/\left[12\left(1-{\nu }^2\right)\right]$ is the flexural rigidity of the thin plate.

The six lowest dimensionless frequencies for two macroplates under three types of meshing are shown in

Table 1. The six lowest dimensionless frequencies for an annular sectorial macroplate with three types of meshing (single attribute).

Plate Type	Source	Dimensionless Frequency
		${\overline{\mathit{\omega }}}_1$	${\overline{\mathit{\omega }}}_2$	${\overline{\mathit{\omega }}}_3$	${\overline{\mathit{\omega }}}_4$	${\overline{\mathit{\omega }}}_5$	${\overline{\mathit{\omega }}}_6$
	Mesh I	81.6282	131.5274	170.5684	204.4813	213.2321	273.2040 (19.7‰)
	Mesh II	83.3317	133.4009	173.9067	207.6397	217.9088	278.5504 (0.54‰)
	Mesh III	83.3762	133.5073	173.9903	207.7985	218.1862	278.6688 (0.11‰)
	Ref. [71]	83.39	133.5	174.0	207.8	218.2	278.7 (0.00‰)
	ABAQUS	83.4149	133.6469	174.2138	208.1828	218.3593	278.9532
	Mesh I	79.7667	139.6404	163.5869	212.3000	215.0767	267.0375 (13.5‰)
	Mesh II	80.7754	142.1242	165.5137	217.2101	218.9190	270.4049 (1.10‰)
	Mesh III	80.9566	142.2275	165.6100	217.3999	219.1845	270.5344 (0.61‰)
	Ref. [71]	81.01	142.2	165.8	217.4	219.3	270.7 (0.00‰)
	ABAQUS	81.1477	142.8058	166.1301	217.8956	220.0316	270.9513

and

Table 2. The six lowest dimensionless frequencies for an equilateral triangular macroplate with three types of meshing (single attribute).

Plate Type	Source	Dimensionless Frequency
		${\overline{\mathit{\omega }}}_1$	${\overline{\mathit{\omega }}}_2$	${\overline{\mathit{\omega }}}_3$	${\overline{\mathit{\omega }}}_4$	${\overline{\mathit{\omega }}}_5$	${\overline{\mathit{\omega }}}_6$
	Mesh I	77.6961	132.1974	132.1974	188.2262	196.6173	196.6173 (1.94‰)
	Mesh II	77.7317	132.2817	132.2817	188.3909	196.8075	196.8075 (0.98‰)
	Mesh III	77.7419	132.3057	132.3057	188.4393	196.8628	196.8628 (0.69‰)
	Ref. [71]	77.79	132.3	132.3	188.6	197.0	197.0 (0‰)
	ABAQUS	77.9299	132.7202	132.7202	189.1298	197.6068	197.6068
	Mesh I	8.6418	31.3613	34.7885	75.3018	76.0219	86.9152 (1.20‰)
	Mesh II	8.6433	31.3884	34.7981	75.3645	76.0880	86.9775 (0.49‰)
	Mesh III	8.6438	31.3968	34.8010	75.3841	76.1084	86.9971 (0.26‰)
	Ref. [71]	8.646	31.41	34.81	75.40	76.15	87.02 (0.00‰)
	ABAQUS	8.6460	31.4342	34.8360	75.5201	76.2471	87.1583

. It is expected that the predicted frequency parameters converge to their counterparts using the pb-2 Ritz method [71] and ABAQUS (8-node curved thick shell element) as mesh density increases. The use of Mesh III can produce good accuracy since the maximum relative error between the proposed method and the reported results in [71] is less than 1‰. However, it can be observed that the change in the boundary conditions affects the convergence error. To further validate the effectiveness of our method, we provide the six lowest vibration mode shapes in the form of deflection contour plots for two macroplates (Figure 5 and Figure 6). As expected, the present contour plots are in agreement with those obtained by ABAQUS.

Table 3. The eight lowest dimensionless frequencies for a simply supported square microplate with different mesh densities (single attribute).

