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Article

Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm

1
School of Science, North China University of Science and Technology, Tangshan 063000, China
2
School of Science, Yanshan University, Qinhuangdao 066004, China
3
School of Artificial Intelligence, Tangshan College, Tangshan 063000, China
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2025, 9(7), 477; https://doi.org/10.3390/fractalfract9070477
Submission received: 28 June 2025 / Revised: 15 July 2025 / Accepted: 16 July 2025 / Published: 21 July 2025

Abstract

This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials.

1. Introduction

In the field of contemporary mechanics of materials [1,2,3,4], the study of dynamic response properties of nanostructured materials has become a focus of much attention. With the rapid development of nanotechnology, the understanding and control of the dynamic behavior of multilayer nanoplates, as a kind of nanostructured material with excellent mechanical properties and extensive application potential [5,6], are of great significance in promoting the innovation of related technologies. Among them, bilayer nanoplates, as a typical multilayer structure [7,8,9,10,11], have dynamic response properties that are crucial for understanding and designing high-performance nanodevices. Pradhan et al. [7] reinterpreted classical plate theory and first-order shear deformation theory, used nonlocal differential constitutive relations, proposed a nonlocal elasticity theory applicable to nanoplate vibrations, and analyzed the effect of nonlocal parameters on the natural frequencies of nanoplates. Pouresmaeeli et al. [8] analyzed the free vibration of a four-sided simply supported two-layer orthotropic nanoplate based on an analytical method of nonlocal theory, gave the expressions for the natural frequency, and investigated the effect of material parameters on the vibration frequencies. This article aims to explore in-depth the response mechanism of a two-layer nanoplate under complex dynamic loads and to provide the theoretical basis for the optimal design of nanoplate structures by introducing advanced mathematical tools and physical models.
Fractional calculus has been extensively applied in the field of mechanics in recent years due to its ability to more accurately characterize the memory effects and nonlocal properties of materials [12,13,14]. There are various definitions of fractional derivatives, such as the Riemann–Liouville definition, the Grünwald–Letnikov definition, the Caputo definition, etc. [15]. The Caputo fractional derivatives have significant advantages over other types of fractional derivatives in terms of maintaining the initial conditions of the original function, having good convergence, and stabilizing properties. In order to further reveal the mechanical properties of bilayer nanoplates, this study accurately captures the viscoelastic behavior of nanomaterials under dynamic loading by introducing the Fractional Kelvin–Voigt (FKV) model under the definition of Caputo fractional derivatives, which has shown significant advantages in describing the mechanical response of polymers and other viscoelastic materials [16,17,18]. Zhang et al. [17] verified that the fractional derivative viscoelastic model is more accurate than the traditional integer viscoelastic model. The paper investigated the guided waves in a viscoelastic FGM hollow cylinder under a fractional Kelvin–Voigt model. Javadi et al. [18] focused on analyzing the nonlinear vibration behavior of viscoelastic beams under harmonic excitation based on the FKV model. Therefore, the application of Caputo-type fractional derivatives in the FKV model can deepen our understanding of the viscoelastic properties of double-layer nanoplates as expressed by the fractional model.
Traditional theories have obvious limitations in describing nanoscale effects, such as the size effect and surface effect. In order to overcome these limitations, the nonlocal elasticity theory has emerged [19], which provides a new perspective to study mechanical behavior of materials at the nanoscale by taking into account the interactions between points within the material. In particular, the differential form of the nonlocal elasticity theory [20] shows a unique advantage in capturing the nonlocal relationship between stresses and strains within materials at the nanoscale [21,22]. Pradhan et al. [21] investigated the vibration analysis of orthotropic graphene sheets embedded in an elastic medium, taking into account the influence of the small-scale effect by utilizing the nonlocal differential constitutive relations of Eringen. Pisano et al. [22] suggested that a fully nonlocal elasticity model has difficulty in solving boundary-value problems. In contrast, both the integral approach and the differential approach yield a unique solution for the small-scale beam problem. The differential form of the nonlocal elasticity theory provides a more accurate theoretical framework for describing the nonlocal mechanical behavior of bilayer nanoplates under dynamic loading.
In terms of numerical solutions, facing complex fractional governing equations, it is often difficult to apply them directly due to the limitations of traditional numerical methods (e.g., finite difference method, finite element method, and spectral methods) [23,24]. Therefore, in this article, shifted Legendre polynomials are chosen for function approximation. Shifted Legendre polynomials, as an efficient numerical method, can significantly simplify the numerical solution process while dealing with high-dimensional problems and ensuring computational accuracy [25,26,27,28]. Compared with other polynomials (e.g., Chebyshev polynomials, Bernstein polynomials, etc.), the shifted Legendre polynomials have higher efficiency and accuracy in function approximation, especially in dealing with problems with complex boundary conditions and multiscale properties [29,30]. Cao et al. [29] proposed an improved fractional viscoelastic model for describing the physical behavior of a polymeric material structure (PMMA Viscoelastic Beam) and numerically solved the fractional governing equations using the shifted Legendre polynomials algorithm. Sun et al. [30] utilized the shifted Legendre polynomial algorithm to conduct a numerical analysis of the governing equations of a fractional viscoelastic plate within the time domain, and validated the algorithm’s validity and accuracy through error analysis and example studies. It is worth noting that although this method performs well in terms of accuracy and simplification, the demand for computing resources may increase when dealing with larger-scale or more complex systems. Matrix compression technology can be used as a mitigation strategy. Aimi et al. [31] applied the partially pivoted version of the Adaptive Cross Approximation technique to reduce the computational time and the memory requirement for the matrix blocks of the resulting global discretization.
Based on the above theoretical foundation, the highlights of this article are mainly reflected in the following aspects: Firstly, the FKV model under the definition of Caputo-type fractional derivatives and the differential form of nonlocal elasticity theory are combined to construct fractional differential equations consisting of the upper and lower plate governing equations, to describe the mechanical behaviors of the bilayered nanoplate. Secondly, the various items of the governing equations are subjected to a systematic convergence analysis, which effectively verifies the convergence property of the ternary displacement function and its fractional derivatives in the whole governing equations, thus ensuring the mathematical rigor and the reliability of the equations. Finally, due to the discretization of the time variables in the computational process, the complicated process in the traditional frequency-domain method is avoided, which facilitates solving the complex fractional differential equations directly in the time domain.
The sections of the full text are organized as follows: Section 2 will introduce the basics of Caputo-type fractional derivatives and shifted Legendre polynomials in detail; Section 3 will construct the fractional constitutive relation of the bilayered nanoplates and derive the governing equations based on the FKV model, the differential form of the theory of nonlocal elasticity, and Kirchhoff’s plate theory; Section 4 will introduce the question of how to use shifted Legendre polynomials for functional approximation, and then convert the governing equations of the double-layer plate into algebraic equations; Section 5 will verify the validity and applicability of the method proposed in this paper through the dimensionless equations and numerical example analyses; Section 6 will explore the effects of the material parameters of the bilayer nanoplates on their dynamic behaviors; and Section 7 will summarize the whole paper and look forward to the future directions of the research.

