Finite-Amplitude Damped Inhomogeneous Waves in a Deformed Blatz-Ko Material

Abstract

In this paper special Blatz-Ko nonlinear elastic materials are considered, which are characterized by a constitutive constant and a constitutive function. We deal with the propagation of finite-amplitude inhomogeneous plane waves in such materials subjected to an arbitrary static homogeneous deformation. Linearly polarized transverse “damped” inhomogeneous plane wave solutions are explicitly obtained. Such waves are attenuated (or amplified) both in space and time (time-harmonic inhomogeneous plane waves obtained previously by Destrade appear as a special case). The properties of the energy flux vector and energy density associated with these wave solutions are investigated. With an appropriate concept of mean, it is seen that the “mean” energy-flux vector and the “mean” energy density satisfy two relations which are independent of the constitutive constant and constitutive function of the model, and of the homogeneous static deformation of the material. These relations are the same as those obtained by Hayes in the general context of time-harmonic inhomogeneous plane waves in linear systems. However, here, the theory is nonlinear, and the finite-amplitude waves are not time-harmonic.