Hardt, Robert2022-09-262022-09-262022-052022-04-21May 2022Chen, Yikai. "Mathematical Results for Michell Trusses." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113381">https://hdl.handle.net/1911/113381</a>.https://hdl.handle.net/1911/113381Given an equilibrated vector force system F of finite mass and bounded support, we investigate the possibility and properties of a cost minimizing structure of given materials that balances F. Our work generalizes and reinterprets results of Michell and Gangbo where the given equilibrated force system occurs on a finite set of points and the balancing structure consists of finitely many stressed bars joining these points. Such a stressed bar corresponds to an interval [a,b] in Rn having a multiplicity lambda in R where |lambda| indicates the stress density on the bar and sgn(lambda) indicates whether it is being compressed or extended. While there exists a finite bar system to balance any given equilibrated finite force system, Michell already observed that a finite cost-minimizing one may not exist. In this thesis, we introduce two new mathematical representations of Michell trusses based on one- dimensional finite mass varifolds and flat Rn-chains. Here one may use a one-dimensional signed varifold to model the balancing structure so that the internal force of the positive (or compressed) part coincides with its first variation of mass while the internal force of the negative (or extended) part coincides with its negative first variation. For the chain model, we use the subspace of structural flat Rn chains in which the coefficient vectors are a.e. colinear with the orientation vectors. The net force then becomes simply the Rn chain boundary and so cost-minimization becomes precisely the mass-minimizing Plateau problem for structural chains. For either model, a known compactness theorem leads to existence of optimal cost-minimizers as well as time-continuous cost-decreasing flows.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Michell TrussesGeometric structuresThesis2022-09-26