2. Fundamental Definition

2.1. Caputo-Type Fractional Derivatives

In this section, a systematic introduction will be given to the Caputo-type fractional derivative definition [15], which was proposed by the Italian mathematician Caputo in the 1960s, and has since been widely used to describe dynamical systems with memory effects due to its intuitive treatment of initial conditions.
Definition 1.
The Caputo-type fractional derivative is defined as the derivative of order α of a function f ( t ) :
D t α c f ( t ) = 1 Γ ( s α ) 0 t f ( s ) ( λ ) ( t λ ) α + 1 s d λ , s 1 < α < s , d s d t s f ( t ) , s = α ,
where D t α c is the Caputo fractional differential operator, α is a fractional order, f is a continuously differentiable function defined on the interval ( 0 , + ) , and s is a positive integer, while Γ ( δ ) is a Gamma function defined as Γ ( δ ) = 0 e t t δ 1 d t .
Based on the above definition of Caputo-type fractional derivatives, we can further obtain
D t α c t s = 0 , s = 0 , Γ ( s + 1 ) Γ ( s + 1 α ) t s α , s = 1 , 2 , 3 ,
For any function f ( t ) , g ( t ) C 1 ( R ) , C is a constant, λ , μ R , 0 < α 1 , and Caputo-type fractional derivatives have the following three properties:
( 1 ) D t α c C = 0 , ( 2 ) D t α c ( C f ( t ) ) = C D t α c f ( t ) , ( 3 ) D t α c [ λ f ( t ) + μ g ( t ) ] = λ D t α c f ( t ) + μ D t α c g ( t ) .

2.2. The Shifted Legendre Polynomials

In this section, the definition of the shifted Legendre polynomials [30] and how they can be expressed in matrix product form will be elaborated in order to provide a theoretical basis and computational means for numerical analysis and practical applications in the subsequent sections.
Definition 2.
The expression for the n t h shifted Legendre polynomial defined within the interval [0, 1] is given as follows:
l n ( x ) = i = 0 n ( 1 ) n + i Γ ( n + i 1 ) Γ ( n i + 1 ) ( Γ ( i + 1 ) ) 2 x i ,
where x [ 0 , 1 ] , i = 0 , 1 , , n .
The column vector φ n ( x ) , consisting of the shifted Legendre polynomials on the interval [0, 1], can be expressed in the following matrix product form:
φ n ( x ) = l i ( x ) , 0 i n T = A X ( x ) ,
where X ( x ) = 1 , x , , x n T , A = a i j i , j = 0 n , a i j = 0 , i < j ( 1 ) i + j Γ ( i + j 1 ) Γ ( i j + 1 ) ( Γ ( j + 1 ) ) 2 , i j .
Extending the definition interval of the shifted Legendre polynomials from [0, 1] to [ 0 , H ] , the n t h polynomials can be expressed as
L n ( x ) = i = 0 n ( 1 ) n + i Γ ( n + i 1 ) Γ ( n i + 1 ) ( Γ ( i + 1 ) ) 2 x H i = i = 0 n ( 1 ) n + i Γ ( n + i 1 ) Γ ( n i + 1 ) ( Γ ( i + 1 ) ) 2 1 H i x i ,
where x [ 0 , H ] , i = 0 , 1 , , n .
Then when x [ 0 , H ] , φ n 1 ( x ) can be expressed as the following matrix product:
φ n 1 ( x ) = M G ( x ) ,
where G ( x ) = 1 , x , , x n 1 T , M = m i j i , j = 0 n 1 , m i j = 0 , i < j ( 1 ) i + j Γ ( i + j 1 ) Γ ( i j + 1 ) ( Γ ( j + 1 ) ) 2 1 H i , i j .
Similarly, when y is within the range [ 0 , S ] , φ n 2 ( y ) can be represented as the following matrix:
φ n 2 ( y ) = N G ( y ) ,
where G ( y ) = 1 , y , , y n 2 T , N = n i j i , j = 0 n 2 , n i j = 0 , i < j ( 1 ) i + j Γ ( i + j 1 ) Γ ( i j + 1 ) ( Γ ( j + 1 ) ) 2 1 S i , i j .
Likewise, for t in the interval [ 0 , K ] , φ n 3 ( t ) can be described as the product of the following matrix:
φ n 3 ( t ) = R G ( t ) ,
where G ( t ) = 1 , t , , t n 3 T , R = r i j i , j = 0 n 3 , r i j = 0 , i < j ( 1 ) i + j Γ ( i + j 1 ) Γ ( i j + 1 ) ( Γ ( j + 1 ) ) 2 1 K i , i j .

3. Establishment of the Governing Equation for Double-Layer Nanoplate

3.1. The Constitutive Relationship

This paper analyses a double-layered and viscoelastic orthotropic isotropic nanoplate structure, where the thickness h ( h = 0.34 nm ) of the plate is considerably less than the length L a and the width L b , and the effect of transverse shear deformation in the theoretical model can be neglected (please see Figure 1 for a schematic of the structure). The two-layer nanoplate is coupled by the viscoelastic medium, and the effect is non-negligible. In addition, this nanoplate structure is subjected to a uniformly distributed transverse load q ( x , y , t ) . Although the two nanoplates may each possess unique physical properties in practical situations, for the sake of simplifying the theoretical analysis and calculations, it is assumed that the nanoplates have the same geometrical dimensions (width and length) and bending stiffness. The density ρ of the two plates is known to be 2250 (kg/m3).
The constitutive relation of the nanoplates is constructed using the FKV model combined with the Caputo fractional derivative, as shown in Figure 2. The viscoelastic medium between the two nanoplates is represented by a spring (assumed to have a stiffness of k 0 ) and a spring-viscous pot connected in parallel to represent the elasticity and viscosity, respectively. In the viscoelastic plate system, the Winkler modulus ( k 1 ) is introduced to simulate the interlayer contact of the two-layer nanoplate, considering both the elastic and viscous properties.
In this paper, the Kirchhoff plate theory and the differential form of the nonlocal elasticity theory are combined to construct the constitutive equations of a bilayer nanoplate. This formula is based on the constitutive relations of the nonlocal continuum theory [19]. The constitutive relations expressed in equivalent differential forms, further proposed by Eringen [20], are used:
1 μ 2 2 σ i j = σ i j ,
where μ is the nonlocal parameter, σ i j is the nonlocal stress tensor, σ i j is the local stress tensor, and 2 = 2 x 2 + 2 y 2 is the Laplace operator.
In order to accurately characterize the vibration and buckling of the nanoplates, we set the neutral plane as z = 0 plane in the Cartesian coordinate system and establish the corresponding governing equations. According to the Kirchhoff plate theory, the displacement component u x , u y , u z of any point on the neutral plane of the plate along the x , y , and z axis directions can be expressed as follows:
u x ( x , y , z , t ) = z w ( x , y , t ) x , u y ( x , y , z , t ) = z w ( x , y , t ) y , u z ( x , y , z , t ) = w ( x , y , t ) ,
where w is the transverse displacement of the plate and t is for time. According to the above equation, the relationship between displacement and strain of the nanoplate can be derived:
ε x x ε y y γ x y = z 2 w x 2 z 2 w y 2 2 z 2 w x y .
According to Equations (11) and (12), combined with the fractional Kelvin–Voigt viscoelastic constitutive relation, and also considering the viscoelastic damping of the plate structure, the stress–strain relationship of the nanoplates can be represented as
σ x x σ y y τ x y = E 1 v 2 E v 1 v 2 0 E v 1 v 2 E 1 v 2 0 0 0 E 1 + v ε x x ε y y γ x y + η D t α c E 1 v 2 E v 1 v 2 0 E v 1 v 2 E 1 v 2 0 0 0 E 1 + v ε x x ε y y γ x y = 1 + η D t α c E 1 v 2 E v 1 v 2 0 E v 1 v 2 E 1 v 2 0 0 0 E 1 + v ε x x ε y y γ x y = 1 + η D t α c D 11 D 12 0 D 12 D 22 0 0 0 D 33 ε x x ε y y γ x y .
where E is Young’s modulus, v is Poisson’s ratio, η is the damping coefficient, and D 11 D 12 0 D 12 D 22 0 0 0 D 33 is the stiffness matrix.

