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Volume 7
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Article

Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod

by
Kai Fan
1,
Jiankang Liu
1,*,
Jinbin Wang
1 and
Chen Jin
2
1
School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China
2
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2023, 7(8), 586; https://doi.org/10.3390/fractalfract7080586
Submission received: 25 June 2023 / Revised: 17 July 2023 / Accepted: 27 July 2023 / Published: 29 July 2023
(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications)
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Abstract

Based on the layered and porous characteristics of functionally graded materials and the finite deformation assumption of solids, the fractal nonlinear propagation equation of longitudinal waves in a functionally graded rod is derived. A large number of exact displacement gradient traveling wave solutions of the fractal equation are obtained by using an equivalent simplified extended (G′/G) expansion method. Three sets of existing and different displacement gradient solutions are obtained by analyzing these exact solutions, and then three corresponding fractal dimension strain waves are derived. The results of numerical simulation of the evolution of these three strain waves with fractal dimension show that when the strain wave propagates in the rod, the smaller the fractal dimension or, the larger the radius of the rod, the higher the tensile strength of the material.
Keywords: fractal derivative; extended (G′/G) expansion method; finite deformation fractal elastic rod; kink solitary wave; fractal dimension bell-shaped strain solitary wave fractal derivative; extended (G′/G) expansion method; finite deformation fractal elastic rod; kink solitary wave; fractal dimension bell-shaped strain solitary wave

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MDPI and ACS Style

Fan, K.; Liu, J.; Wang, J.; Jin, C. Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod. Fractal Fract. 2023, 7, 586. https://doi.org/10.3390/fractalfract7080586

AMA Style

Fan K, Liu J, Wang J, Jin C. Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod. Fractal and Fractional. 2023; 7(8):586. https://doi.org/10.3390/fractalfract7080586

Chicago/Turabian Style

Fan, Kai, Jiankang Liu, Jinbin Wang, and Chen Jin. 2023. "Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod" Fractal and Fractional 7, no. 8: 586. https://doi.org/10.3390/fractalfract7080586

APA Style

Fan, K., Liu, J., Wang, J., & Jin, C. (2023). Evolution Analysis of Strain Waves for the Fractal Nonlinear Propagation Equation of Longitudinal Waves in a Rod. Fractal and Fractional, 7(8), 586. https://doi.org/10.3390/fractalfract7080586

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