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.

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3D 磁电弹性方程基本解的新形式

摘要 本文报道了 3D 磁电弹性方程基本解的新形式。论文中提出的最新技术表明，磁电弹性方程的基本解具有复杂的形式，特别是考虑到它们在边界元方法中的应用。在论文中，磁电弹性连续体作为压电-压电复合材料的均质模型。该模型是通过 Mori-Tanaka 微机械方法获得的。应用所谓的动力学准静态近似来获得由双曲运动方程和两个关于电荷和磁荷守恒定律的椭圆方程组成的偏微分方程组。动机示例显示了从经典线性控制理论已知的常微分方程线性系统的脉冲响应与偏微分方程线性系统的基本解之间的类比。空间傅立叶变换将磁电弹性的双曲椭圆方程的耦合系统改变为 k 空间和时域中的微分代数方程。与消除电磁静势约束方程的经典方法不同，引入半状态向量并获得克罗内克指数等于1的描述符系统。对所得系统进行结构分析，并对规则和奇异矩阵铅笔进行分析。准静态近似中主要声学模式的慢度面是为三相复合构造的。此外，应用移位方法并获得半状态方程的一般解作为脉冲响应解的类比。所得结果为获得三维空间动态磁电弹性基本解的新形式以及磁电弹性连续介质的阶跃响应和稳态基本解提供了基础。