3.2. Derivation of the Governing Equations

Substituting the differential form Equation (10) of the nonlocal elasticity theory into the fractional constitutive relation Equation (13), we have
1 μ 2 2 σ x x σ y y τ x y = 1 + η D t α c D 11 D 12 0 D 12 D 22 0 0 0 D 33 ε x x ε y y γ x y .
The governing equations of plates are reasoned by the principle of virtual work, and the equation is shown below:
δ K δ U + δ W = 0 ,
where δ K denotes the kinetic energy of the nanoplates, δ U denotes the strain energy of the nanoplates, and δ W denotes the work done by the nanoplates by the external force.
The strain energy δ U pertaining to the nanoplate is defined as
δ U = ν σ x x δ ε x x + σ y y δ ε y y + τ x y δ γ x y d V = ν σ x x δ z 2 w x 2 + σ y y δ z 2 w y 2 + τ x y δ 2 z 2 w x y d V = A M x x 2 δ w x 2 M y y 2 δ w y 2 2 M x y 2 δ w x y d A ,
where M x x = h / 2 h / 2 z σ x x d z , M y y = h / 2 h / 2 z σ y y d z , and M x y = h / 2 h / 2 z τ x y d z .
The kinetic energy δ K associated with the nanoplate is given by
δ K = V ρ u x t δ u x t + u y t δ u y t + u z t δ u z t d V = A m 0 w t δ w t + m 2 2 w x t 2 δ w x t + 2 w y t 2 δ w y t d A ,
where m 0 = ρ h , m 2 = 1 12 ρ h 3 .
The work done δ W by the external force on the nanoplate can be expressed as
δ W = A q δ w d A .
So for the nanoplates subjected to transverse loading, substituting Equations (16)–(18) into Equation (15), the nanoplates governing equation is obtained using the principle of virtual work as follows:
2 M x x x 2 + 2 2 M x y x y + 2 M y y y 2 + q = m 0 2 w t 2 m 2 4 w x 2 t 2 + 4 w y 2 t 2 .
According to the definition of M x x , M y y , M x y , although it is not possible to solve M x x , M y y , M x y directly, 1 μ 2 2 can be multiplied on both sides of the equation:
1 μ 2 2 M x x = h / 2 h / 2 z 1 μ 2 2 σ x x d z = h / 2 h / 2 z 1 + η D t α c ( E 1 v 2 ε x x + E v 1 v 2 ε y y ) d z ,
1 μ 2 2 M y y = h / 2 h / 2 z 1 μ 2 2 σ y y d z = h / 2 h / 2 z 1 + η D t α c ( E v 1 v 2 ε x x + E 1 v 2 ε y y ) d z ,
1 μ 2 2 M x y = h / 2 h / 2 z 1 μ 2 2 τ x y d z = h / 2 h / 2 z 1 + η D t α c E 1 + v γ x y d z .
In conjunction with the displacement–strain relationship for the plate of Equation (12), Equations (20)–(22) can be further calculated:
1 μ 2 2 M x x = E h 3 12 ( 1 v 2 ) 1 + η D t α c 2 w x 2 + v 2 w y 2 ,
1 μ 2 2 M y y = E h 3 12 ( 1 v 2 ) 1 + η D t α c v 2 w x 2 + 2 w y 2 ,
1 μ 2 2 M x y = h 3 12 · 2 E 1 + v 1 + η D t α c 2 w x y .
The governing equation Equation (19) can be obtained by shifting the terms and then multiplying each term together by 1 μ 2 2 :
1 μ 2 2 2 M x x x 2 + 2 2 M x y x y + 2 M y y y 2 = 1 μ 2 2 m 0 2 w t 2 m 2 4 w x 2 t 2 + 4 w y 2 t 2 q ,
where
1 μ 2 2 2 M x x x 2 = E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w x 4 + v 4 w x 2 y 2 ,
1 μ 2 2 2 M y y y 2 = E h 3 12 ( 1 v 2 ) 1 + η D t α c v 4 w x 2 y 2 + 4 w y 4 ,
1 μ 2 2 2 M x y x y = E h 3 6 ( 1 + v ) 1 + η D t α c 4 w x 2 y 2 .
Then the final governing equation Equation (26) can be rewritten as
E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w i x 4 + E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w i y 4 + v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) 1 + η D t α c 4 w i x 2 y 2 + 1 μ 2 2 m 0 2 w i t 2 m 2 4 w i x 2 t 2 + 4 w i y 2 t 2 q i ( x , y , t ) = 0 ,
where q = q i ( i = 1 , 2 ) is the equivalent acting transverse load of the viscoelastic medium acting on the upper and lower plates, respectively, reflecting the overall behavior of the material during the transverse force. w i ( i = 1 , 2 ) are the transverse displacements of the two plates, respectively.
The plates studied in this paper are in a four-sided simply supported state, so the boundary conditions can be stated as
w i ( 0 , y , t ) = w i ( L x , y , t ) = 0 , i = 1 , 2 , w i ( x , L y , t ) = w i ( x , 0 , t ) = 0 , i = 1 , 2 , M x x ( 0 , y , t ) = M x x ( L x , y , t ) = 0 , M y y ( x , L y , t ) = M y y ( x , 0 , t ) = 0 ,
where the boundary conditions for M x x , M y y are equivalent to the boundary conditions for 1 μ 2 2 M x x , 1 μ 2 2 M y y .
The boundary conditions are modified as follows:
w i ( 0 , y , t ) = w i ( L x , y , t ) = 0 , i = 1 , 2 , w i ( x , L y , t ) = w i ( x , 0 , t ) = 0 , i = 1 , 2 , 1 μ 2 2 M x x ( 0 , y , t ) = 1 μ 2 2 M x x ( L x , y , t ) = 0 , 1 μ 2 2 M y y ( x , L y , t ) = 1 μ 2 2 M y y ( x , 0 , t ) = 0 ,
where
1 μ 2 2 M x x = E h 3 12 ( 1 v 2 ) 1 + η D t α c 2 w x 2 + v 2 w y 2 ,
1 μ 2 2 M y y = E h 3 12 ( 1 v 2 ) 1 + η D t α c v 2 w x 2 + 2 w y 2 ,
1 μ 2 2 M x y = h 3 12 · 2 E 1 + v 1 + η D t α c 2 w x y .

