Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm
Abstract
1. Introduction
2. Fundamental Definition
2.1. Caputo-Type Fractional Derivatives
2.2. The Shifted Legendre Polynomials
3. Establishment of the Governing Equation for Double-Layer Nanoplate
3.1. The Constitutive Relationship
3.2. Derivation of the Governing Equations
4. Numerical Algorithm
4.1. Function Approximation
4.2. Differential Operator Matrix
4.3. Discretization of the Governing Equations for Viscoelastic Double-Layer Nanoplate
5. Convergence Analysis
6. Dynamics Analysis
6.1. Dimensionless Numerical Examples
6.2. Practical Numerical Examples
6.2.1. Influence of Nonlocal Parameters on Dynamic Behavior of Double-Layer Nanoplate
6.2.2. Influence of Stiffness Coefficients on Dynamic Behavior of Double-Layer Nanoplate
7. 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 ), 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.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
FKV | Fractional Kelvin–Voigt |
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Performances | Material 1 | Material 2 |
---|---|---|
Young’s modulus | 1765 | 1060 |
Poisson’s ratio v | 0.3 | 0.25 |
Winkler modulus | 10 | 10 |
Damping factor | 0.1 | 0.1 |
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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
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 StyleFan, 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 StyleFan, 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