The main topic of the Thesis is the optimization of the compliance of thin elastic structures.The problem consists in finding the most robust configurations, when an infinitesimal amount of elastic material is subjected to a fixed force, and contained within a region having infinitesimal volume.The resistance to a load can be measured by computing a shape functional, the compliance, in which the shape represents the volume occupied by the elastic material. Thus we are led to study a minimization problem of a shape functional, under suitable constraints.In particular, we treat the case in which the design region is a thin rod, represented by a cylinder with infinitesimal cross section. The study finds its motivation in engineering problems: thin structures are very convenient to be used in practical applications.The approach we adopt draws inspiration from some recent works by I. Fragalà, G. Bouchitté and P. Seppecher, in which the authors deal with the case of thin elastic plates [G. Bouchitté, I. Fragalà, P. Seppecher: Structural optimization of thin plates: the three dimensional approach., Arch. Rat. Mech. Anal. (2011)]. We point out that these two problems are not merely technical variants one of the other, due to the substantial difference between the limit passages 3d-1d and 3d-2d, namely from 3 to 1 and from 3 to 2 dimensions.The study of optimal configurations led us to face another interesting variational problem: actually to establish whether homogenization phenomena occur in bars in pure torsion regime turns out to be equivalent to solve a nonstandard free boundary problem in the plane. This new problem is very challenging and, besides the link with torsion rods, it has mathematical interest in itself. One of the tools which can be employed to attack the problem is shape derivative for minima of integral functionals. The theory of shape derivatives is a widely studied topic (see e.g. the monograph by A. Henrot and M.Pierre Variations et Optimisation de Formes. Une Analyse Géométrique, Springer Berlin (2005), and the references therein), but the approach we propose in new and relies on assumptions which are weaker that the classical ones.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00845182 |
| Date | 26 February 2013 |
| Creators | Lucardesi, Ilaria |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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