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

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.

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 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

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: 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: 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 where 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: 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: 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 $t\in \left(t_1, t_2\right)$:

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.