Asymptotic homogenization of magnetoelectric reinforced shells: Effective coefficients and influence of shell curvature

Abstract

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.