Normalized Coordinate Equations and an Energy Method for Predicting Natural Curved-Fold Configurations

Abstract

Of the many valid configurations that a curved fold may assume, it is of particular interest to identify natural—or lowest energy—configurations that physical models will preferentially assume. We present normalized coordinate equations—equations that relate fold surface properties to their edge of regression—to simplify curved-fold relationships. An energy method based on these normalized coordinate equations is developed to identify natural configurations of general curved folds. While it has been noted that natural configurations have nearly planar creases for curved folds, we show that nonplanar behavior near the crease ends substantially reduces the energy of a fold.

Bistability and equilibria of creased annular sheets and strips

“…In other words, the crease will intersect with the nearby generators. This may require partitioning the deformed facet into several developable pieces with the generators bounded by different space curves [22], which is beyond the scope of this study. A recent study of the creased disk through FE modeling shows that near the crease, the lines of smallest principal curvature could intersect with the crease [63].…”

“…For engineering applications, discrete crease patterns have been introduced to both thin and thick plates to achieve different functions and forms, such as the foldability and free-form surfaces in rigid and curved origami [7][8][9][10][11][12][13][14][15][16] and sheet metals [17], energy absorption in crash tubes [18][19][20], and the redistribution of bending stiffness [21]. Introducing flexibility to the facets of creased thin sheets leads to the creation of new equilibria, which extend the configuration space of the traditional rigid origami [22][23][24][25][26].…”

“…It is the competition between the mechanics of creases and the flexibility of the facets that determines the mechanics of creased thin structures [14,22,27]. Thin sheets prefer to bend rather than to stretch due to the large ratio of stretching to bending stiffness.…”

“…Thin sheets prefer to bend rather than to stretch due to the large ratio of stretching to bending stiffness. Various continuum theories have been employed to study the mechanics of thin sheets and strips, e.g., Föppl-von Kármán theory [28], 1-director Cosserat plate theory [29], small-deflection inextensible plate theory [30][31][32][33], and geometrically exact inextensible strip model [22,27,34]. Under the inextensible theory, a flat sheet will be deformed into a developable surface.…”

“…The inextensible strip model is known to have singular behaviors for geometries where the local stretch of the surface would be preferred to be incorporated [39][40][41][42][43][44]. On the other hand, it works well to capture the mechanical behaviors of thin sheets with singularity-free and stretching-free geometries [22,27,37,45]. With the proper choice of materials and lightning, it is possible to approximately see the generators (i.e., unbent lines), and potential singularities and stretching areas in deformed thin sheets, which help determine if the inextensible strip model could be applied to the whole geometry or only part of it [1,3].…”