4. Numerical Algorithm

4.1. Function Approximation

When analyzing vibration problems, the three-dimensional function of displacement over time can be split into a product of a two-dimensional displacement function and a one-dimensional time function [32]:
w x , y , t = w ( x , y ) w ( t ) ,
where w ( x , y ) L 2 ( [ 0 , H ] × [ 0 , S ] ) , w t L ( [ 0 , K ] ) .
For any one-dimensional continuous function w t L ( [ 0 , K ] ) , it can be approximated by shifted Legendre polynomials:
w ( t ) k = 0 n 3 v k L k ( t ) = V T φ n 3 ( t ) .
For any two-dimensional continuous function w ( x , y ) L 2 ( [ 0 , H ] × [ 0 , S ] ) , it can likewise be approximated by shifted Legendre polynomials:
w ( x , y ) i = 0 n 1 j = 0 n 2 u i j L i ( x ) L j ( y ) = φ n 1 T ( x ) U φ n 2 ( y ) .
Furthermore, Equations (37) and (38) can be expressed in the form of a matrix product, respectively:
w ( t ) V T φ n 3 ( t ) = v 1 , v 2 , , v n 3 L 0 ( t ) L 1 ( t ) L n 3 ( t ) ,
w x , y φ n 1 T ( x ) U φ n 2 ( y ) = L 0 ( x ) , L 1 ( x ) , , L n 1 ( x ) u 00 u 01 u 0 n 2 u 10 u 11 u 1 n 2 u n 1 0 u n 1 1 u n 1 n 2 L 0 y L 1 y L n 2 y .
The displacement function is approximated by shifted Legendre polynomials, and the matrix product is formulated as
w x , y , t w x , y · w t = φ n 1 T ( x ) U φ n 2 ( y ) V T φ n 3 ( t ) ,
where U and V represent coefficient matrix, respectively.
The double-layer nanoplate structure studied in this paper will involve two displacement functions w 1 x , y , t and w 2 x , y , t . Thus, w 1 x , y , t and w 2 x , y , t are approximated as matrix products by shifted Legendre polynomials, and their respective expressions are as follows:
w 1 x , y , t w 1 x , y · w 1 t = φ n 1 T ( x ) U 1 φ n 2 ( y ) V 1 T φ n 3 ( t ) ,
w 2 x , y , t w 2 x , y · w 2 t = φ n 1 T ( x ) U 2 φ n 2 ( y ) V 2 T φ n 3 ( t ) .
Due to the displacement functions satisfying w i ( x , y ) L 2 ( [ 0 , H ] × [ 0 , S ] ) , according to Equations (7)–(9), it can be seen that φ n 1 T ( x ) φ n 2 ( y ) φ n 3 ( t ) has the same expression in both displacement functions, where U i and V i ( i = 1 , 2 ) are coefficient matrices.

4.2. Differential Operator Matrix

The governing equation, Equation (30), involves the integer partial differentiation of the displacement function w i ( x , y , t ) and the fractional partial differentiation. The differential operator matrix to be derived in this subsection will replace these partial differentiations.
Definition 3.
D x m is referred to as the m-order differential operator matrix of the shifted Legendre polynomials if the matrix D x m enables the existence of φ n 1 ( m ) ( x ) = D x m φ n 1 ( x ) .
Let m = 1 ; Equation (7) is as follows:
φ n 1 ( x ) = M G ( x ) = M G ( x ) = M 1 x ( x n 1 ) = M 0 1 n 1 x n 1 1 = M A G ( x ) ,
where A = [ a i j ] i , j = 0 n 1 , a i j = 0 , i 1 i , i = j + 1 .
According to Definition 3, the above equation can be written as
φ n 1 ( x ) = D x φ n 1 ( x ) = D x M G ( x ) .
The equality of equations Equations (44) and (45) leads to D x = M A M 1 .
Thus, φ n 1 ( x ) can be formulated as follows:
φ n 1 ( x ) = D x φ n 1 ( x ) = M A M 1 φ n 1 ( x ) ,
where D x = M A M 1 is called the first-order differential operator matrix of the shifted Legendre polynomials.
Let m = 2 , φ n 1 ( x ) can be expressed as
φ n 1 ( x ) = M A M 1 2 φ n 1 ( x ) = D x 2 φ n 1 ( x ) ,
where D x 2 = M A M 1 2 is the second-order differential operator matrix.
In the same vein, combining Equations (7)–(9), the differential operator matrix of the shifted Legendre polynomial can be derived as
φ n 1 ( m ) ( x ) = D x m c φ n 1 ( x ) = M A M 1 m φ n 1 ( x ) ,
φ n 2 ( m ) ( y ) = D y m c φ n 2 ( y ) = N A N 1 m φ n 2 ( y ) ,
φ n 3 ( m ) ( t ) = D t m c φ n 3 ( t ) = R A R 1 m φ n 3 ( t ) .
By combining Equations (48)–(50) with the functional approximation in Section 4.1 and substituting into the integral partial differential of the governing equation, we obtain the following result:
m w i ( x , y , t ) x m m φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) x m = m φ n 1 T ( x ) x m U i φ n 2 ( y ) V i T φ n 3 ( t ) = φ n 1 T ( x ) M A M 1 T m U i φ n 2 ( y ) V i T φ n 3 ( t ) ,
m w i ( x , y , t ) y m m φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) y m = φ n 1 T ( x ) U i m φ n 2 ( y ) y m V i T φ n 3 ( t ) = φ n 1 T ( x ) U i N A N 1 m φ n 2 ( y ) V i T φ n 3 ( t ) ,
m w i ( x , y , t ) t m m φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) t m = φ n 1 T ( x ) U i φ n 2 ( y ) V i T m φ n 3 ( t ) t m = φ n 1 T ( x ) U i φ n 2 ( y ) V i T R A R 1 m φ n 3 ( t ) ,
m + n w i ( x , y , t ) x m y n m + n φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) x m y n = m φ n 1 T ( x ) x m U i n φ n 2 ( y ) y n V i T φ n 3 ( t ) = φ n 1 T ( x ) M A M 1 T m U i N A N 1 n φ n 2 ( y ) V i T φ n 3 ( t ) ,
m + n w i ( x , y , t ) x m t n m + n φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) x m t n = m φ n 1 T ( x ) x m U i φ n 2 ( y ) V i T n φ n 3 ( t ) t n = φ n 1 T ( x ) M A M 1 T m U i φ n 2 ( y ) V i T R A R 1 n φ n 3 ( t ) ,
m + n w i ( x , y , t ) y m t n m + n φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) y m t n = φ n 1 T ( x ) U i m φ n 2 ( y ) y m V i T n φ n 3 ( t ) t n = φ n 1 T ( x ) U i N A N 1 m φ n 2 ( y ) V i T R A R 1 m φ n 3 ( t ) ,
m + n + r w i ( x , y , t ) x m y n t r m + n + r φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) x m y n t r = m φ n 1 T ( x ) x m U i n φ n 2 ( y ) y n V i T r φ n 3 ( t ) t r = φ n 1 T ( x ) M A M 1 T m U i N A N 1 n φ n 2 ( y ) V i T R A R 1 r φ n 3 ( t ) .
Definition 4.
K t is called the fractional operator matrix of the shifted Legendre polynomials if the matrix K t enables the existence of D t α φ n 3 ( t ) = K t φ n 3 ( t ) .
Utilizing the concept of Caputo derivatives, we can obtain that D t α c t n 3 = Γ ( n 3 + 1 ) Γ ( n 3 + 1 α ) t n 3 α .
Combined with Equation (9), D t α c φ n 3 ( t ) is expressed as follows:
D t α c φ n 3 ( t ) = D t α c ( R G ( t ) ) = R D t α c G ( t ) = R D t α c 1 t t n 3 = R 0 Γ ( 2 ) Γ ( 2 α ) t 1 α Γ ( n 3 + 1 ) Γ ( n 3 + 1 α ) t n 3 α = R B G ( t ) .
Since G ( t ) = R 1 φ n 3 ( t ) , D t α c φ n 3 ( t ) can be rewritten as
D t α c φ n 3 ( t ) = R B R 1 φ n 3 ( t ) = K t φ n 3 ( t ) .
Combining this formula with the approximate function from Section 4.1 and substituting it into the fractional partial differential terms in the governing equation, the results are as follows:
D t α c w i x , y , t D t α c φ n 1 T ( x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) = φ n 1 T ( x ) U i φ n 2 ( y ) V i T D t α c φ n 3 ( t ) = φ n 1 T ( x ) U i φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) ,
D t α c m w i ( x , y , t ) x m D t α c φ n 1 T ( x ) M A M 1 T m U i φ n 2 ( y ) V i T φ n 3 ( t ) = φ n 1 T ( x ) M A M 1 T m U i φ n 2 ( y ) V i T D t α c φ n 3 ( t ) = φ n 1 T ( x ) M A M 1 T m U i φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) ,
D t α c m w i ( x , y , t ) y m D t α c φ n 1 T ( x ) U i N A N 1 m φ n 2 ( y ) V i T φ n 3 ( t ) = φ n 1 T ( x ) U i N A N 1 m φ n 2 ( y ) V i T D t α c φ n 3 ( t ) = φ n 1 T ( x ) U i N A N 1 m φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) ,
D t α c m + n w i ( x , y , t ) x m y n D t α c φ n 1 T ( x ) M A M 1 T m U i N A N 1 n φ n 2 ( y ) V i T φ n 3 ( t ) = φ n 1 T ( x ) M A M 1 T m U i N A N 1 n φ n 2 ( y ) V i T D t α c φ n 3 ( t ) = φ n 1 T ( x ) M A M 1 T m U i N A N 1 n φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) .

