A Cosserat Theory for Solid Crystals – with Application to Fiber-Reinforced Plates

The focus of this thesis is to understand the behavior of composite plates reinforced withrigid bars that are free to twist and bend with respect to the medium. Such composites areubiquitous in nature and in industry, especially with increased interest in modeling biological elements as well as nano-technology. Structured fabrics abound in nature and industry – from cytoskeletons to kevlar sheets.In the first part of this thesis existing theory on such materials is reviewed. A particularmicrostructure - that of a fiber-reinforced medium - is the subject of further study. Sucha medium is treated as a special case of a nematic elastomer with constrained directors.The salient feature of such a material is the presence of only along-fiber derivatives in theproblem, precluding certain boundary data.The second part of the thesis focuses on the development of a two-dimensional platetheory from the three-dimensional fiber reinforced medium studied previously. The resulting two-director model shows behavior similar to a nematic elastomer with directors that are unable to shear. This is then specialized to the case of a laminate with a single family of fibers. To obtain an understanding of the theory it is applied to a simple controllable deformation; that of a fiber-reinforced plate bent into a cylindrical shell. The orientation of the fibers is selected to result in both bending and twist. This controllable deformation helps us understand the possible boundary data and its e↵ect on the solution. The effect of boundary data on existence and uniqueness of solution is highlighted through the example problem