Mode	Mesh Density						Navier Method
	$4\times 4$	$8\times 8$	$12\times 12$	$16\times 16$	$20\times 20$	$24\times 24$
${\overline{\omega }}_1$	1.7737	1.7747	1.7747	1.7747	1.7747	1.7747	1.7747
${\overline{\omega }}_2$	3.9638	3.9502	3.9488	3.9485	3.9485	3.9484	3.9484
${\overline{\omega }}_3$	6.0028	5.9862	5.9839	5.9833	5.9831	5.9830	5.9830
${\overline{\omega }}_4$	7.1447	7.0213	7.0113	7.0094	7.0088	7.0086	7.0085
${\overline{\omega }}_5$	9.0701	8.9722	8.9628	8.9607	8.9601	8.9598	8.9595
${\overline{\omega }}_6$	11.7753	10.8629	10.8185	10.8101	10.8077	10.8068	10.8063
${\overline{\omega }}_7$	12.0071	11.8812	11.8680	11.8649	11.8638	11.8634	11.8630
${\overline{\omega }}_8$	13.5871	12.7407	12.7007	12.6928	12.6904	12.6895	12.6883

presents the eight lowest dimensionless frequencies of a square epoxy resin microplate with $L/h=10$, $\mathrm{E}=1.44 \mathrm{GPa}$, $\rho =1220 \mathrm{kg}/{\mathrm{m}}^3$, $\nu =0.38$, $l_s/h=1$, and $l_d/h=1$. It is noted that numerical and analytical frequencies can achieve consistency with the increase in the mesh density.

The variations in the logarithms of the 1-norms of reduced stiffness and mass matrices (after imposing essential boundary conditions) against strain gradient and inertia gradient parameters are illustrated in Figure 7 and Figure 8, respectively. With the increase in gradient parameters, ${\log }_{10}\left(\mathrm{Cond}\left(\mathit{K}, 1\right)\right)$ and ${\log }_{10}\left(\mathrm{Cond}\left(\mathit{M}, 1\right)\right)$ both decrease. The flattened curves (see Figure 7 and Figure 8) indicate that the increasing gradient parameters can improve the convergence of the elements.

Table 4. Comparison of dimensionless fundamental frequency for a square nanoplate with different gradient parameters (single attribute).

Plate Type	${\mathit{l}}_{\mathit{d}}$	${\mathit{l}}_{\mathit{s}}$
		0.2 nm		1.0 nm
		Ref. [68]	Present	Ref. [68]	Present
SFSF	0.0 nm	0.9774	0.9802	0.9878	0.9920
	0.2 nm	0.9735	0.9761	0.9839	0.9864
	0.5 nm	0.9539	0.9553	0.9640	0.9688
	1.0 nm	0.8922	0.8945	0.9017	0.9079
SCSF	0.0 nm	3.6907	3.6967	4.6619	4.6668
	0.2 nm	3.6556	3.6620	4.6160	4.6198
	0.5 nm	3.4858	3.4887	4.3949	4.3968
	1.0 nm	3.0268	3.0302	3.8032	3.8046

lists the dimensionless fundamental frequencies presented by the present gradient elastic Mindlin plate model and the available gradient elastic Kirchhoff plate model in [68]. For comparison, the plate dimensions are $h=0.34 \mathrm{nm}$ and $L_X=L_Y=10 \mathrm{nm}$. It can be seen that the present predictions are consistent with those in the literature when the plate is very thin.

4.2. Parameter Settings

The solution method is then used to analyze the free vibration of gradient elastic annular sectorial and triangular microplates made of epoxy resin. Here, the material property parameters are assumed as [31] $E=1.44\mathrm{GPa}$, $\rho =1220\mathrm{kg}/{\mathrm{m}}^2$, $K_s=5/6$, and $\nu =0.38$.

Table 5. The six lowest dimensionless frequencies for an annular sectorial microplate with different boundary conditions and gradient parameters (single attribute).

