A new structure that has just been found in this present, named to the convex-concave shell. Based on the change of radius according to the sine or cosine function, a type of rotating circular shell made of functionally graded materials (FGM) has appeared and been investigated in the paper. Characteristics of dynamics of the FGM convex-concave shell in thermal environment and surrounded by the elastic medium are analyzed by an analytical approach. The piezoelectric layers are bonded in the internal and external surfaces of the shells. By using the classical thin shell theory, Von Karman-Donnell geometrical nonlinearity assumption, the governing equations are established and the Galerkin's method is applied to investigate the nonlinear dynamical behaviors of the shells. The numerical results are calculated and validated to the previous literatures of the cylindrical shells-a special case of this structure. In addition, Finite Element Analysis (FEA) is also used to compare with these of the convex-concave shells in two types I and II. The effects of the electro-thermal environment, geometrical parameters and the elastic foundation are scrutinized by the numerical and 2D and 3D graphical results. The achieved results of the structure promise to contribute greatly to significant applications of aerospace, civil and mechanical engineering.

在本发明中刚刚发现的一种新结构被命名为凸凹壳。根据半径根据正弦或余弦函数的变化，出现了一种由功能梯度材料（FGM）制成的旋转圆形壳体，并已对其进行了研究。通过解析的方法分析了FGM凸凹壳在热环境中并被弹性介质包围的动力学特性。压电层结合在壳体的内表面和外表面中。利用经典的薄壳理论，通过Von Karman-Donnell几何非线性假设，建立了控制方程，并采用Galerkin方法研究了壳的非线性动力学行为。计算得出的数值结果已得到圆柱壳的先前文献的验证，这是这种结构的特例。此外，还使用有限元分析（FEA）来与两种类型I和II中的凸凹壳进行比较。电热环境，几何参数和弹性基础的影响通过数值以及2D和3D图形结果进行仔细检查。该结构的成功成果有望为航空航天，土木和机械工程的重要应用做出巨大贡献。几何参数和弹性基础由数值以及2D和3D图形结果进行检查。该结构的成功成果有望为航空航天，土木和机械工程的重要应用做出巨大贡献。几何参数和弹性基础由数值以及2D和3D图形结果进行检查。该结构的成功成果有望为航空航天，土木和机械工程的重要应用做出巨大贡献。