Extension of Koiter’s linear shell theory to materials exhibiting arbitrary symmetry

Koiter’s linear shell theory applies to isotropic elastic materials and to anisotropic materials that exhibit reflection symmetry of the elastic properties with respect to the shell mid-surface. To the extent that such shells are exceptional, classical linear shell theory is incomplete. This lacuna is addressed here through a systematic procedure, applicable equally to all kinds of material symmetry, which entails an expansion of the potential energy of the shell in powers of its thickness in a manner reminiscent of Koiter’s work. The variables are the displacement field of the shell mid-surface and certain director fields that arise in the course of the expansion procedure. The directors are constrained in accordance with necessary conditions arising in the exact three-dimensional theory, yielding the optimal expression for the potential energy among those of third order in the small thickness. For materials lacking reflection symmetry, it is found that the strain energy of the shell is sensitive to tangential gradients of strain in addition to the usual strain and bending strain of the conventional theory.