Plate Type	$\left({\overline{\mathit{l}}}_{\mathit{s}}, {\overline{\mathit{l}}}_{\mathit{d}}\right)$	Dimensionless Frequency
		${\overline{\mathit{\omega }}}_1$	${\overline{\mathit{\omega }}}_2$	${\overline{\mathit{\omega }}}_3$	${\overline{\mathit{\omega }}}_4$	${\overline{\mathit{\omega }}}_5$	${\overline{\mathit{\omega }}}_6$
	(0, 0)	81.5207	130.4961	168.5017	202.1629	211.4280	268.4294
	(0.5, 0)	95.3423	153.3994	207.0074	256.5577	272.0939	364.4674
	(0, 0.5)	72.2138	104.7030	131.3718	146.1192	152.6934	182.6297
	(1, 0)	124.5375	203.0303	281.0181	366.8073	393.5932	534.6659
		1.5277	1.5558	1.6677	1.8144	1.8616	1.9918
	(0, 1)	55.9479	72.3129	88.2711	93.1260	97.2287	110.6023
		0.6863	0.5541	0.5239	0.4606	0.4599	0.4120
	(1, 1)	87.4697	112.4565	143.7237	165.9530	178.1079	203.0451
	(0, 0)	74.2015	138.7872	149.6016	208.7868	211.5255	251.4064
	(0.5, 0)	86.4273	165.6917	178.8613	266.6957	273.2989	334.9031
	(0, 0.5)	65.5494	111.5257	116.0503	150.6620	153.5407	169.7088
	(1, 0)	111.5916	220.7434	239.4153	380.8189	393.0436	493.3882
		1.5039	1.5905	1.6004	1.8240	1.8581	1.9625
	(0, 1)	50.6369	76.9966	77.9073	95.9221	98.1906	104.1261
		0.6824	0.5548	0.5208	0.4642	0.4594	0.4142
	(1, 1)	76.1093	121.2415	125.4321	170.5555	175.3253	194.4725
	(0, 0)	79.2224	138.5925	161.0288	210.7184	212.4034	261.4489
	(0.5, 0)	93.0171	165.6100	195.9620	270.9050	275.0648	353.2054
	(0, 0.5)	69.8781	111.4121	125.3498	152.8662	153.3883	177.4652
	(1, 0)	122.1085	221.8823	266.1490	387.9548	399.3468	519.6088
		1.5413	1.6010	1.6528	1.8411	1.8801	1.9874
	(0, 1)	53.7549	76.9914	84.3009	97.4848	97.8276	109.4238
		0.6785	0.5555	0.5235	0.4626	0.4606	0.4185
	(1, 1)	84.5940	124.0212	138.3856	174.6448	181.5263	198.7258

and

Table 6. The six lowest dimensionless frequencies for an equilateral triangular microplate with different boundary conditions and gradient parameters (single attribute).

Plate Type	$\left({\overline{\mathit{l}}}_{\mathit{s}}, {\overline{\mathit{l}}}_{\mathit{d}}\right)$	Dimensionless Frequency
		${\overline{\mathit{\omega }}}_1$	${\overline{\mathit{\omega }}}_2$	${\overline{\mathit{\omega }}}_3$	${\overline{\mathit{\omega }}}_4$	${\overline{\mathit{\omega }}}_5$	${\overline{\mathit{\omega }}}_6$
	(0, 0)	76.2249	128.9764	128.9764	183.0821	191.1030	191.1030
	(0.5, 0)	84.1899	152.6676	152.6676	228.2820	238.8048	238.8048
	(0, 0.5)	68.9814	106.9641	106.9641	140.7625	143.9840	143.9840
	(1, 0)	101.3952	200.9779	200.9779	317.4252	331.8159	331.8159
		1.3302	1.5583	1.5583	1.7338	1.7363	1.7363
	(0, 1)	55.0620	76.0369	76.0369	94.0234	94.1282	94.1282
		0.7224	0.5895	0.5895	0.5136	0.4926	0.4926
	(1, 1)	75.4742	120.1357	120.6292	158.4080	158.7861	165.5730
	(0, 0)	66.3187	119.6351	119.8490	174.2392	183.0867	183.8417
	(0.5, 0)	72.0779	138.4393	140.1666	213.9839	225.1839	227.1890
	(0, 0.5)	59.7884	98.9798	99.1781	133.2975	137.7091	138.4922
	(1, 0)	84.7178	176.6627	182.8545	292.5272	308.8696	313.9850
		1.2774	1.4767	1.5257	1.6789	1.6870	1.7079
	(0, 1)	47.5424	70.3561	70.5703	88.5408	90.0805	90.7414
		0.7169	0.5881	0.5888	0.5082	0.4920	0.4936
	(1, 1)	61.6449	104.5859	108.3691	147.1023	148.1694	153.0898
	(0, 0)	8.4766	30.0886	33.8771	72.8753	73.1805	85.1658
	(0.5, 0)	11.6780	32.8524	38.9259	84.5572	84.8922	94.7538
	(0, 0.5)	8.2626	25.6136	30.7860	56.8479	60.8161	67.2185
	(1, 0)	16.5950	37.9233	48.0417	104.3431	105.8601	114.3535
		1.9577	1.2604	1.4181	1.4318	1.4466	1.3427
	(0, 1)	7.7009	18.7560	24.9306	38.7446	42.6922	46.7500
		0.9085	0.6234	0.7359	0.5317	0.5834	0.5489
	(1, 1)	15.2310	23.1984	34.8584	54.2883	55.8119	67.2840

