Coarse-Grained Fundamental Forms for Characterizing Isometries of Trapezoid-based Origami Metamaterials

Investigations of origami tessellations as effective media reveal the ability to program the components of their elasticity tensor. However, existing efforts focus on crease patterns that are composed of parallelogram faces where the parallel lines constrain the quasi-static elastic response. In this work, crease patterns composed of more general trapezoid faces are considered and their low-energy linear response is explored. Deformations of such origami tessellations are modeled as linear isometries that do not stretch individual panels at the small scale yet map to non-isometric changes of coarse-grained fundamental forms that quantify how the effective medium strains and curves at the large scale. Two distinct mode shapes, a rigid breathing mode and a nonrigid shearing mode, are identified in the continuum model. A specific example, called Morph-derivative trapezoid-based origami, is presented with analytical expressions for its deformations in both the discrete and continuous models. A developable specimen is fabricated and tested to validate the analytical predictions. This work advances the continuum modeling of origami tessellations as effective media with the incorporation of more generic faces and ground states, thereby enabling the investigation of novel designs and applications.