Hyperbolicity of velocity–stress–electromagnetic field equations for waves in anisotropic magnetoelectroelastic solids with hexagonal symmetry

Abstract

This paper reports on a theoretical framework to analyse coupled wave propagation in magnetoelectroelastic solids of hexagonal symmetry. The constitutive equations contain the effective properties of micromechanics to model piezoelectric–piezomagnetic composites. The Mori–Tanaka scheme is used. The governing equations include the elastodynamic equations of motions and unique forms of the Maxwell equations for electromagnetics. The result is a set of fifteen first-order, fully coupled, hyperbolic partial differential equations with velocities, elastic stress components, and electromagnetic fields as the unknowns and as the components of the state vector. The structural analysis of the introduced equation set is performed, and the hyperbolicity is formally proved. The physics of coupled acoustic–electromagnetic wave propagation are fully described by the eigenstructure of matrices included in the considered system of equations. In particular, the eigenvalues of the main matrix pencil are the wave speeds, and a part of the left eigenvectors represents the coupled-wave polarization. The spectrum of the main matrix pencil shows the two-scale structure with the gap between mainly electromagnetic and mainly acoustic coupled modes. The analysis of the two-scale system dynamics is also performed. The small parameter is identified, and the slow–fast decomposition is obtained. The slow subsystem in the two-time-scale model is identified as the so-called quasi-static approximation of the magnetoelectroelastic continuum dynamics.