Michell-like grillages and structures with locking (Q2774051)

The study of microstructure and the development of theory of locking materials goes back to the 1970s, and is closely related to studies of homogenization carried out by R. V. Kohn and his coauthors in that era. William Prager exhibited much interest in the Michell optimal designs, and produced several publications, first working alone, and later in cooperation with Rozvany. It was discovered that, as the stiffness conditions (or minimal compliance) are imposed, the shape degenerates into massive amount of very thin ribs, or grillages which distribute the load. One recalls the beautiful shapes derived by Michell in the first decade of the 20th century. What we call Michell structures are solutions to finding an infimum to the integral \(\int_\Omega\{|\sigma_I|+ |\sigma{II}|\} dA\), where \(\sigma_I\) and \(\sigma_{II}\) are the principal (admissible) stresses.NEWLINENEWLINENEWLINEOne possible approach is to start with composite two-material design and watch as one of the materials disappears and is replaced by micro-voids, as the optimization process proceeds. Here, the authors investigate the locking limit in this optimization process as the volume decreases. The locking locus is shown to converge to a square as the Poisson ratio varies between \(0\) and \(1\). The so-called \(G\)-limit sought here was probably first proposed by K. A. Lurie, and then investigated by \textit{K. A. Lurie} and \textit{A. V. Cherkaev} [Proc. R. Edinb., Sect. A 104, 21-38 (1986; Zbl 0623.73011)]. Their results for shells are compared both with Koiter's theory of this shells and with classical results of Rockafellar-Temam convexity theory. The authors conclude finally that the shape optimization for isotropic plates with small volumes leads to locking, representing a solution of dual variational problems.NEWLINENEWLINENEWLINEThis is an important article, revealing an impressive progress in understanding of some rather mysterious at first glance behavior, which has played earlier structural optimization processes. The reviewer recalls correspondence with William Prager in the late sixties, in which he received an advice to carry optimization for a few steps, and then to quit, because of strange properties emerging as one approaches the ``optimum''. Particularly interesting is the authors' result establishing duality between the process described here and the transition to plasticity, and also a remak that the tensors that the reviewer always associated with specific stress and strain tensors, assume an interpretation of rates of these tensors in this limit analysis. However, the reviewer found some parts of this paper rather difficult to read without reading several earlier papers.