Abstract
The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation.
Highlights
We investigate the propagation and interaction of weakly nonlinear elastic plane waves in transversely isotropic media
It is known that a quadratic nonlinearity is present only in quasilongitudinal waves when we restrict ourselves to isotropic materials [1]
The aim of this paper is to demonstrate the difference between isotropic and transversely isotropic models with respect to the propagation and interaction of weakly nonlinear elastic waves
Summary
We investigate the propagation and interaction of weakly nonlinear elastic plane waves in transversely isotropic media. We are interested in weakly nonlinear effects. It is known that a quadratic nonlinearity is present only in quasilongitudinal waves when we restrict ourselves to isotropic materials [1]. The “weakest” type of nonlinearity which is manifested by shear waves propagation is cubic in such materials [2,3]. Anisotropy and in particular the presence of fibres changes this situation. The coupling of shear waves on a quadratically nonlinear level is possible when a special direction of propagation in anisotropic materials is chosen [3,4,5]. We demonstrate that the quadratic nonlinearity may show up even for a single shear wave propagation due to the presence of fibres
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