Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity
Abstract
Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern's soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami's creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes  origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a platelike theory for parallelogram origami patterns with an explicit coarsegrained quadratic energy depending on the gradient of the creaseactuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two wellknown origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.
 Publication:

arXiv eprints
 Pub Date:
 November 2023
 DOI:
 10.48550/arXiv.2311.10870
 arXiv:
 arXiv:2311.10870
 Bibcode:
 2023arXiv231110870X
 Keywords:

 Condensed Matter  Soft Condensed Matter
 EPrint:
 minor improvements to exposition