4.3. Discretization of the Governing Equations for Viscoelastic Double-Layer Nanoplate

The matrix product form of the integer and fractional differential partial differentiation obtained above is brought into the governing equation, Equation (30), and the upper plate governing equation containing the matrix of differential operators is given by
E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) M A M 1 T 4 U 1 φ n 2 ( y ) V 1 T φ n 3 ( t ) + η E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) M A M 1 T 4 U 1 φ n 2 ( y ) V 1 T R B R 1 φ n 3 ( t ) + E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) U 1 N A N 1 4 φ n 2 ( y ) V 1 T φ n 3 ( t ) + η E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) U 1 N A N 1 4 φ n 2 ( y ) V 1 T R B R 1 φ n 3 ( t ) + v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) φ n 1 T ( x ) M A M 1 T 2 U 1 N E N 1 2 φ n 2 ( y ) V 1 T φ n 3 ( t ) + η v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) φ n 1 T ( x ) M A M 1 T 2 U 1 N E N 1 2 φ n 2 ( y ) V 1 T R B R 1 φ n 3 ( t ) + m 0 φ n 1 T ( x ) U 1 φ n 2 ( y ) V 1 T R A R 1 2 φ n 3 ( t ) + ( μ 2 m 0 m 2 ) φ n 1 T ( x ) M A M 1 T 2 U 1 φ n 2 ( y ) V 1 T R A R 1 2 φ n 3 ( t ) + ( μ 2 m 0 m 2 ) φ n 1 T ( x ) U 1 N A N 1 2 φ n 2 ( y ) V 1 T R A R 1 2 φ n 3 ( t ) + μ 2 m 2 φ n 1 T ( x ) M A M 1 T 4 U 1 φ n 2 ( y ) V 1 T R A R 1 2 φ n 3 ( t ) + 2 μ 2 m 2 φ n 1 T ( x ) M A M 1 T 2 U 1 N A N 1 2 φ n 2 ( y ) V 1 T R A R 1 2 φ n 3 ( t ) + μ 2 m 2 φ n 1 T ( x ) U 1 N A N 1 4 φ n 2 ( y ) V 1 T R A R 1 2 φ n 3 ( t ) ( 1 μ 2 2 ) q i ( x , y , t ) = 0 .
The lower plate governing equation is as follows:
E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) M A M 1 T 4 U 2 φ n 2 ( y ) V 2 T φ n 3 ( t ) η E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) M A M 1 T 4 U 2 φ n 2 ( y ) V 2 T R B R 1 φ n 3 ( t ) + E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) U 2 N A N 1 4 φ n 2 ( y ) V 2 T φ n 3 ( t ) + η E h 3 12 ( 1 v 2 ) φ n 1 T ( x ) U 2 N A N 1 4 φ n 2 ( y ) V 2 T R B R 1 φ n 3 ( t ) + v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) φ n 1 T ( x ) M A M 1 T 2 U 2 N E N 1 2 φ n 2 ( y ) V 2 T φ n 3 ( t ) + η v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) φ n 1 T ( x ) M A M 1 T 2 U 2 N E N 1 2 φ n 2 ( y ) V 2 T R B R 1 φ n 3 ( t ) + m 0 φ n 1 T ( x ) U 2 φ n 2 ( y ) V 2 T R A R 1 2 φ n 3 ( t ) + ( μ 2 m 0 m 2 ) φ n 1 T ( x ) M A M 1 T 2 U 2 φ n 2 ( y ) V 2 T R A R 1 2 φ n 3 ( t ) + ( μ 2 m 0 m 2 ) φ n 1 T ( x ) U 2 N A N 1 2 φ n 2 ( y ) V 2 T R A R 1 2 φ n 3 ( t ) + μ 2 m 2 φ n 1 T ( x ) M A M 1 T 4 U 2 φ n 2 ( y ) V 2 T R A R 1 2 φ n 3 ( t ) + 2 μ 2 m 2 φ n 1 T ( x ) M A M 1 T 2 U 2 N A N 1 2 φ n 2 ( y ) V 2 T R A R 1 2 φ n 3 ( t ) + μ 2 m 2 φ n 1 T ( x ) U 2 N A N 1 4 φ n 2 ( y ) V 2 T R A R 1 2 φ n 3 ( t ) ( 1 μ 2 2 ) q i ( x , y , t ) = 0 .
These structural boundary conditions are rewritten as a matrix product of the form
φ n 1 T ( 0 ) U i φ n 2 ( y ) V i T φ n 3 ( t ) = φ n 1 T ( L x ) U i φ n 2 ( y ) V i T φ n 3 ( t ) = 0 ,
φ n 1 T ( x ) U i φ n 2 ( 0 ) V i T φ n 3 ( t ) = φ n 1 T ( x ) U i φ n 2 ( L y ) V i T φ n 3 ( t ) = 0 ,
φ n 1 T ( 0 ) M A M 1 T 2 U i φ n 2 ( y ) V i T φ n 3 ( t ) + v φ n 1 T ( 0 ) U i N A N 1 2 φ n 2 ( y ) V i T φ n 3 ( t ) + η φ n 1 T ( 0 ) M A M 1 T 2 U i φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) + η v φ n 1 T ( 0 ) U i N A N 1 2 φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) = 0 ,
φ n 1 T ( L x ) M A M 1 T 2 U i φ n 2 ( y ) V i T φ n 3 ( t ) + v φ n 1 T ( L x ) U i N A N 1 2 φ n 2 ( y ) V i T φ n 3 ( t ) + η φ n 1 T ( L x ) M A M 1 T 2 U i φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) + η v φ n 1 T ( L x ) U i N A N 1 2 φ n 2 ( y ) V i T R B R 1 φ n 3 ( t ) = 0 ,
v φ n 1 T ( x ) M A M 1 T 2 U i φ n 2 ( 0 ) V i T φ n 3 ( t ) + φ n 1 T ( x ) U i N A N 1 2 φ n 2 ( 0 ) V i T φ n 3 ( t ) + η v φ n 1 T ( x ) M A M 1 T 2 U i φ n 2 ( 0 ) V i T R B R 1 φ n 3 ( t ) + η φ n 1 T ( x ) U i N A N 1 2 φ n 2 ( 0 ) V i T R B R 1 φ n 3 ( t ) = 0 ,
v φ n 1 T ( x ) M A M 1 T 2 U i φ n 2 ( L y ) V i T φ n 3 ( t ) + φ n 1 T ( x ) U i N A N 1 2 φ n 2 ( L y ) V i T φ n 3 ( t ) + η v φ n 1 T ( x ) M A M 1 T 2 U i φ n 2 ( L y ) V i T R B R 1 φ n 3 ( t ) + η φ n 1 T ( x ) U i N A N 1 2 φ n 2 ( L y ) V i T R B R 1 φ n 3 ( t ) = 0 .
The variable ( x , y , t ) is discretized into grid points ( x i , y j , t k ) by applying the collocation method, and then substituted into Equations (64)–(71). The MATLAB program obtains the numerical solution of the original system of equations. All numerical calculations were executed using MATLAB R2017b on a Laptop equipped with an Intel Core i9-12900H processor running at 2.26 GHz and 16 GB of RAM.

