Open Access

ARTICLE

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

Ran Wang¹, Xuegang Yuan^1,2, Hongwu Zhang¹, Jing Zhang³, Na Lv^2,*

1 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China.
2 School of Science, Dalian Minzu University, Dalian 116600, China .
3 College of Mathematics and Information, North Minzu University, Yinchuan Ningxia 750021, China.

* Corresponding author: Na Lv. Email: .

Computers, Materials & Continua 2018, 55(2), 345-357. https://doi.org/10.3970/cmc.2018.00233

Abstract

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.

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Keywords

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