summarize the six lowest dimensionless frequencies for an annular sectorial microplate and an equilateral triangular microplate, where ${\overline{l}}_s=l_s/h$, ${\overline{l}}_d=l_d/h$, $R_{\mathrm{out}}/h=10$, and $L/h=10$, and the underlined values in Table 5 are the ratios of the frequency for the case of ${\overline{l}}_s=1$ (or ${\overline{l}}_d=1$) to the frequency for the case of ${\overline{l}}_s={\overline{l}}_d=0$. As expected, an increased ${\overline{l}}_s$ or a decreased ${\overline{l}}_d$ can lead to the increasing vibration frequencies of microplates, especially for the higher-order modes. This is because the strain gradient and inertia gradient play roles in enhancing the structural bending rigidity and inertia. By comparing the underlined values in Table 6, we observe that strain gradient and inertia gradient effects become significant when increasing the order of the vibration mode, but they are slightly affected by the boundary conditions. However, variations in the frequency ratio against the order of the vibration mode are different between equilateral triangular and annular sectorial microplates. In Table 5 and Table 6, the boundary condition has a more remarkable impact on the frequency ratio in the equilateral triangular case than it does in the annular sectorial case. By comparing the results shown in bold (Table 5 and Table 6) between the cases of ${\overline{l}}_s={\overline{l}}_d=0$ and ${\overline{l}}_s={\overline{l}}_d=1$, we find that the inertia gradient has a greater influence on the frequencies than the strain gradient for the second vibration mode.

The six lowest vibration mode shapes of an annular sectorial microplate with SSCC edges and an equilateral triangular microplate with SCC edges are illustrated in Figure 9 and Figure 10, respectively. In the figures, we observe that the first mode shape is almost unaffected by gradient parameters, while others are not. It is indicated that the introduced gradient effects change the vibration mode shapes, along with the frequency values. Similar observations have been previously reported in [57], where the authors show the four lowest eigenmodes of square and annular gradient elastic microplates with in-plane free vibrations.

Figure 11 presents the 7th to 12th vibration mode shapes and the related frequencies of an annular sectorial microplate in terms of two outer radius-to-thickness ratios and two sets of gradient parameters, respectively. A comparison between Cases 1 and 3 (or Cases 2 and 4) shows that the transverse shear deformation can considerably change the higher-order vibration frequencies and mode shapes. The comprehensive effect of the strain gradient and inertia gradient causes a decrease in the vibration frequencies of the thick microplate. Moreover, the size dependence of vibration mode shapes is also shown by comparing Case 1 with Case 2 (or Case 3 with 4).

In addition, another comparative test has been conducted to confirm whether the normal curvature has an influence on the vibration mode shapes of an annular sectorial microplate with SSCC edges. As shown in Figure 12, considering the normal curvature, there are significant changes in the vibration mode shapes and increasing vibrational frequencies, especially for higher-order modes.

5. Conclusions

In this article, we formulate the equations of motion and appropriate boundary conditions for Mindlin microplates with arbitrary shapes based on the negative second-order gradient elastic theory and the energy variational principle. A C¹-type four-node DQFE is proposed to analyze the resulting higher-order boundary value problems. Then, the DQ rule is properly used to convert the equation from natural coordinates into Cartesian coordinate systems in the straight-sided quadrilateral domain. The element stiffness matrix, mass matrix, and load vector are derived using the minimum-potential-energy principle. The effectiveness of our theoretical model and solution method is demonstrated in comparison to other methods. Finally, the free vibration of annular sectorial and triangular microplates is analyzed using the new solution method. Several interesting points are observed as follows.

(1) The element convergence can be improved by increasing two types of gradient effects.

(2) The strain gradient and inertia gradient can cause the frequency stiffening and softening effects of microplates, respectively.

(3) The vibration mode shapes can be changed to a certain extent by introducing two types of gradient effects.

(4) The high-order vibration frequencies and mode shapes are more sensitive to boundary conditions, transverse shear deformation, and gradient parameters.