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.

本文对弱非线性平面波在各向同性和横向各向同性介质中的传播和相互作用进行了研究。它始于对考虑的超弹性模型的存储能量函数的定义。提供了平面波的弹性动力学方程以及一阶拟线性双曲系统。确定该系统的本征系统以研究三波相互作用系数。本文的主要部分涉及对这些系数的讨论。应用弱非线性渐近方法表明，在横向各向同性的情况下，无粘性的伯格斯方程描述了单个准剪切波的演化。该结果与各向同性的情况相矛盾，在各向同性的情况下，具有二次非线性的方程无法描述任何剪切波的传播。