The present study investigates the thermoelastic response of a heterogeneous piezoelectric sphere under thermal shock loading. Boundary conditions as well as loading are considered as symmetric; thus, the response of the sphere is expected to be symmetric. All of the properties of the thick-walled sphere, including mechanical, electrical and thermal properties, are considered dependent on the radial position, except for the relaxation time, which is considered a constant value along the radius. The governing equations of the sphere have been derived under heterogeneous anisotropic assumptions. The general form of the second law of thermodynamics, which is nonlinear in nature, and is called nonlinear energy equation is used. The number of the established equations is three, which includes the motion equation, energy equation and Maxwell electrostatic equation of Maxwell. These equations are obtained in terms of radial displacement, temperature difference and electric potential. The energy equation is derived based on Lord and Shulman theory with a single relaxation time. In the next step, by introducing dimensionless variables, the governing equations are provided in dimensionless presentation. Then these equations have been discretized using generalized differential quadrature method. Also, in order to follow the solution of the equations in time domain, Newmark method has been used. Since the system of equations is nonlinear, Picard algorithm is applied as a predictor-corrector mechanism to solve the nonlinear system of equations. Then numerical results are presented to investigate the propagation of mechanical, thermal and electric waves inside the heterogeneous sphere and also their reflection from the outer surface of the sphere. By examining the results, it can be seen that mechanical and thermal waves propagate with a limited speed, while the speed of electric wave propagation is infinite.

中文翻译：

---

由功能梯度压电材料制成的球形血管的非线性广义压电热弹性


本研究研究了异质压电球在热冲击载荷下的热弹性响应。边界条件和载荷被认为是对称的;因此，球体的响应应该是对称的。厚壁球体的所有特性，包括机械、电气和热特性，都被认为取决于径向位置，但松弛时间除外，它被认为是沿半径的常数值。球体的控制方程是在异构各向异性假设下推导出来的。使用了热力学第二定律的一般形式，它本质上是非线性的，称为非线性能量方程。建立的方程数为三个，包括运动方程、能量方程和麦克斯韦静电方程。这些方程是根据径向位移、温差和电势获得的。能量方程是根据 Lord 和 Shulman 理论推导出来的，具有单一的弛豫时间。在下一步中，通过引入无量纲变量，以无量纲表示形式提供控制方程。然后，使用广义微分正交法对这些方程进行离散化。此外，为了在时域中跟踪方程的解，使用了 Newmark 方法。由于方程组是非线性的，因此将 Picard 算法用作预测器-校正器机制来求解非线性方程组。然后，给出了数值结果，以研究机械波、热波和电波在异质球体内的传播以及它们从球体外表面的反射。 通过检查结果，可以看出机械波和热波以有限的速度传播，而电波的传播速度是无限的。