Selected Classes of Spatial Michell’s Structures

The spatial Michell structures are solutions to the problem ( 3.85) in the kinematic setting and to the dual problem ( 3.95) in the stress-based setting. The optimality condition ( 3.84) referring to the spatial setting (\(n=3\)) differs from the plane setting (\(n=2\)), since in the plane problem, the number of the bounds is equal to the number of the adjoint principal strains and in some regimes the following equalities can be fulfilled \(\displaystyle \varepsilon _{II}=-\frac{\sigma _0}{\sigma _C}\), \(\displaystyle \varepsilon _{I}=\frac{\sigma _0}{\sigma _T}\), while in the spatial problem we may have \(\displaystyle \varepsilon _{III}=-\frac{\sigma _0}{\sigma _C}\), \(\displaystyle \varepsilon _{I}=\frac{\sigma _0}{\sigma _T}\), with \(\varepsilon _{II}\) attaining neither of the bounds. Thus, we expect that in many spatial problems the solutions will be either composed of fibrous membrane shells, or will be collections of planar structures not linked in the direction transverse to their planes. The well-known Michell’s sphere as well as other optimal shells of revolution subject to the pure torsion load belong to the first class mentioned, see Sects. 5.2 and 5.3. The rotationally symmetric Hemp’s structures (Sect. 5.1) belong to the second class.