5. Convergence Analysis

This section deeply explores the convergence properties of the displacement function w ( x , y , t ) and its α -order derivative in the governing equations, which can be found in [33].
Theorem 1.
Suppose that the function w x , y , t Δ is a continuously differentiable function with a continuous M t h derivative, and the M + 1 t h derivative exists, where the region Δ = 0 , 1 × 0 , 1 × 0 , 1 . w n ( x , y , t ) is the best approximation of w ( x , y , t ) , and q n ( x , y , t ) is a shifted Legendre interpolating polynomial of the function w ( x , y , t ) at the point x i , y j , t k . Then, the following inequality relation holds:
w ( x , y , t ) w n ( x , y , t ) 2 w ( x , y , t ) q n ( x , y , t ) 2 .
The error inequality relation is formulated as follows:
w ( x , y , t ) w n ( x , y , t ) 2 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) x M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) y M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) t M + 1 1 64 1 M 3 M + 3 · max ( x , y , t ) Δ 3 M + 3 w ( x , y , t ) x M + 1 y M + 1 y M + 1 .
However, Theorem 1 concerns the convergence analysis of the displacement function in the governing equation. The convergence analysis of the fractional partial differentiation of the displacement function should not be neglected, and we further analyze the convergence of the fractional partial differentiation of the displacement function based on the above theorem [34].
The conclusion of Theorem 1 is simplified to the boundedness of the function, so it can be abbreviated as ε > 0 , w ( x , y , t ) q n ( x , y , t ) ε . According to Theorem 1, it can be seen that w ( x , y , t ) q n ( x , y , t ) is also a continuously differentiable function and is defined in the compact domain. Therefore, w ( x , y , t ) q n ( x , y , t ) is also bounded. By function boundedness, K > 0 , making w ( x , y , t ) q n ( x , y , t ) K ε .
According to the definition of Caputo fractional derivative in Equation (2), the fractional order α 0 , 1 . Let s = 1 ; Equation (2) can be rewritten as
D t α c f ( t ) = 1 Γ ( 1 α ) 0 t f ( λ ) ( t λ ) α d λ , 0 < α < 1 , f ( t ) , α = 1 .
Combining Equations (2) and (73), one obtains the following inequality:
D t α c w ( x , y , t ) q n ( x , y , t ) 1 Γ ( 1 α ) 0 t 1 ( t λ ) α w ( x , y , λ ) q n ( x , y , λ ) d λ = 1 Γ ( 1 α ) 0 t 1 ( t λ ) α w ( x , y , λ ) q n ( x , y , λ ) d λ 1 Γ ( 1 α ) K · [ 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) x M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) y M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) t M + 1 1 64 1 M 3 M + 3 · max ( x , y , t ) Δ 3 M + 3 w ( x , y , t ) x M + 1 y M + 1 y M + 1 ] · 0 t 1 ( t λ ) α d λ .
Since 0 t 1 ( t λ ) α d λ = t 1 α 1 α , where t 1 α is bounded, assuming that the upper bound is P, the function satisfies t 1 α P . So the following can be further obtained:
D t α c w ( x , y , t ) q n ( x , y , t ) t 1 α ( 1 α ) Γ ( 1 α ) K · [ 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) x M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) y M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) t M + 1 1 64 1 M 3 M + 3 · max ( x , y , t ) Δ 3 M + 3 w ( x , y , t ) x M + 1 y M + 1 y M + 1 ] P K ( 1 α ) Γ ( 1 α ) · [ 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) x M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) y M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) t M + 1 1 64 1 M 3 M + 3 · max ( x , y , t ) Δ 3 M + 3 w ( x , y , t ) x M + 1 y M + 1 y M + 1 ] .
Again according to the inequality in Equation (72), the convergence of the fractional partial differential of the displacement function is proved:
D t α c w ( x , y , t ) w n ( x , y , t ) 2 P K ( 1 α ) Γ ( 1 α ) · [ 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) x M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) y M + 1 + 1 4 1 M M + 1 · max ( x , y , t ) Δ M + 1 w ( x , y , t ) t M + 1 1 64 1 M 3 M + 3 · max ( x , y , t ) Δ 3 M + 3 w ( x , y , t ) x M + 1 y M + 1 y M + 1 ] .
So, by proving the convergence property of the displacement function w ( x , y , t ) and its α -order derivative in the governing equations, it can be shown that the governing equations of the double-layer nanoplate studied in the article all have the convergence property. These studies provide the theoretical basis for subsequent numerical calculations.

6. Dynamics Analysis

The purpose of this section is to deeply explore the dynamic behavior of a double-layer viscoelastic plate under the influence of different material properties. The vibration characteristics of the structure are systematically analyzed through a combination of theoretical modeling and numerical simulation, with a view to providing theoretical support and optimization suggestions for engineering applications.

