A Symplectic Method for Analyzing the Nonlocal Modal Behavior of Kirchhoff Plates and Numerical Validation

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This paper proposes a symplectic-based numerical solver to account for nonlocal effects in the modeling and design of MEMS/NEMS nanoplates. Eringen’s integral constitutive relation serves as the theoretical foundation for this solver, which supports custom kernel functions and mixture parameters. The resulting finite element implementation provides a numerical framework for nonlocal free vibration analysis of Kirchhoff plates. The numerical examples show that the method captures stiffness softening and frequency shifts associated with size effects in nanostructures. This formulation is relevant for the structural design and frequency calibration of micro- and nano-devices, such as nanomechanical resonators, mass sensors, and radio frequency (RF) filters.

Abstract

Eringen’s integral constitutive relation is more general than its differential counterpart for modeling small-scale effects in micro- and nanostructures; however, it leads to integro-differential governing equations that are difficult to solve, which has limited the practical use of integral formulations. To directly address this gap, this paper introduces a novel symplectic-based numerical method that efficiently and accurately analyzes the free vibration of small-scale Kirchhoff plates governed by Eringen’s integral nonlocal model. The method discretizes the nonlocal integral operator by introducing inter-belt elements for long-range interactions and adopting a truncated influence domain, while balancing computational efficiency and accuracy. The effects of the nonlocal parameter, two-phase mixture parameter, mode numbers, kernel types, and geometric parameters on the natural frequencies are systematically investigated. The results indicate stiffness softening. For a simply supported square nanoplate with side length $a=10 \mathrm{n}\mathrm{m}$, the first-order frequency parameter decreases by approximately 25% as the nonlocal parameter increases from 0 to 4 nm, and higher-order modes exhibit substantially greater sensitivity to nonlocal effects. Convergence and accuracy are validated against published continuum-level solutions and molecular dynamics simulations; relative deviations are below 2% in most cases, and the local limit $\left(l_a=0\right)$ yields errors on the order of ${10}^{-3}$.

1. Introduction

MEMS/NEMS nanoplates are being adopted rapidly across engineering applications. Examples range from ultra-sensitive mass sensors and high-frequency RF filters to biomedical devices such as single-molecule detection chips and DNA sequencing platforms. This trend is driven in part by the high surface-area-to-volume ratio and favorable mechanical properties of nanoscale structures. Common nanostructures include nanorods, nanobeams, and nanoplates. At the nanoscale, thermal, electrical, mechanical, and chemical properties can deviate markedly from their macroscopic counterparts [1,2,3,4], creating a need for dedicated modeling and characterization approaches. The behavior of nanostructures can be studied by experiments [5], molecular dynamics (MD) simulations [6], and non-classical continuum mechanics methods [7]. Because experiments and MD simulations can be costly, non-classical continuum models are often used to predict the mechanical response of nanostructures.

A widely adopted non-classical continuum mechanics theory was proposed by Eringen and colleagues [8,9]. In this theory, the stress at a reference point depends on the strain at that point and the strains at other points in the medium. A kernel function is introduced to characterize the attenuation of long-range interactions as a function of distance. Eringen’s nonlocal elasticity theory includes both integral and differential constitutive relations. Because integral constitutive equations account for global interactions, they typically yield integro-differential equations that are computationally demanding; as a result, differential constitutive equations have often been preferred in nonlocal studies [10,11,12,13]. However, differential-type constitutive equations are derived under restrictive assumptions, and their applicability is therefore limited [14]. Since Eringen’s theory was originally founded on an integral constitutive relation, the differential type generally lacks the generality of the integral type.

Integral constitutive relations have been successfully applied to problems of nonlocal Euler–Bernoulli beams, resolving issues arising from the use of differential constitutive relations [15,16,17,18]. Compared with one-dimensional structures, nanoplate structures are more complex, and their governing equations are more challenging to handle. Consequently, irrespective of the constitutive relation employed, research on the free vibration of nanoplates is relatively scarce. Some representative studies are briefly reviewed below.

Lu et al. proposed Kirchhoff and Mindlin plate element models based on the nonlocal theory [19]. In the subsequent analysis of nonlocal plate vibration, researchers have adopted a wide range of numerical and analytical solution methods. These include the Rayleigh–Ritz Method [20,21], the Finite Element Method (FEM) [22,23,24], the Finite Strip Method [25,26], the Galerkin Method [27,28], the Differential Quadrature (DQ) Method [29,30,31,32], the meshless kp-Ritz Method [33], the Chebyshev Collocation Method [34], the Galerkin Strip Distributed Transfer Function Method [35], and the Discrete Singular Convolution Method [36,37].

Numerical methods are of paramount importance in the analysis of the mechanical behavior of nonlocal plates. Inter-belt analysis, a numerical method, was first proposed by Zhong and co-workers [38,39]. Conventional local finite element methods generally consider the connection between elements, or between an element and the external environment, as the displacement at their interface, where the interface is assumed to have no thickness. However, in the context of nonlocal theory, the influence of long-range interactions between elements must be taken into account. In this specific context, the connection zone between two non-adjacent yet mutually influencing elements is no longer a surface but a strip of elements. Since its boundary is no longer a plane but has a finite width, it is termed an inter-belt. Since its proposal, this analysis method has been applied to the nonlocal analysis of carbon nanotubes [40] and subsequently verified for nonlocal rods, Euler–Bernoulli beams, Timoshenko beams, and plane-strain problems [41,42,43]. However, all existing inter-belt studies have been confined to one-dimensional structures or two-dimensional in-plane problems; extension of the formulation to plate bending, which involves the curvature tensor and requires a compatible Kirchhoff plate element, has not yet been attempted.

The present work addresses this gap by extending the inter-belt method, for the first time, to the free vibration analysis of two-dimensional nonlocal Kirchhoff plates. Within the Eringen two-phase framework, a nonlocal element coupling stiffness matrix is derived for the Kirchhoff plate element using a center-point approximation. A systematic comparison between the exponential and Gaussian kernel functions is conducted, and the optimal two-phase mixture parameter is independently calibrated for each kernel. The computed frequencies are verified against both published continuum-level solutions [10,22] and molecular dynamics simulations of single-layered graphene sheets [4], thereby providing cross-scale validation that was absent in prior inter-belt studies. Furthermore, the effects of nonlocal parameters, two-phase mixture parameters, mode numbers, kernel functions, and geometric parameters on vibration frequencies under nonlocal conditions are examined, and the advantages of this algorithm in handling integro-differential equations are demonstrated.

Section 2 reviews the fundamental form of Eringen’s nonlocal elasticity theory and discusses the methodology for handling integral constitutive relations. Section 3 presents the nonlocal finite element formulation for Kirchhoff plates. Section 4 provides numerical results and a comprehensive parametric study. The findings are discussed in Section 5, and conclusions are drawn in Section 6. The nomenclature used throughout this paper is summarized in Appendix A.

