Least-weight design of perforated elastic plates for given compliance: Nonzero Poisson's ratio
In earlier papers of the authors and \textit{R. Sandler}, \textit{N. Olhoff} and \textit{M. P. Bendsaee} [Int. J. Solids Struct. 23, 521-536 (1987; Zbl 0606.73105) and ibid. 537-550 (1987; Zbl 0606.73106)] least-weight solutions were derived for perforated elastic plates having (a) a constant thickness (b) a prescribed compliance value and (c) a Poisson ratio of zero value. In view of earlier mathematical studies by Kohn, Stand, Lurie et al., the above results were based on a microstructure in which ribs of first- and second-order infinitesimal spacing run in the two principal directions. It was found, however, that the optimal solution for axially symmetric plates always consists of (i) unperforated regions or/and (ii) regions with ribs in only one (i.e. the radial) direction. In view of extensive recent study of this problem by leading mathematicians, the above conclusions have important implications. In the current paper, the foregoing results are extended to plates having a nonzero Poisson ratio and the theory is illustrated with examples. Tow solutions given are valid for both prescribed compliance and prescribed defections. This investigation shows that in some unconstrained shape optimization problems the optimal solution cannot be obtained by conventional numerical (e.g. finite element) methods.
simply supported uniformly loaded circular plates
ribs of first- and second-order infinitesimal spacing
scientific article; zbMATH DE number 4003689
scientific article