Frequency domain Green’s function and boundary integral equations for multifield materials and quasicrystals

Abstract

Present paper considers anisotropic elastic, magnetoelectroelastic and quasicrystal solids. It is shown that the equations of time-harmonic motion and constitutive relations for considered materials can be presented in the compact and unified form. The matrix approach is proposed for derivation of 3D time-harmonic Green’s functions for considered materials. The cases of each material type are studied separately. Two models of phason field dynamics in quasicrystals are considered in details. It is shown that in the case of phason field models based on the use of hydrodynamic formulations, phonon oscillations have an elastic nature, while the phason field leads to their decay due to phonon-phason interaction. The strict proof is given to the statement that the eigenvalues of the time-harmonic magnetoelectroelaticity problem are all positive. The paper also shows the application of the obtained time-harmonic Green’s functions in obtaining the boundary integral equations for the considered classes of problems. For this purpose, the novel approach is proposed, which utilizes only the symmetry of the unified material property tensor and the inertia matrix. Derived boundary integral equations can be used in the solution of boundary value problems for considered classes of materials.