Energy balance and fundamental relations in anisotropic-viscoelastic media

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The first part of this work analyses the energy balance equation for inhomogeneous time-harmonic waves propagating in a linear anisotropic-viscoelastic medium whose constitutive equation is described by a general time-dependent relaxation matrix of 21 independent components. This matrix includes most linear anisotropic-viscoelastic rheologies and the generalized Hooke's law. The balance of energy allows the identification of the potential and loss energy densities, which are related to the real and imaginary parts of the complex stiffness matrix. The second part establishes some fundamental relations valid for inhomogeneous viscoelastic plane waves. The scalar product between the complex wavenumber and the complex power flow vector is a real quantity proportional to the time-average kinetic energy density. As in the anisotropic-elastic case, it is confirmed that the phase velocity is the projection of the energy velocity vector onto the propagation direction. A similar equation is obtained by replacing the energy velocity with a velocity related to the dissipated energy. Finally, as in isotropic-viscoelastic media, the time-average energy density can be obtained from the projection of the average power flow vector onto the propagation direction, and the time-average dissipated energy density from the projection of the average power flow vector onto the attenuation direction.