2. Nonlocal Elasticity Theory

2.1. Eringen’s Nonlocal Elasticity Theory

In accordance with the nonlocal elasticity theory proposed by Eringen [8,9], the stress at a point $\mathit{x}$ in a continuum is determined not only by the strain at that point, but also by the strains at all other points ${\mathit{x}}^{\prime }$ within the body. A constitutive relation in integral form describes this nonlocal effect. In the case of a homogeneous, isotropic linear elastic solid, the constitutive equation can be expressed as

$$t_{kl}(x)={\int }_V\alpha \left(\right|{\mathit{x}}^{\prime }-\mathit{x}|,\kappa ){\sigma }_{kl}({\mathit{x}}^{\prime })\mathrm{d}v({\mathit{x}}^{\prime }),$$

(1)

where $t_{kl}$ is the nonlocal stress tensor; ${\sigma }_{kl}$ is the classical local macroscopic stress tensor, given by the generalized Hooke’s law.

$${\sigma }_{kl}\left({\mathit{x}}^{\prime }\right)=\lambda e_{rr}\left({\mathit{x}}^{\prime }\right){\delta }_{kl}+2\mu e_{kl}\left({\mathit{x}}^{\prime }\right),$$

(2)

where $\lambda$ and $\mu$ are Lamé constants, $e_{kl}$ is the linear strain tensor, and ${\delta }_{kl}$ is the Kronecker delta.

The equation of motion reads

$$t_{kl,k}+\rho (f_l-{\ddot{u}}_l)=0.$$

(3)

The kernel function $\alpha \left(\right|{\mathit{x}}^{\prime }-\mathit{x}|,\kappa )$ describes the attenuation characteristics of nonlocal interactions, where $\kappa =e_0{a}_0/l_0$ is the dimensionless nonlocal parameter, $a_0$ is the internal characteristic length (e.g., lattice constant), $l_0$ is the external characteristic length (e.g., wavelength or crack length), and $e_0$ is a constant determined by material properties.

Although the integral constitutive relation can be transformed into a differential form for specific kernel functions, the integral form is more broadly applicable, as it can accommodate more complex boundary conditions and physical phenomena [14,44].

2.2. Inter-Belt Analysis Theory

The inter-belt analysis method is a numerical approach developed based on symplectic mathematical theory and computational structural mechanics. In the Hamiltonian description of conservative systems, symplectic (structure-preserving) algorithms are designed to retain the key geometric structure and thereby provide improved long-time stability; in this sense, symplecticity is often interpreted as preserving the intrinsic structure of conservative systems in computation [39,40]. Within this Hamiltonian framework, structural mechanics is reformulated in terms of generalized displacements and their energy-conjugate dual variables as the state vector, with the governing equations derived from the stationarity of the energy functional. The inter-belt method is grounded in this symplectic–Hamiltonian framework, employing it as the organizational basis for discretizing nonlocal interactions. In the present study, this viewpoint is used primarily to define energy-conjugate state variables and organize the discretization of long-range coupling; the resulting free-vibration computation ultimately reduces to assembling the finite element matrices and solving a generalized eigenvalue problem, rather than performing a symplectic time-marching integration. In traditional finite element theory, the connection between elements is generally treated as an interface of zero thickness, for which only $C^0$ continuity of displacement is required. However, in the context of nonlocal elasticity theory, the presence of long-range forces extends the interaction between material points beyond immediately adjacent elements, encompassing a finite region termed the influence domain. Consequently, the boundary between interacting elements is no longer an idealized geometric surface, but rather a belt-like region of finite width— hence the designation inter-belt.

Within the integral nonlocal constitutive framework [41], the inter-belt analysis method has been demonstrated to be an effective approach for numerically treating the governing integro-differential equations through systematic discretization. Specifically, the total deformation energy associated with nonlocal interactions is decomposed into a sum of contributions from a finite set of elemental substructures.

Upon discretizing the nonlocal interactions within the influence domain, the nonlocal global stiffness matrix is no longer a banded, sparse matrix as in traditional finite elements. Rather, it is a generalized stiffness matrix that includes coupling terms between non-adjacent nodes. The construction of an element stiffness matrix that incorporates inter-belt effects facilitates the transformation of complex integro-differential dynamic equations into standard linear algebraic eigenvalue problems, thereby enabling efficient computation of the modal frequencies and dynamic response of nonlocal plate structures.

3. Kirchhoff Plate Element and Nonlocal Finite Element Formulation

In Section 2, the fundamental form of Eringen’s nonlocal elasticity theory was reviewed, and the fundamentals of inter-belt analysis in the discretization of integral nonlocal constitutive relations were introduced. On this basis, the present section commences with a concise review of the classical Kirchhoff plate element and its local finite element formulation. Subsequently, Eringen’s two-phase nonlocal theory is applied to the Kirchhoff plate bending problem. The nonlocal strain energy expression based on the center-point approximation is presented, and the corresponding nonlocal element stiffness matrix form is derived, providing a theoretical basis for subsequent nonlocal inter-belt analysis, discretization, and numerical examples.

3.1. Classical Kirchhoff Plate Bending Theory and Local Finite Element Formulation

The kinetic energy expression of a classical local Kirchhoff plate is

$$T_e={\displaystyle \frac{1}{2}}{\dot{\mathit{q}}}_e{\left(t\right)}^{\mathrm{T}}\mathit{m}{\dot{\mathit{q}}}_e\left(t\right),$$

(4)

where $\mathit{m}$ is the mass matrix of the plate element, and ${\mathit{q}}_e$ is the element degree-of-freedom vector.

$$\mathit{m}=\rho h{\int }_{-b}^b{\int }_{-c}^c{\mathit{N}}_e{\left(z\right)}^{\mathrm{T}}{\mathit{N}}_e\left(z\right)d\zeta d\eta ,$$

(5)

Assembling the local element stiffness matrices ${\mathit{k}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ yields the local global stiffness matrix ${\mathit{K}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$. Similarly, assembling the local element mass matrices $\mathit{m}$ yields the global mass matrix, $\mathit{M}$. The assembled free-vibration equation reads

$$\mathit{M}\ddot{\mathit{q}}+{\mathit{K}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}\mathit{q}=0.$$

(6)

The eigenvalue equation corresponding to Equation (6) is

$$\mathrm{d}\mathrm{e}\mathrm{t}\left({\widehat{\mathit{K}}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}-{\omega }^2\widehat{\mathit{M}}\right)=0,$$

(7)

where ${\widehat{\mathit{K}}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ and $\widehat{\mathit{M}}$ denote the reduced matrices after imposing boundary conditions. The natural frequencies of the local Kirchhoff plates can be obtained by solving the above equation [45]. Boundary conditions are imposed by eliminating rows and columns from the stiffness matrix, and the eigenvalue problem is solved using MATLAB_R2023b.

