We develop in this paper a general asymptotic homogenization model pertaining to magnetoelectric shells and sandwich shells composed of an arbitrary distribution of reinforcements or actuators attached to the surface of a thin shell or sandwiched between two shells. The constituents of the reinforced shells are orthotropic and may exhibit piezoelectric and/or piezomagnetic behavior. A set of twenty unit cell problems are solved for a basic reinforced shell that acts as a building block from which other, more comprehensive structures, can be designed and analyzed. The unit cell problems permit the computation of the so-called effective coefficients of the homogenized shell structure. Of particular interest among the effective coefficients are the product properties which manifest themselves in piezoelectric/piezomagnetic composites but not (usually) in the individual constituents. The work highlights the strong influence of the curvature of the shell’s middle surface on the effective properties; thus, after homogenization the effective shell may be characterized by structural inhomogeneity even if the original structure was homogeneous. The model is illustrated via a number of interesting and practically important examples which include spherical, paraboloidal, ellipsoidal and cylindrical shells reinforced with ribs, wafers, or triangularly arranged reinforcements. It is shown in this paper that in the case of a purely elastic shell the results of the model converge to those of previously published works; in the more comprehensive case however pertaining to a shell reinforced with piezoelectric and piezomagnetic constituents, it is shown that the effective elastic (and most other) coefficients are functions of not only the elastic properties (as predicted by some other models) but also of the remaining properties (such as electric and magnetic properties) of the constituents. In this sense therefore, the model developed represents an important refinement over other works.

我们在本文中开发了一个关于磁电壳和夹心壳的一般渐近均匀化模型，该模型由任意分布的增强物或致动器组成，这些增强物或致动器附着在薄壳的表面或夹在两个壳之间。增强壳的成分是正交各向异性的并且可以表现出压电和/或压磁行为。解决了一组 20 个单元格问题，用于作为构建块的基本增强壳，可以设计和分析其他更全面的结构。晶胞问题允许计算均质壳结构的所谓有效系数。有效系数中特别令人感兴趣的是产品特性，这些特性在压电/压电复合材料中表现出来，但（通常）不在单个成分中表现出来。该工作突出了壳体中间表面曲率对有效性能的强烈影响；因此，在均质化后，即使原始结构是均质的，有效壳的特征也可能是结构不均质。该模型通过许多有趣且实际重要的示例进行说明，这些示例包括用肋条、晶片或三角形排列的加固物加固的球形、抛物面、椭圆形和圆柱形壳。本文表明，在纯弹性壳的情况下，模型的结果收敛于先前发表的作品的结果；然而，在更全面的情况下，关于用压电和压电成分增强的壳，它表明有效弹性（和大多数其他）系数不仅是弹性特性（如某些其他模型所预测的）的函数，而且是成分的剩余特性（如电和磁特性）。因此，从这个意义上说，开发的模型代表了对其他作品的重要改进。