6.1. Dimensionless Numerical Examples

In this subsection, we construct the following mathematical numerical example where the coefficients of each term in the equation, while not having direct physical significance, can be used to validate the effectiveness of the shifted Legendre polynomial approach to this class of problems. The numerical example concerns the dynamic response of a bilayer nanoplate structure under a given external excitation.
1.1 1 + 0.1 D t α c 4 w i x 4 + 1.1 4 w i y 4 + 1.1 1 + 0.1 D t α c 4 w i x 2 y 2 = 1 2 2 2 w i t 2 + 4 w i x 2 t 2 + 4 w i y 2 t 2 + q i ( x , y , t ) .
Firstly, assume the exact solutions of the following structure:
ω 1 x , y , t = 0.001 sin π x sin π y t 2 ,
ω 2 x , y , t = 0.001 sin π x sin π y t 2 .
These solutions represent the displacement distribution of the double-layer plate. Subsequently, we substituted the exact solutions into Equation (78) and derived the external forces q 1 and q 2 applied to the upper and lower plates by backward derivation:
q 1 = 2 m 2 v 1 π 2 sin π x l sin π y m l 2 2 m 0 v 1 sin π x l sin π y m + 2 m 2 v 1 π 2 sin π x l sin π y m m 2 + a 1 t 2 v 1 π 4 sin ( π x l ) sin ( π y m ) l 4 + a 1 t 2 v 1 π 4 sin ( π x l ) sin ( π y m ) m 4 2 μ 2 m 0 v 1 π 2 sin ( π x l ) sin ( π y m ) l 2 + 2 μ 2 m 2 v 1 π 4 sin ( π x l ) sin ( π y m ) l 4 2 μ 2 m 0 v 1 π 2 sin ( π x l ) sin ( π y m ) m 2 + 2 μ 2 m 2 v 1 π 4 sin ( π x l ) sin ( π y m ) m 4 + a 2 t 2 v 1 π 4 sin ( π x l ) sin ( π y m ) l 2 m 2 + 4 μ 2 m 2 v 1 π 4 sin ( π x l ) sin ( π y m ) l 2 m 2 + 24 η a 1 t 2 t ( 4 a t ) v 1 π 4 sin ( π x l ) sin ( π y m ) l 4 Γ ( 5 a t ) + 24 η a 1 t 2 t ( 4 a t ) v 1 π 4 sin ( π x l ) sin ( π y m ) m 4 Γ ( 5 a t ) + 24 η a 2 t 2 t ( 4 a t ) v 1 π 4 sin ( π x l ) sin ( π y m ) l 2 m 2 Γ ( 5 a t ) ,
q 2 = 2 m 2 v 2 π 2 sin π x l sin π y m l 2 2 m 0 v 2 sin π x l sin π y m + 2 m 2 v 2 π 2 sin π x l sin π y m m 2 + a 1 t 2 v 2 π 4 sin ( π x l ) sin ( π y m ) l 4 + a 1 t 2 v 2 π 4 sin ( π x l ) sin ( π y m ) m 4 2 μ 2 m 0 v 2 π 2 sin ( π x l ) sin ( π y m ) l 2 + 2 μ 2 m 2 v 2 π 4 sin ( π x l ) sin ( π y m ) l 4 2 μ 2 m 0 v 2 π 2 sin ( π x l ) sin ( π y m ) m 2 + 2 μ 2 m 2 v 2 π 4 sin ( π x l ) sin ( π y m ) m 4 + a 2 t 2 v 2 π 4 sin ( π x l ) sin ( π y m ) l 2 m 2 + 4 μ 2 m 2 v 2 π 4 sin ( π x l ) sin ( π y m ) l 2 m 2 + 24 η a 1 t 2 t ( 4 a t ) v 2 π 4 sin ( π x l ) sin ( π y m ) l 4 Γ ( 5 a t ) + 24 η a 1 t 2 t ( 4 a t ) v 2 π 4 sin ( π x l ) sin ( π y m ) m 4 Γ ( 5 a t ) + 24 η a 2 t 2 t ( 4 a t ) v 2 π 4 sin ( π x l ) sin ( π y m ) l 2 m 2 Γ ( 5 a t ) .
Next, we use the shifted Legendre polynomial algorithm with n = 4 to solve the mathematical example and evaluate the accuracy of the numerical solution using absolute error. The absolute error is calculated by the formula
E ( x , y , t ) = w n ( x , y , t ) w ( x , y , t ) ,
where w n ( x , y , t ) is the numerical solution and w ( x , y , t ) is the exact solution.
Through the calculation, we obtained the numerical solutions for the upper plate and the lower plate, respectively, as well as the absolute error plots, as shown in Figure 3 and Figure 4. We found that the numerical solutions were in high agreement with the exact solutions for both the upper plate and the lower plate, and the absolute errors were all of a small order of magnitude ( 1 0 5 ), which fully proves the reliability and practicability of this algorithm in the field of vibration analysis of the double-layer nanoplate structure.

6.2. Practical Numerical Examples

In the previous subsection, we verified the accuracy and feasibility of the adopted methodology through convergence analysis and dimensionless numerical examples. Based on these, the aim of this subsection is to delve into the specific effects of the material parameters (nonlocal parameter μ and stiffness coefficient k 0 ) on the dynamic response of the bilayer nanoplate structure when two different materials are used under specific loading conditions. Through numerical analysis, we expect to reveal how these parameters modulate the dynamic behavior of the structure and provide a theoretical basis for designing high-performance nanostructures.
In the analysis in this subsection, we consider a two-layer nanoplate structure with identical material properties and dimensions for the upper and lower plates, which is subjected to an equivalent transverse load q 1 for the upper plate and q 2 for the lower plate [35].
q 1 = k 0 w 1 w 2 + k 1 w 1 + f cos ( ω t ) ,
q 2 = k 0 w 2 w 1 + k 1 w 2 .
In particular, q 1 contains an external excitation f cos ( ω t ) , where the magnitude f = 0.01 and the frequency ω = π . A negative magnitude indicates that the external excitation is in the opposite direction to the pre-determined positive direction, which is due to internal pre-stressing of the structure, external constraints, or a specific loading method, and a discussion of negative magnitude is covered in [36], which can be used as a support.
Substituting the equivalent transverse loads applied to the upper and lower plates into the governing equations, respectively, the upper plate governing equation is formulated as follows:
E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w 1 x 4 + E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w 1 y 4 + v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) 1 + η D t α c 4 w 1 x 2 y 2 = 1 μ 2 2 m 0 2 w 1 t 2 + m 2 4 w 1 x 2 t 2 + 4 w 1 y 2 t 2 k 0 w 1 w 2 + k 1 w 1 + f cos ( ω t ) .
The lower plate governing equation is as follows:
E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w 2 x 4 + E h 3 12 ( 1 v 2 ) 1 + η D t α c 4 w 2 y 4 + v E h 3 6 ( 1 v 2 ) + E h 3 3 ( 1 + v ) 1 + η D t α c 4 w 2 x 2 y 2 = 1 μ 2 2 m 0 2 w 2 t 2 + m 2 4 w 2 x 2 t 2 + 4 w 2 y 2 t 2 k 0 w 2 w 1 + k 1 w 2 .
The material parameters of two different materials are particularly considered in this study. The parameters of two materials [8,37] are given in the following Table 1:

6.2.1. Influence of Nonlocal Parameters on Dynamic Behavior of Double-Layer Nanoplate

In order to fully assess the effect of nonlocal parameters, two different materials (Material 1 and Material 2) were selected for numerical analysis. For each material, we considered the case of nonlocal parameter μ = 0.5 nm, 1 nm, 1.5 nm, and 2 nm, respectively. Figure 5 demonstrates the amplitude response of the upper and lower plates of Material 1 for different nonlocal parameters at a certain moment in time. The left diagram shows that the amplitude of the upper plate remains relatively stable with the change in the nonlocal parameters, and there is no obvious increasing or decreasing trend. The right diagram shows that the amplitude of the lower plate is also not significantly affected by the nonlocal parameter and is always smaller than that of the upper plate. This pattern is repeated in the analysis of Material 2 (please see Figure 6).
For bilayer nanoplate structure, the high degree of symmetry between the upper and lower plates in terms of material properties, geometrical dimensions, and boundary conditions, as well as the strong coupling effects between them (the interfacial connections of the bilayer have a high stiffness, and the effects of nonlocal effects on the single plate are homogenized), together result in the fact that certain properties of the structure in terms of its dynamical behavior (e.g., the amplitude) are insensitive to the variations of the nonlocal parameters. This finding is consistent with some of the conclusions in the literature [37] and further confirms the reliability of our numerical results.
In addition, the purpose of the analysis of two different materials is to verify by comparison that this law is not a chance phenomenon and is consistent under the conditions in the paper. This provides an important basis for our understanding of the dynamical behavior of nanoscale structures.