Under the classical local Kirchhoff plate theory, generalized stress resultants are introduced [46]

$$\mathit{F}={\mathit{D}}_0\mathit{\zeta }={\displaystyle \frac{E{h}^3}{12(1-{\nu }^2)}}\left[\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& {\displaystyle \frac{1-\nu }{2}}\end{array}\right]\left[\begin{array}{c}{\displaystyle \frac{\partial {\theta }_y}{\partial x}}\\ -{\displaystyle \frac{\partial {\theta }_x}{\partial y}}\\ -({\displaystyle \frac{\partial {\theta }_y}{\partial y}}+{\displaystyle \frac{\partial {\theta }_x}{\partial x}})\end{array}\right].$$

(8)

The local Kirchhoff plate element stiffness matrix can be obtained through the principle of minimum potential energy as

$${\mathit{k}}_{local}={\int }_{A_E}{\mathit{B}}^{\mathrm{T}}(x,y) {\mathit{D}}_0 \mathit{B}(x,y) \mathrm{d}A,$$

(9)

where $\mathit{B}(x,y)\in {\mathbb{R}}^{3\times 12}$ is the geometric matrix of the Kirchhoff plate element. The above theory provides a reference local model for the subsequent introduction of nonlocal effects.

3.2. Nonlocal Stiffness Derivation Based on the Two-Phase Model

To characterize the long-range interactions at the nanoscale, the Eringen nonlocal model introduced in Section 2.1 is applied to the Kirchhoff plate bending problem. For a rectangular plate divided into several standard Kirchhoff plate elements, let one non-boundary element be $E0$, with the origin of the local coordinate system placed at the centroid of the element. The standard element has dimensions $2b$ and $2c$ in the $x$ and $y$ directions, respectively, and thickness $h$. Within the framework of the generalized stress–strain relationship, the nonlocal strain energy of element $E0$ is written as

$$U_{nonlocal}={\displaystyle \frac{1}{2}}{\int }_{A_{E0}}{\mathit{\zeta }}^{\mathrm{T}}\left(x,y\right)\left[{\mathit{D}}_0{\int }_{A_{plate}}\alpha \left(s,\kappa \right) \mathit{\zeta }\left(x^{\prime },y^{\prime }\right) \mathrm{d}{A}_{plate}\right]\mathrm{d}{A}_{E0},$$

(10)

where $A_{plate}$ is the area of the middle surface of the entire plate, $s=\sqrt{(x^{\prime }-x{)}^2+(y^{\prime }-y{)}^2}$ is the two-dimensional Euclidean distance between two integration points (which, after the center-point approximation introduced below, reduces to the centroid-to-centroid distance between two elements). $\alpha (s,\kappa )$ is a nonlocal kernel function that satisfies normalization and monotonic decay conditions, and ${\mathit{D}}_0$ is the bending stiffness matrix defined in the previous subsection. Discretizing the global integral by elements yields

$${\mathit{D}}_0{\int }_{A_{plate}}\alpha (s,\kappa ) \mathit{\zeta }(x^{\prime },y^{\prime }) \mathrm{d}{A}_{plate}={\mathit{D}}_0\sum _{k=0}^n{\int }_{A_{Ek}}{\alpha }_k \mathit{\zeta }(x^{\prime },y^{\prime }) \mathrm{d}{A}_{Ek},$$

(11)

where $A_{Ek}$ is the area of the $k$-th element. ${\alpha }_k$ is the value of the kernel function $\alpha (s,\kappa )$ evaluated under a fixed $\kappa$, reflecting the strength of long-range interaction between elements. To further simplify, the generalized strain integral over each element is approximated by a midpoint quadrature rule, evaluating the strain at the geometric center of the element and multiplying by the element area.

$${\int }_{A_{Ek}}\mathit{\zeta }(x^{\prime },y^{\prime }) \mathrm{d}{A}_{Ek}\approx {\mathit{\zeta }}_{Ek} A_{Ek},$$

(12)

where ${\mathit{\zeta }}_{Ek}$ is the generalized strain at the geometric center point of element $Ek$. For a standard Kirchhoff rectangular element, it can be expressed through the center point geometric operator $\mathit{T}$,

$${\mathit{\zeta }}_{Ek}=\mathit{B}(0,0) {\mathit{q}}_{Ek}=\mathit{T}{\mathit{q}}_{Ek},$$

(13)

where $\mathit{B}(0,0)$ is the geometric matrix at the origin of the element’s natural coordinates. For a standard element of size $2b\times 2c$, $\mathit{T}$ is explicitly written as

$$\mathit{T}=\mathit{B}\left(0,0\right)=\left[\begin{array}{cccccccccccc}0& 0& {\displaystyle \frac{1}{4b}}& 0& 0& -{\displaystyle \frac{1}{4b}}& 0& 0& -{\displaystyle \frac{1}{4b}}& 0& 0& {\displaystyle \frac{1}{4b}}\\ 0& -{\displaystyle \frac{1}{4c}}& 0& 0& -{\displaystyle \frac{1}{4c}}& 0& 0& {\displaystyle \frac{1}{4c}}& 0& 0& {\displaystyle \frac{1}{4c}}& 0\\ -{\displaystyle \frac{1}{bc}}& {\displaystyle \frac{1}{4b}}& -{\displaystyle \frac{1}{4c}}& {\displaystyle \frac{1}{bc}}& -{\displaystyle \frac{1}{4b}}& -{\displaystyle \frac{1}{4c}}& -{\displaystyle \frac{1}{bc}}& -{\displaystyle \frac{1}{4b}}& {\displaystyle \frac{1}{4c}}& {\displaystyle \frac{1}{bc}}& {\displaystyle \frac{1}{4b}}& {\displaystyle \frac{1}{4c}}\end{array}\right].$$

(14)

Substituting Equations (12) and (13) into Equation (11) yields

$${\mathit{D}}_0{\int }_{A_{plate}}\alpha (s,\kappa ) \mathit{\zeta }(x^{\prime },y^{\prime }) \mathrm{d}{A}_{plate}\approx {\mathit{D}}_0\sum _{k=0}^n{\alpha }_k {\mathit{\zeta }}_{Ek}{A}_{Ek}={\mathit{D}}_0\sum _{k=0}^n{\alpha }_k \mathit{T}{\mathit{q}}_{Ek}{A}_{Ek}.$$

(15)

When the plate is uniformly divided into standard rectangular elements of equal size, their areas satisfy

$$A_{Ek}=A_{E1}=\cdots =A_{En}=A_E=4bc.$$

(16)

