Origami-kirigami approach to materials structures modelling

Abstract

Additive manufacturing and origami/kirigami principles fit together like a coffee cup and saucer. Recently derived translational surface with an elliptic curve (not an ellipse), as the governing one, led to constructing origami-like object. Set of eight surfaces, each with a square-shaped orthogonal projection, turned out to be enchantingly similar to heaven-and-hell-origami. In presented work engaged starting translational surface (STS) is originally subjected to the selected linear and nonlinear transformations, in two ways. The first one is dedicated to spacial placement of STS replicas. Shape STS variations represent the second approach. Square-shape of STS orthogonal projection is preserved during all the mappings, it does not change. On the other hand, especially in this point, the similarity with origami/kirigami structures is broken; generally, square-shaped sheet of paper changes via folding. In our paper, preserved orthogonal projection offers suitable opportunities to illustrate a structure of the hypothetical material. Explicit and parametric equations of geometrical elements, transformational matrices and MATLAB application MuPAD serve as the useful mathematical and computational tools for formal and graphical representation of the modeled structures.