The Physics of Origami
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McInerney, James
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Abstract
The physics of origami concerns the relationship between the introduction of creases to a thin sheet and their influence on the mechanical response of the sheet. In many cases, the kinematics are well-approximated by treating the origami sheet as completely rigid and imposing geometric compatibility conditions, similar to Lagrange multipliers, that permit folding of the sheet provided the faces are not deformed. In the linear regime, these rigid modes are governed by a compatibility matrix, similar to a quantum Hamiltonian, whose symmetry properties govern the existence and features of the modes. Most of the existing literature has focused on particular examples of crease patterns and there is a lack of general statements on the relationship between crease geometry and mechanical response.
In this thesis, I explore two broad classes of periodic origami sheet: those whose unit cells are composed of all (any number) triangular faces and those whose unit cells are com- posed of four parallelogram faces. For the triangulations I show that a hidden symmetry guarantees the existence of a two-dimensional space of degenerate ground states, through which the sheet can rigidly fold, and enforces a realness condition on the compatibility matrix so that periodically triangulated origami belongs to a special topological class of materials. For the parallelograms I show that a discrete symmetry divides the compatibility matrix into symmetric and antisymmetric sectors of modes and moreover, that these mode symmetries lead to distinct mechanical responses: the symmetric mode generates exclusively in-plane strains while the antisymmetric mode generates exclusively out-of-plane curvatures. These efforts form a basis for the classification of periodic origami by the symmetries of their compatibility matrices that may be extended to other systems such as spin magnets and mechanical metamaterials.
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Date
2021-08-06
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Dissertation