Substituting the aforementioned results back into the nonlocal strain energy expression for element $E0$ yields

$$U_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}={\displaystyle \frac{1}{2}}{\int }_{A_{E0}}{\mathit{\zeta }}^{\mathrm{T}}\left(x,y\right)\left[{\mathit{D}}_0\sum _{k=0}^n{\alpha }_k \mathit{T}{\mathit{q}}_{Ek}\right]\mathrm{d}{A}_{E0},$$

(17)

$$\mathit{\zeta }\left(x,y\right)=\mathit{B}\left(x,y\right){\mathit{q}}_{E0}.$$

(18)

Substituting Equations (18) and (15) into Equation (10) yields,

$$U_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}=\sum _{k=0}^n{\displaystyle \frac{A_E{\alpha }_k}{2}}{\int }_{A_{E0}}{\mathit{q}}_{E0}^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\left(x,y\right){\mathit{D}}_0\mathit{T}{\mathbf{q}}_{Ek}\mathrm{d}{A}_{E0}.$$

(19)

Let

$${\mathit{k}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}={\int }_{A_{E0}}{\mathit{B}}^{\mathrm{T}}\left(x,y\right){\mathit{D}}_0\mathit{T}\mathrm{d}{A}_{E0}.$$

(20)

The integrand ${\mathit{B}}^{\mathrm{T}}(x,y) {\mathit{D}}_0\mathit{T}$ in Equation (20) is, in general, not symmetric. To restore the symmetry required by the strain energy formulation, the standard symmetrization procedure [44] is adopted by averaging the original form and its transpose:

$${\mathit{k}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}={\displaystyle \frac{1}{2}}{\int }_{A_{E0}}\left[{\mathit{B}}^{\mathrm{T}}(x,y){\mathit{D}}_0\mathit{T}+{\mathit{T}}^{\mathrm{T}}{\mathit{D}}_0\mathit{B}\left(x\right)\right]\mathrm{d}{A}_{E0}.$$

(21)

This averaging guarantees reciprocity of the coupling stiffness between any element pair, consistent with the energy symmetry of the variational principle [45]. The symmetrized ${\mathit{k}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ serves as the element coupling stiffness contribution used to assemble the nonlocal global stiffness matrix ${\mathit{K}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$. For the entire plate structure, ${\mathit{K}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ is linearly combined with the local global stiffness matrix ${\mathit{K}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ according to the Eringen two-phase model,

$$\mathit{K}=\left(1-{\xi }_2\right){\mathit{K}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}+{\xi }_2{\mathit{K}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}},$$

(22)

where ${\xi }_2\in \left[0,1\right]$ is the two-phase mixture parameter (also referred to as the mixing parameter in parts of the literature) that weights the relative contributions of the local and nonlocal phases.

The model under consideration naturally degenerates into the classical local Kirchhoff plate finite element when the dimensionless nonlocal parameter $\kappa \to 0$, or when ${\xi }_2\to 0$, while still reflecting the long-range coupling effects between elements under finite nonlocal parameters.

The above derivation is based on the element center-point approximation (Equation (12)) and the equal-area assumption (Equation (16)), allowing the nonlocal integral constitutive relation to be embedded in the standard Kirchhoff plate finite element framework in the form of a matrix. Mathematically, Equation (12) corresponds to the two-dimensional midpoint quadrature rule [45], whose truncation error is $O\left(h_e^2\right)$ for a smooth integrand on a rectangular element of characteristic size $h_e$. This error converges to zero under mesh refinement, as confirmed by the convergence study in Section 4.1.

3.3. Nonlocal Kernel Functions and the Selection of Their Influence Domains

In the nonlocal integral constitutive relation (Equation (1)) and the aforementioned finite element discretization format (Equation (11)), the kernel function $\alpha \left(|{\mathit{x}}^{\prime }-\mathit{x}|,\kappa \right)$ plays a decisive role. Physically, it describes the nonlocal weighted influence of the strain field at the source point ${\mathit{x}}^{\prime }$ on the stress state at the reference point $\mathit{x}$. To ensure the physical completeness of the nonlocal theory, the kernel function must satisfy three basic properties: The kernel function reaches a maximum at ${\mathit{x}}^{\prime }=\mathit{x}$ and decays monotonically as the Euclidean distance $s=|{\mathit{x}}^{\prime }-\mathit{x}|$ between the two points increases. This reflects the physical fact that long-range interactions between microscopic particles weaken with increasing distance; when the dimensionless nonlocal parameter $\kappa \to 0$, the kernel function should degenerate into the Dirac $\delta$ function, at which point the nonlocal elasticity theory reverts to the classical local elasticity theory. Therefore, classical elasticity theory can also be viewed as a special case of the nonlocal theory when long-range forces are ignored; to ensure the completeness of the constitutive relation, the integral of the kernel function over the entire domain should be 1:

$${\int }_V\alpha \left(\right|{\mathit{x}}^{\prime }-\mathit{x}|,\kappa )\mathrm{d}V\left({\mathit{x}}^{\prime }\right)=1.$$

(23)

While any function that satisfies the aforementioned three properties can theoretically serve as a kernel function, different kernel function forms result in different constitutive responses. In the present study, the two most commonly used two-dimensional isotropic kernel functions are examined: the exponential and Gaussian kernels. These functions are selected for analysis of the nonlocal Kirchhoff plate problem. Their mathematical expressions are given by

$$\alpha \left(s,l_a\right)={\displaystyle \frac{1}{2\pi l_a^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{s}{l_a}}\right),$$

(24)

$$\alpha \left(s,l_a\right)={\displaystyle \frac{1}{4\pi l_a^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{s^2}{4l_a^2}}\right),$$

(25)

where $l_a=e_0{a}_0$ is the nonlocal parameter.

In numerical implementation, owing to the rapid decay of the kernel function, its value becomes negligible when the distance $s$ exceeds a certain threshold. To reduce the computational cost, the kernel is truncated by introducing a finite influence domain. As demonstrated by Abdollahi and Boroomand [47], who conducted a systematic sensitivity study on the truncation radius in the context of Eringen’s nonlocal integral model, truncating the kernel beyond a finite influence radius can significantly reduce the computational burden while retaining over $1-e^{-9}\approx 99.99\mathrm{\%}$ of the total kernel weight when the radius is chosen as three times the characteristic length. Following this rationale, the two-dimensional nonlocal influence domain in the present study is defined as a circular region centered at the reference point $x$ with a radius $R=3l_a$, and the kernel function outside this region is set to zero, i.e., $\alpha (s,l_a)=0$ for $s>R$. The truncation error introduced by discarding contributions beyond this radius is compensated by the renormalization procedure described below. The normalization condition in Equation (23) is defined over the full domain $V$. After truncation, the integration domain is reduced to the circular region $V_R=\{\left|s\right|\le 3l_a\}$, and the truncated kernel no longer integrates to unity over $V_R$. To restore the normalization condition, the truncated kernel is renormalized by dividing it by its integral over $V_R$:

$$\tilde{\alpha }\left(s,l_a\right)={\displaystyle \frac{\alpha \left(s,l_a\right)}{{\int }_{V_R}\alpha (s,l_a)\mathrm{d}V\left({\mathit{x}}^{\prime }\right)}}, s\le R$$

(26)

This renormalization ensures that the total nonlocal weighting within the truncated domain equals unity, thereby preserving the self-consistency of the nonlocal constitutive relation after domain truncation. In all subsequent numerical computations, the kernel functions (Equations (24) and (25)) are replaced by their renormalized counterparts obtained via Equation (26). When taking $l_a=1 \mathrm{n}\mathrm{m}$, the three-dimensional plots of the two renormalized kernel functions within the influence domain are shown in Figure 1 and Figure 2, respectively.

