Doctoral Thesis/Year of 2022 https://oist.repo.nii.ac.jp/?action=repository_rss&index_id=235 Year of 20222022-10-03T20:49:16+09:00Mathematical Modeling and Numerical Analysis of Unstretchable Elastic Ribbons https://oist.repo.nii.ac.jp/?action=repository_uri&item_id=2792 Being able to model stable states of elastic ribbons can be beneficial for understanding various natural phenomena in physics, biology and chemistry. Ribbons also find their application in engineering. However, identifying stable state of an unstretchable elastic ribbon is a hard task. When modeling a material that bends easily but cannot extend or contract much without tearing or creasing as a two-dimensional elastic body, its resistance to elongation and contraction can be incorporated considering only deformations under which the distances between material points are preserved. The primary objective of this thesis is to develop numerical methods for finding stable equilibria of an unstretchable two-dimensional elastic material bent so that its short edges are joined, with or without twist, to form a closed band. For certain parametrizations of the reference and deformed surfaces, a dimension reduction converts the problem to one involving a system of ordinary differential equations for a pair of vector fields satisfying certain constraints that derive from the requirement that the material be unstretchable and periodicity or antiperiodicity conditions that incorporate the way in which the short edges of the strip are joined. We discretize this problem to obtain a multi-dimensional constrained optimization problem that is solved numerically. To incorporate the discrete isometry constraints, we use Lagrange multipliers approach and minimize an accordingly augmented version of the bending energy. Additionally, we introduce an alternative constraint to ensure that the deformation is injective. The new constraint is bilateral and obviates the need to impose inequality constraints, removes a certain singular feature of the energy density and circumvents symmetry assumptions that have been imposed in all previous studies of this kind.2022-09-22T00:00:00+09:00