On shear motions in nonlinear transverse isotropic elastodynamics

We study the propagation of shear waves in an anisotropic incompressible medium composed of an elastic fiber-reinforced material. We first derive the general determining equations of such motions when the fibers are arbitrarily arranged with respect to the direction of propagation of the waves. Then, we examine the standard reinforced model so as to root the nonlinearity of the dynamic problem directly to the presence of the reinforcing fibers, and finally, we consider an asymptotic model in terms of the limit of finite but small amplitude of the waves. Clearly, if the fibers are arranged along the direction of propagation of the waves, the mechanical behavior of the material will be isotropic. When the fibers are tilted an amount of the same order of the amplitude of the wave with respect to this direction, the asymptotic first-order reduced system is characterized by a peculiar mixed second-/third-order nonlinearity. This contrasts to what occurs for larger tilting angles where only quadratic nonlinearities must be considered. In the latter case, as we may expect, in the asymptotic limit, we reduce the system to an inviscid classical Burger's equation. Our results are fundamental to the study of the dynamics of fiber-reinforced materials and equally when we consider dissipative and/or dispersive effects.

Automated summary

This study investigates the propagation of shear waves in an anisotropic incompressible medium composed of an elastic fiber-reinforced material. The determining equations for such motions are derived, and the nonlinearity of the dynamic problem is attributed to the presence of reinforcing fibers. An asymptotic model is considered for waves with finite but small amplitude. The results show that the material behaves isotropically when the fibers are arranged along the direction of wave propagation. However, when the fibers are tilted, a peculiar mixed second-/third-order nonlinearity is observed for small tilting angles, while a quadratic nonlinearity is sufficient for larger tilting angles. The study's findings are fundamental for understanding the dynamics of fiber-reinforced materials, including dissipative and/or dispersive effects.