Figure 3 presents a representative patch of the nanoplate finite element domain, illustrating the uniform structured mesh and the element selection scheme for the nonlocal influence domain, where the circles indicate the centroids of the respective elements. In the numerical implementation, the influence domain of each target element is identified by a circle of radius $R=3l_a$ centered at its geometric centroid; any neighboring element whose centroid falls within this circle contributes to the nonlocal stiffness via the renormalized kernel function (Equation (26)). Among the selected elements, those whose entire area lies within the influence domain are classified as interior elements, whereas those whose centroid lies inside the influence domain but whose area extends partially beyond it are classified as boundary elements. However, the mismatch between the circular influence boundary and the rectilinear element edges introduces a staircase-type geometric approximation error that is negligible in practice, because the kernel functions decay to near zero at $s=R$ (Figure 1 and Figure 2), boundary elements contribute minimally to the nonlocal stiffness, and the residual approximation error diminishes further under mesh refinement.

When the influence domain $V_R$ of an element lies entirely within the plate, all elements within $V_R$ contribute fully to the nonlocal stiffness through the renormalized kernel (Equation (26)). When $V_R$ extends beyond the plate edge, however, no material exists in the exterior region, and consequently, no strain energy contribution arises from that portion, thereby introducing a stiffness deficit in boundary elements relative to interior ones. Physically, this reflects the boundary-layer effect inherent in integral nonlocal elasticity on finite domains [47]. This stiffness deficit arises solely from the geometric truncation of the influence domain at the plate boundary and is therefore independent of the specific boundary condition imposed. Nevertheless, the boundary condition type governs how this deficit manifests in the structural response: under simply supported conditions, the rotational degrees of freedom at boundary nodes remain active and participate in the assembly of the nonlocal stiffness matrix, allowing the stiffness deficit to be directly reflected in the computed modal frequencies; under clamped conditions, all boundary-node degrees of freedom are kinematically suppressed and excluded from the assembled eigenvalue problem, rendering the effect of this deficit less discernible in the overall dynamic response. The magnitude of the boundary-layer effect diminishes as the ratio of the nonlocal characteristic length to the structural dimension decreases [47], and the two-phase formulation (Equation (22)) restores well-posedness through the inclusion of the local phase (1 − ${\xi }_2$) [48,49], which simultaneously attenuates the weight assigned to ${\mathit{K}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ and thereby reduces the overall amplitude of the boundary-layer effect.

4. Numerical Results

In the preceding sections, a nonlocal Kirchhoff plate finite element model was established based on inter-belt analysis. This model was used to derive the stiffness matrix and governing equations. To verify the correctness, convergence, and effectiveness of this algorithm in handling nonlocal effects, a series of numerical examples are presented in this section.

In the numerical examples of this section, unless otherwise stated, the two-dimensional exponential kernel function is selected. The geometric and material parameters of the plate are set as follows: elastic modulus $E=1.06\times {10}^{12} \mathrm{Pa}$, Poisson’s ratio $\nu =0.25$, and mass density $\rho =2250{ \mathrm{kg}/\mathrm{m}}^3$. The frequency parameter is defined as ${\omega }_0=\omega a^2\sqrt{\rho h/D_0}$, where $a=10 \mathrm{n}\mathrm{m}$ is the side length of the square plate, $h=0.34 \mathrm{n}\mathrm{m}$ is the plate thickness, and $D_0$ is the bending stiffness of the plate.

4.1. Algorithm Convergence and Accuracy Verification

To verify the accuracy of the nonlocal finite element algorithm based on inter-belt analysis proposed in the present study, this section examines mesh-refinement convergence, compares results with the literature, and determines the optimal two-phase mixture parameter for the considered kernel functions.

As demonstrated in Figure 4, the frequency parameter of the Kirchhoff plates converges with increasing element number under all-edges simply supported (SSSS) boundary conditions, using the exponential kernel function under varying nonlocal parameter values. Figure 4a illustrates the convergence trend of the first-order frequency parameter. The variables $m$ and $n$ denote the number of half-waves along the $x$ and $y$ axes, respectively, and their values jointly determine the vibration mode order. The three curves correspond to the local case ($l_a=0 \mathrm{n}\mathrm{m}$) and two nonlocal cases ($l_a=1 \mathrm{n}\mathrm{m}$ and $2 \mathrm{n}\mathrm{m}$), with the mesh refined sequentially from $10\times 10$ to $60\times 60$.

Results indicate that, under local conditions, the first-order frequency parameter increases gradually with mesh refinement and stabilizes beyond $30\times 30$ elements, indicating good convergence. Upon introducing the nonlocal effect, the convergence behavior changes markedly. For $l_a=1 \mathrm{n}\mathrm{m}$, the frequency parameter decreases monotonically as the number of elements increases, stabilizing once the mesh reaches $40\times 40$. For $l_a=2 \mathrm{n}\mathrm{m}$, the same decreasing trend is observed, but with a slightly faster convergence rate, with stabilization achieved at $30\times 30$ elements. This contrasting convergence pattern can be attributed to the discretization characteristics of the nonlocal integral. As the mesh is refined, the inter-belt integration captures an increasingly complete portion of the nonlocal kernel interaction, and the softening effect is more accurately reflected, causing the frequency parameters to decrease progressively toward the converged solution. Conversely, in the local case, the coarse-mesh discretization underestimates the local stiffness contribution, resulting in initially lower frequency parameters that gradually increase with mesh refinement. In both cases, the present algorithm attains stable converged solutions at adequate mesh densities, as evidenced by the plateau regions in Figure 4.

Figure 4b provides a detailed illustration of the convergence of the fourth-order frequency parameter under otherwise identical conditions. The findings indicate that, under the examined nonlocal parameters, the fourth-order frequency parameters can also converge to stable values once the element number reaches $30\times 30$. Unless stated otherwise, a $30\times 30$ mesh (the standard element size is ${\displaystyle \frac{1}{3}}$ nm) is employed in subsequent numerical validations. This ensures a satisfactory balance between accuracy and computational efficiency.

