Energy flux and dissipation of inhomogeneous plane waves in hereditary viscoelasticity

Abstract

Inhomogeneous small-amplitude plane waves of (complex) frequency ω are propagated through a linear dissipative material which displays hereditary viscoelasticity. The energy density, energy flux and dissipation are quadratic in the small quantities, namely, the displacement gradient, velocity and velocity gradient, each harmonic with frequency ω , and so give rise to attenuated constant terms as well as to inhomogeneous plane waves of frequency 2 ω . The quadratic terms are usually removed by time averaging but we retain them here as they are of comparable magnitude with the time-averaged quantities of frequency ω . A new relationship is derived in hereditary viscoelasticity that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2 ω . It is shown that the complex group velocity is related to the amplitudes of the terms with frequency 2 ω rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.

Preface to a special feature dedicated to the memory of Prof. Peter Chadwick FRS

“…The paper by Fu et al originates from Peter's representation of the surface wave of arbitrary profile in terms of a single harmonic function, and presents a reduced model equation describing the surface dynamics of a generally anisotropic elastic half-space. Visco-and electroelastic phenomena are investigated in the papers by Scott [5] and Dorfmann & Ogden [6]. The first paper is focused on the analysis of energy flux, while the second one considers waves and vibrations in a finitely deformed flexible tube.…”