6.2.2. Influence of Stiffness Coefficients on Dynamic Behavior of Double-Layer Nanoplate

Further exploration is required to examine the impact of material parameters on the dynamic response of double-layer nanoplates. We performed numerical analyses for two different materials (Material 1 and Material 2) to analyze the specific effect of the variation of stiffness coefficients k 0 on the structural response under different materials.
We have considered the cases with stiffness coefficients of 20, 40, 60, and 100 and calculated the amplitude response of the upper and lower plates at a certain point in time, respectively. Diagrams (a) and (b) of Figure 7 show the amplitude variation curves of the upper and lower plates, respectively, for different stiffness coefficients of Material 1 (plots (c) and (d) correspond to Material 2). It can be seen from the figures that the amplitudes of both the upper and lower plates show a decreasing trend as the stiffness factor increases.
In order to further verify this regular pattern, we calculated the displacement difference between the upper and lower plates under different stiffness coefficients of the two materials and plotted Figure 8 and Figure 9. These plots visually show that the upper-plate displacement is always larger than the lower-plate displacement, and with the increase in the stiffness coefficient, the vibration displacements of both the upper plate and the lower plate show a tendency to decrease.
The stiffness coefficient, as an important material property, reflects the ability of a structure to resist deformation. In this subsection, we find that an increase in stiffness coefficient can significantly reduce the amplitude of the plate, which may be due to the fact that the structure with higher stiffness has higher energy dissipation efficiency during the vibration process, thus effectively suppressing the amplitude of vibration.
This discovery has important implications for the design of high-performance nanostructures. By systematically adjusting the stiffness coefficient of the material, precise control over the dynamic behavior of the structure can be achieved to fulfill specific engineering requirements.

7. Conclusions

In this article, systematic research has been carried out for the modeling and numerical analysis of the fractional dynamics of bilayer nanoplate structures. Below are the main points summarized from the conclusions:
(1)
Based on the FKV model and the nonlocal elasticity theory, the fractional governing equations describing the coupled vibration behavior of bilayer nanoplates were successfully constructed, which were solved by applying the shifted Legendre polynomial algorithm.
(2)
The convergence analysis shows that the method exhibits good mathematical convergence regarding the displacement function and its fractional derivatives. Moreover, the dimensionless numerical example shows that the numerical and exact solutions are in high agreement in both the upper and lower plate vibration responses (with absolute errors up to 10 5 ), which verifies the effectiveness of the algorithm.
(3)
When the nonlocal parameter is varied in the range of 0–2 nm, the vibration displacements of the double-layer plate do not show significant sensitivity. However, as the stiffness coefficient increases, the vibration displacements of the upper and lower plates show a tendency to decrease.
The numerical analysis method in this study provides a theoretical tool for the dynamic design of nanoscale multilayer structures. Future research can be further extended to analyze the dynamics of multilayer nanostructures under complex working conditions, such as materials with functional gradients, temperature-coupled fields, and electromagnetic coupled fields.

Author Contributions

Conceptualization, Y.C. (Yiming Chen) and Q.F.; methodology, Y.C. (Yiming Chen) and Q.F.; software, J.Q. and Q.F.; validation, L.W. and Q.F.; formal analysis, Q.F.; investigation, Q.F.; resources, Q.L., Y.C. (Yuhuan Cui) and J.Q.; data curation, Q.F.; writing—original draft preparation, Q.F.; writing—review and editing, Q.F., L.W. and Q.L.; visualization, Q.F.; supervision, Y.C. (Yuhuan Cui), J.Q. and L.W.; project administration, Q.L., Y.C. (Yuhuan Cui) and J.Q.; funding acquisition, Q.L., Y.C. (Yuhuan Cui) and J.Q. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (52074126) and the Natural Science Foundation of Hebei Province (E2022209110) in China.

Data Availability Statement

Data sharing is not applicable. No data was used for the research described in the article.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Abbreviations

The following abbreviation is used in this manuscript:
FKVFractional Kelvin–Voigt

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Figure 1. Double-layer viscoelastic nanoplate.
Figure 1. Double-layer viscoelastic nanoplate.
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Figure 2. FKV model.
Figure 2. FKV model.
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Figure 3. Solutions and errors of the upper plate: (a) Exact solution. (b) Numerical solution. (c) Absolute error.
Figure 3. Solutions and errors of the upper plate: (a) Exact solution. (b) Numerical solution. (c) Absolute error.
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Figure 4. Solutions and errors of the lower plate: (a) Exact solution. (b) Numerical solution. (c) Absolute error.
Figure 4. Solutions and errors of the lower plate: (a) Exact solution. (b) Numerical solution. (c) Absolute error.
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Figure 5. Upper- and lower-plate amplitudes for different nonlocal parameters for Material 1: (a) Upper plate. (b) Lower plate.
Figure 5. Upper- and lower-plate amplitudes for different nonlocal parameters for Material 1: (a) Upper plate. (b) Lower plate.
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Figure 6. Upper- and lower-plate amplitudes for different nonlocal parameters for Material 2: (a) Upper plate. (b) Lower plate.
Figure 6. Upper- and lower-plate amplitudes for different nonlocal parameters for Material 2: (a) Upper plate. (b) Lower plate.
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Figure 7. Upper- and lower-plate amplitudes with different stiffness factors: (a) Upper plate of Material 1. (b) Lower plate of Material 1. (c) Upper plate of Material 2. (d) Lower plate of Material 2.
Figure 7. Upper- and lower-plate amplitudes with different stiffness factors: (a) Upper plate of Material 1. (b) Lower plate of Material 1. (c) Upper plate of Material 2. (d) Lower plate of Material 2.
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Figure 8. Displacement of upper and lower plates for different stiffness coefficients for Material 1: (a) Displacement of Material 1 with k 0 = 20 . (b) Displacement of Material 1 with k 0 = 40 . (c) Displacement of Material 1 with k 0 = 60 . (d) Displacement of Material 1 with k 0 = 100 .
Figure 8. Displacement of upper and lower plates for different stiffness coefficients for Material 1: (a) Displacement of Material 1 with k 0 = 20 . (b) Displacement of Material 1 with k 0 = 40 . (c) Displacement of Material 1 with k 0 = 60 . (d) Displacement of Material 1 with k 0 = 100 .
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Figure 9. Displacement of upper and lower plates for different stiffness coefficients for Material 2: (a) Displacement of Material 2 with k 0 = 20 . (b) Displacement of Material 2 with k 0 = 40 . (c) Displacement of Material 2 with k 0 = 60 . (d) Displacement of Material 2 with k 0 = 100 .
Figure 9. Displacement of upper and lower plates for different stiffness coefficients for Material 2: (a) Displacement of Material 2 with k 0 = 20 . (b) Displacement of Material 2 with k 0 = 40 . (c) Displacement of Material 2 with k 0 = 60 . (d) Displacement of Material 2 with k 0 = 100 .
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Table 1. Material properties.
Table 1. Material properties.
PerformancesMaterial 1Material 2
Young’s modulus E ( G p a ) 17651060
Poisson’s ratio v0.30.25
Winkler modulus k 1 1010
Damping factor η 0.10.1
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Fan, Q.; Liu, Q.; Chen, Y.; Cui, Y.; Qu, J.; Wang, L. Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm. Fractal Fract. 2025, 9, 477. https://doi.org/10.3390/fractalfract9070477

AMA Style

Fan Q, Liu Q, Chen Y, Cui Y, Qu J, Wang L. Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm. Fractal and Fractional. 2025; 9(7):477. https://doi.org/10.3390/fractalfract9070477

Chicago/Turabian Style

Fan, Qianqian, Qiumei Liu, Yiming Chen, Yuhuan Cui, Jingguo Qu, and Lei Wang. 2025. "Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm" Fractal and Fractional 9, no. 7: 477. https://doi.org/10.3390/fractalfract9070477

APA Style

Fan, Q., Liu, Q., Chen, Y., Cui, Y., Qu, J., & Wang, L. (2025). Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm. Fractal and Fractional, 9(7), 477. https://doi.org/10.3390/fractalfract9070477

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