The verification of algorithm convergence serves as a foundation for the subsequent assessment of algorithmic accuracy. A key step in this assessment is the determination of the two-phase mixture parameter ${\xi }_2$, which, within Eringen’s two-phase local/nonlocal framework [44] based on his nonlocal elasticity theory [8], controls the relative contribution of the local and nonlocal phases in the constitutive relation. At present, no consensus exists on a universally applicable value of this parameter. Wang et al. [48] and Zhu et al. [50] derived exact analytical solutions for static bending and buckling of Euler–Bernoulli beams using the two-phase model, respectively, and both demonstrated that the structural response is highly sensitive to ${\xi }_2$. Fernández-Sáez and Zaera [49] investigated beam vibrations under the two-phase nonlocal elasticity theory and similarly treated the mixture parameter as a quantity to be calibrated rather than prescribed a priori. More recently, Tuna et al. [51] compared the deformation predicted by Eringen’s two-phase continuum model with corresponding discrete atomic lattice results and confirmed that achieving continuum–discrete consistency requires the mixture parameter to be fitted to reference data. These findings collectively indicate that the value of ${\xi }_2$ is problem-dependent, and its determination typically relies on calibration against benchmark analytical solutions, finite element results, or molecular dynamics simulations. Accordingly, in the present study, ${\xi }_2$ is calibrated by comparing the computed frequency parameters with the finite element results obtained by Shahidi et al. (2013) [22], as illustrated in Figure 5.

As demonstrated in Figure 5, the dimensionless first-order frequency parameters of the nonlocal Kirchhoff plate vary appreciably with the two-phase mixture parameter under different nonlocal parameters, indicating that the structural response is highly sensitive to ${\xi }_2$. To determine the optimal value rigorously, a nonlinear least-squares fitting is performed by minimizing the residual sum of squares between the computed frequency parameters and the finite element results of Shahidi et al. [22] across all examined nonlocal parameters. The fitting yields ${\xi }_2=0.564$ with a minimum residual sum of squares of $1.372\times {10}^{-1}$. This result establishes the optimal two-phase mixture parameter for the exponential kernel function.

Although the theoretical value range of ${\xi }_2$ is [0, 1], as ${\mathrm{\xi }}_2\to 1$ the model degenerates to the purely integral nonlocal formulation, which has been demonstrated to give rise to ill-posed problems within bounded domains. Therefore, to ensure physical consistency and numerical stability, it is recommended in related studies to constrain ${\xi }_2$ to values around 0.5 [48,50]. The calibrated value of ${\xi }_2=0.564$ obtained above falls within this recommended range, thereby further supporting the reliability of the present calibration.

As illustrated in Figure 6, the frequency parameters under the Gaussian kernel function are likewise dependent on the two-phase mixture parameter ${\xi }_2$. Applying the same nonlinear least-squares fitting procedure, the optimal value under the Gaussian kernel is determined to be ${\xi }_2=0.523$, with a minimum residual sum of squares of $1.247\times {10}^{-1}$. Although the optimal values of ${\xi }_2$ for different kernel functions exhibit slight discrepancies (0.564 for the exponential kernel, 0.523 for the Gaussian kernel), both lie in close proximity to 0.5, thereby corroborating the physical plausibility of this parameter range. While the Gaussian kernel function possesses superior smoothness, the exponential kernel function plays a more fundamental role in nonlocal theory owing to its distinctive mathematical properties. Specifically, the exponential kernel can accurately reproduce the dispersion curves of atomic lattice dynamics. Furthermore, it serves as the Green’s function of a specific linear differential operator, thereby establishing a mathematical equivalence between the nonlocal integral constitutive relation and higher-order differential equations [52]. This equivalence enables the transformation of complex integro-differential equations into differential forms that are more amenable to analytical treatment. In contrast, the Gaussian kernel function does not provide such a mathematical bridge for integro-differential transformation and typically necessitates purely numerical solutions. Therefore, to facilitate comparison with existing differential-form nonlocal studies, the subsequent parametric analysis is primarily based on the exponential kernel function, with ${\xi }_2=0.564$.

Based on the preceding convergence analysis and the calibrated optimal two-phase mixture parameters,

Table 1. Comparison of the first-order frequency parameter ${\omega }_0$ for simply supported (SSSS) Kirchhoff plates between the present method and the reference FEM solution [22] (exponential kernel).

${\mathit{l}}_{\mathit{a}}$ (nm)	Present	Reference (FEM) [22]	Relative Error
0	19.7152	19.7275	0.06%
1	18.3679	18.0282	1.88%
2	16.6946	16.7039	0.05%
3	15.4762	15.6342	1.01%
4	14.7710	14.7468	0.16%

,

Table 2. Comparison of the first-order frequency parameter ${\omega }_0$ for simply supported (SSSS) Kirchhoff plates between the present method and the reference Navier solution [10] (exponential kernel).

${\mathit{l}}_{\mathit{a}}$ (nm)	Present	Reference (Navier) [10]	Relative Error
0	19.7152	19.7392	0.12%
1	18.3679	18.0390	1.82%
2	16.6946	16.7138	0.11%
3	15.4762	15.6435	1.06%
4	14.7710	14.7556	0.10%

,

Table 3. Comparison of the first-order frequency parameter ${\omega }_0$ for simply supported (SSSS) Kirchhoff plates between the present method and the reference FEM solution [22] (Gaussian kernel).

${\mathit{l}}_{\mathit{a}}$ (nm)	Present	Reference (FEM) [22]	Relative Error
0	19.7166	19.7275	0.05%
1	18.3329	18.0282	1.69%
2	16.6820	16.7039	0.13%
3	15.4741	15.6342	1.02%
4	14.8482	14.7468	0.68%

and

Table 4. Comparison of the first-order frequency parameter ${\omega }_0$ for simply supported (SSSS) Kirchhoff plates between the present method and the reference Navier solution [10] (Gaussian kernel).

${\mathit{l}}_{\mathit{a}}$ (nm)	Present	Reference (Navier) [10]	Relative Error
0	19.7166	19.7392	0.11%
1	18.3329	18.0390	1.62%
2	16.6820	16.7138	0.19%
3	15.4741	15.6435	1.08%
4	14.8482	14.7556	0.62%

present a comparison of the frequency parameters calculated in this study with existing literature results under the exponential kernel function (Table 1 and Table 2) and the Gaussian kernel function (Table 3 and Table 4), respectively.

