New forms of the fundamental solutions for 3D magnetoelectroelasticity equations

Abstract

The paper reports the new forms of the fundamental solutions for 3D magnetoelectroelasticity equations. The state-of-art presented in the paper shows that the fundamental solutions for magnetoelectroelasticity equations take complex forms, especially considering their applications in the boundary element method. In the paper, the magnetoelectroelastic continuum serves as a homogenized model of the piezoelectric-piezomagnetic composites. The model is obtained by the Mori-Tanaka micromechanical approach. The so-called quasi-static approximation of dynamics is applied to obtain the set of the partial differential equations consists of the hyperbolic equation of motion and two elliptic equations for the conservation laws of the electric and magnetic charge. The motivation example shows the analogy between the impulse response of the linear system of ordinary differential equations, known from the classical linear control theory, and the fundamental solution of the linear system of partial differential equations. The spatial Fourier transform changes the coupled system of hyperbolic-elliptic equations of magnetoelectroelasticity into differential-algebraic equations in the k-space and the time domain. Instead of the classic approach eliminating the constraint equations for the electromagnetic static potentials, the semi-state vector is introduced and the descriptor system with the Kronecker index equals 1 is obtained. The structural analysis is performed for the resultant system and the regular and singular matrix pencils were analyzed. The slowness surfaces of the mainly acoustic modes in the quasi-static approximation are constructed for the 3-phase composite. Further, the shifting method is applied and the general solution of the semi-state equations is obtained as an analogon of the impulse response solution. The obtained result serves as a base for obtaining the new form of the fundamental solution of dynamic magnetoelectroelasticity in 3-dimensional space and also for the step response and steady-state fundamental solution for magnetoelectroelastic continuum.