Work

Fractal Modeling of Nonlinear Flexural Wave Propagation in Functionally Graded Beams: Solitary Wave Solutions and Fractal Dimensional Modulation Effects

Year: 2025

Type: article

Abstract: In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to the fractal dimension index. In the framework of the fractal derivative, a fractal nonlinear wave governing equation integrating geometric nonlinear effects and microstructure characteristics was derived, and the coupling effect of finite deformation and fractal characteristics was clarified. Four groups of deflection gradient traveling wave analytical solutions were obtained by solving the equation through the extended minimal (G′/G) expansion method. Compared with the traditional (G′/G) expansion method, the new method, which is concise and expands the solution space, generates additional csch2 soliton solutions and csc2 singular-wave solutions. Numerical simulations showed that the spatiotemporal fractal dimension can dynamically modulate the amplitude attenuation, waveform steepness, and phase rotation characteristics of kink solitary waves in beams. At the same time, it was found that the decrease in the spatial fractal dimension will make the deflection curve of the beam more gentle, revealing that the fractal characteristics of the microstructure have an active control effect on the geometric nonlinearity. This model provides theoretical support for the prediction and regulation of the wave behavior of fractal microstructure FGM components, and has application potential in acoustic metamaterial design and engineering vibration control.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract9090553

Authors: Kai Fan, Zhongqing Ma, Cunlong Zhou, Jiankang Liu, Huaying Li

Institution: Taiyuan University of Science and Technology

Language: English

Topic: Nonlocal and gradient elasticity in micro/nano structures

Subfield: Materials Chemistry

Field: Materials Science

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funder: National Natural Science Foundation of China