Table 1, Table 2, Table 3 and Table 4 show that, based on the two-phase mixture parameter determined previously (${\xi }_2=0.564$ for exponential kernel, ${\xi }_2=0.523$ for Gaussian kernel), the frequency parameters calculated using the inter-belt analysis method in the present study agree well with the results in the literature, with relative deviations in most cases being below 2%. In the local case where the nonlocal parameter is 0, the relative error is only of the order of ${10}^{-3}$, thereby verifying the accuracy of the algorithm’s degeneration. Further observations of the error distribution under different nonlocal parameters revealed that both kernel functions showed remarkably high agreement when the nonlocal parameter $l_a=2 \mathrm{n}\mathrm{m}$. This was evidenced by relative errors falling below 0.2%, thereby demonstrating the model’s high consistency under this particular parameter. In the context of larger nonlocal parameters ($l_a=3.4 \mathrm{n}\mathrm{m}$), the relative error of the exponential kernel function results (approximately 0.1–1.1%) is marginally lower than that of the Gaussian kernel function (approximately 0.6–1.1%). This finding suggests that the exponential kernel function is numerically more stable in this range. Overall, for the two kernel functions considered and the tested range of nonlocal parameters, the relative errors remain within a reasonable range. This confirms the accuracy and reliability of the proposed integral-form nonlocal finite element algorithm for the two kernel functions considered in this study and the nonlocal parameters examined.

To further validate the proposed method against atomistic simulation data,

Table 5. Fundamental frequency comparison between the present method (exponential kernel, $l_a=1.41 \mathrm{n}$m) and MD simulations [4] for zigzag SLGS (SSSS).

Side Length (nm)	Present (THz)	Molecular Dynamics (THz) [4]	Relative Error
10	0.0584121	0.0587725	0.61%
15	0.0271221	0.0273881	0.97%
20	0.0155628	0.0157524	1.2%
25	0.0100699	0.009984	0.86%
30	0.0070408	0.0070655	0.34%
35	0.0051967	0.0052982	1.91%

,

Table 6. Fundamental frequency comparison between the present method (exponential kernel, $l_a=1.34 \mathrm{n}$m) and MD simulations [4] for armchair SLGS (SSSS).

Side Length (nm)	Present (THz)	Molecular Dynamics (THz) [4]	Relative Error
10	0.0588269	0.0595014	1.13%
15	0.0272443	0.0277928	1.97%
20	0.0156129	0.0158141	1.27%
25	0.0100951	0.0099750	0.97%
30	0.0070553	0.0070712	0.22%
35	0.0052059	0.0052993	1.76%

,

Table 7. Fundamental frequency comparison between the present method (Gaussian kernel, $l_a=1.41 \mathrm{n}$m) and MD simulations [4] for zigzag SLGS (SSSS).

Side Length (nm)	Present (THz)	Molecular Dynamics (THz) [4]	Relative Error
10	0.058372	0.0587725	0.68%
15	0.0271347	0.0273881	0.92%
20	0.0155779	0.0157524	1.10%
25	0.0100822	0.009984	0.98%
30	0.0070505	0.0070655	0.21%
35	0.0052044	0.0052982	1.77%

and

Table 8. Fundamental frequency comparison between the present method (Gaussian kernel, $l_a=1.34 \mathrm{n}$m) and MD simulations [4] for armchair SLGS (SSSS).

Side Length (nm)	Present (THz)	Molecular Dynamics (THz) [4]	Relative Error
10	0.0587807	0.0595014	1.21%
15	0.0272525	0.0277928	1.94%
20	0.0156248	0.0158141	1.19%
25	0.0101053	0.0099750	1.30%
30	0.0070634	0.0070712	0.11%
35	0.0052123	0.0052993	1.64%

compare the fundamental frequencies obtained by the present formulation with molecular dynamics (MD) results for simply supported (SSSS) square single-layered graphene sheets (SLGS) reported by Ansari et al. [4]. To ensure a consistent comparison, the nonlocal parameters are set to the values identified in [4] by fitting the nonlocal continuum model to MD data, namely $l_a=1.41 \mathrm{n}\mathrm{m}$ for zigzag SLGS and $l_a=1.34 \mathrm{n}\mathrm{m}$ for armchair SLGS, with the side length of the square plate varying from 10 nm to 35 nm. Both exponential (Table 5 and Table 6) and Gaussian (Table 7 and Table 8) kernel functions are examined. Across all cases, the relative errors remain below 2%, which is comparable to the accuracy achieved in the continuum-level benchmarks of Table 1, Table 2, Table 3 and Table 4. The computed fundamental frequencies decrease monotonically with increasing side length, reproducing the size-dependent trend observed in the MD simulations. For a given side length, the armchair configuration yields slightly higher frequencies than the zigzag configuration (e.g., 0.0595 THz vs. 0.0588 THz at 10 nm in the MD reference), reflecting the chirality dependence of the effective mechanical properties of graphene. Moreover, the two kernel functions produce comparable levels of accuracy across the examined parameter range, with neither exhibiting a clear systematic advantage over the other. Together with the continuum-level benchmarks in Table 1, Table 2, Table 3 and Table 4, this atomistic-level comparison provides additional evidence supporting the reliability and applicability of the proposed integral nonlocal finite element formulation.

4.2. Parametric Study

4.2.1. Influence of the Nonlocal Parameter and the Two-Phase Mixture Parameter

As demonstrated in Figure 7, the variation in the percentage reduction in the first five frequency parameters with the nonlocal parameter $l_a$ is presented under the four-sided simply supported (SSSS) boundary conditions. These results show that as the nonlocal parameter increases, the percentage reduction in frequency for each mode exhibits a marked upward trend. This finding suggests that an increase in the nonlocal parameter reduces the plate’s overall stiffness, thereby decreasing its natural frequency. Furthermore, the magnitude of frequency reduction for higher-order modes is significantly larger than that for lower-order modes, indicating that high-frequency vibration is more sensitive to nonlocal effects, which is consistent with the general principle of nonlocal elasticity that short-wavelength deformation is more affected by micro-scale effects.

Figure 8 further illustrates the effect of the two-phase mixture parameter ${\xi }_2$ on the percentage reduction in the frequency parameters of the first five vibration modes when the nonlocal parameter, $l_a=1 \mathrm{n}\mathrm{m}$, is constant. The results demonstrate that as the parameter ${\xi }_2$ increases from 0.1 to 0.9, the frequency-reduction percentage increases linearly. Since ${\xi }_2$ governs the weight of the nonlocal phase in the two-phase mixture, an increase in its value directly amplifies the nonlocal softening effect, thereby leading to a further reduction in structural stiffness and vibration frequency.

4.2.2. Influence of Aspect Ratio

Figure 9 shows the variation in the first-order frequency parameter with the aspect ratio for nanoplates of different side lengths under a fixed nonlocal parameter $l_a=1 \mathrm{n}\mathrm{m}$. In this analysis, to balance accuracy with computational efficiency, standard elements with a side length of 0.5 nm are used, and the plate thickness and nonlocal parameter are kept constant. It has been observed that as the aspect ratio increases, the first-order natural frequency decreases monotonically. When the aspect ratio is small, the frequency declines sharply (as illustrated by the steeper curve in the figure); conversely, when the aspect ratio exceeds a certain threshold, the frequency change curve tends to flatten. This phenomenon shows that, at the nanoscale, alterations in the plate’s geometric configuration exert a substantial influence on its dynamic characteristics. However, as the plate becomes more slender, the marginal impact of this geometric effect diminishes concomitantly. It is notable that when the aspect ratio exceeds 5, the influence of further increases in aspect ratio on the frequency becomes less significant.

4.2.3. Influence of Thickness Ratio

As illustrated in Figure 10, the thickness ratio significantly influences the first-order frequency parameter of nanoplates of varying dimensions, with the nonlocal parameter fixed at 1 nm. The thickness ratio range for the numerical example is set to $\left[0.034,0.1\right]$. The lower limit corresponds to the benchmark thickness ratio for single-layer graphene, with the aim of verifying the model’s effectiveness for single-atomic-layer nanomaterials. The upper limit corresponds to the applicability limit of Kirchhoff thin-plate theory, which requires ($h/a\le 1/10$) to ensure the validity of the negligible-transverse-shear assumption. This ensures the validity of the assumption of neglecting transverse shear deformation. The results show that, as the thickness ratio increases, the first-order natural frequency of the plate increases approximately linearly. This behavior arises from the direct enhancement of bending stiffness with increasing thickness, thereby increasing the natural frequency of the structure. It is noteworthy that in the presence of nonlocal effects, the influence of thickness ratio variation on frequency still adheres to the fundamental principles of classical mechanics. Specifically, stiffness enhancement results in an increase in frequency.

5. Discussion

The inter-belt discretization framework extends Eringen’s integral nonlocal constitutive relation to plate-bending problems, retaining full generality with respect to kernel choice. Unlike differential-form nonlocal models, which require specific kernel assumptions to admit equivalence with higher-order differential equations [52], the present integral formulation imposes no such restriction, and the resulting nonlocal operator is embedded within the standard finite element assembly framework in a manner that preserves the form of the generalized eigenvalue problem. The cross-scale verification against continuum benchmarks [10,22] and molecular dynamics data [4] yields relative errors below 2% in most cases, indicating that the center-point approximation (Equation (12)) and influence-domain truncation introduce limited discretization error for the investigated cases and mesh densities.

The observed stiffness-softening trend is in line with the nature of long-range interactions in integral nonlocal elasticity: contributions from distant material points reduce the effective bending stiffness of the assembled system, thereby lowering natural frequencies relative to the classical local model. The parametric results further indicate that higher-order modes exhibit greater sensitivity to nonlocal effects for the investigated cases. Notably, both calibrated mixture parameters, ${\xi }_2=0.564$ for the exponential kernel and ${\xi }_2=0.523$ for the Gaussian kernel, cluster near 0.5, consistent with the range recommended in beam-level two-phase studies [48,50]. This consistency across the two kernel types and the investigated plate configurations indicates that values around 0.5 are a robust outcome of the present calibration and are compatible with prior two-phase applications; nevertheless, ${\xi }_2$ remains a phenomenological, problem-dependent parameter, and its broader generality requires further theoretical and experimental/atomistic evidence.

The calibrated model carries direct practical relevance for MEMS/NEMS device design, where frequency predictions must account for size-dependent softening. The quantified sensitivity relationships—specifically, the approximately 25% reduction in the first-order frequency parameter as $l_a$ increases from 0 to 4 nm, and the approximately linear dependence of frequency reduction on ${\xi }_2$—provide a systematic basis for inverse identification workflows in which experimentally measured resonant frequencies can be used to determine effective nonlocal parameters for a given device geometry and boundary condition, thereby supporting frequency calibration and material characterization in nanomechanical resonators, mass sensors, and RF filters.

Despite these contributions, the present formulation retains certain inherent limitations. It is restricted to linear elasticity and regular rectangular geometries; geometric nonlinearity, multi-physics coupling, and irregular boundaries are not addressed. Although the influence-domain truncation to $R=3l_a$ [47], the reuse of kernel weights on the uniform structured mesh, and the separate treatment of interior and boundary elements collectively reduce computational overhead, and the assembled ${\mathit{K}}_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}$ remains inherently denser than its local counterpart. This density is a fundamental consequence of the long-range coupling structure of integral nonlocal elasticity [47]. For a structured two-dimensional mesh with characteristic element size (e.g., typical element edge length) $h_e$, each element couples to approximately $O\left(\left(R\right|h_e{)}^2\right)$ neighbors within the influence disk, so memory and assembly costs increase accordingly compared with the local case (where the connectivity is $O\left(1\right)$ per element). The associated memory and factorization costs, therefore, grow noticeably with mesh refinement or increasing $l_a$. Computational efficiency thus remains a practical limitation, particularly for three-dimensional configurations. Future extensions to nonlinear vibration and integration with isogeometric or meshfree discretizations would further broaden the framework’s scope and applicability.

6. Conclusions

A symplectic inter-belt finite element formulation is developed for free vibration analysis of Kirchhoff plates governed by Eringen’s integral nonlocal elasticity. By discretizing long-range interactions as element-to-element couplings within a finite influence domain, the integral operator is incorporated into a standard finite element framework, yielding a generalized eigenvalue problem for modal analysis.

Mesh-refinement studies show stable convergence of the frequency parameters for the investigated cases. With calibrated two-phase mixture parameters, benchmark comparisons indicate that the present results agree well with published solutions: relative deviations are below 2% in most cases, and the local limit $\left(l_a=0\right)$ yields errors on the order of ${10}^{-3}$. The calibration based on nonlinear least-squares fitting yields ${\xi }_2=0.564$ for the exponential kernel and ${\xi }_2=0.523$ for the Gaussian kernel, both are close to the literature-recommended range around 0.5.

Parametric results confirm that stiffness softening and frequency reduction occur as the nonlocal length scale increases, with higher-order modes showing greater sensitivity to nonlocal effects. The frequency-reduction percentage increases approximately linearly with ${\xi }_2$ (from 0.1 to 0.9). Furthermore, geometric studies show a monotonic decrease in the first-order frequency with increasing aspect ratio (with reduced sensitivity beyond an aspect ratio of approximately 5) and an approximately linear increase with thickness ratio over $h/a\in \left[0.034,0.1\right]$.

The inter-belt framework has progressively expanded from one-dimensional structures and two-dimensional in-plane problems [41,42,43] to the plate-bending formulation presented here, suggesting that extension to more complex configurations such as multi-layered nanoplates or functionally graded microstructures is a viable next step. The framework’s compatibility with standard finite element assembly and its support for integral nonlocal constitutive relations indicate potential utility for frequency modeling of MEMS/NEMS devices, including nanomechanical resonators and mass sensors.