Origami lattices with free-form surface ornaments

Abstract

Lattice structures are used in the design of metamaterials to achieve unusual physical, mechanical, or biological properties. The properties of such metamaterials result from the topology of the lattice structures, which are usually three-dimensionally (3D) printed. To incorporate advanced functionalities into metamaterials, the surface of the lattice structures may need to be ornamented with functionality-inducing features, such as nanopatterns or electronic devices. Given our limited access to the internal surfaces of lattice structures, free-form ornamentation is currently impossible. We present lattice structures that are folded from initially flat states and show that they could bear arbitrarily complex surface ornaments at different scales. We identify three categories of space-filling polyhedra as the basic unit cells of the cellular structures and, for each of those, propose a folding pattern. We also demonstrate "sequential self-folding" of flat constructs to 3D lattices. Furthermore, we folded auxetic mechanical metamaterials from flat sheets and measured the deformation-driven change in their negative Poisson's ratio. Finally, we show how free-form 3D ornaments could be applied on the surface of flat sheets with nanometer resolution. Together, these folding patterns and experimental techniques present a unique platform for the fabrication of metamaterials with unprecedented combination of physical properties and surface-driven functionalities.

Fig. 1. Folding kinematics.

(Category 1) To unfold category 1 lattices, every floor needs to be sliced at its boundary with the adjacent floor. Within this context, a floor is a row of the lattice structure. The unfolded floors (A₁, A₂,…) are connected in series. Here, a cubic lattice is illustrated as an example of category 1 lattices. Dark and light green denote the same type of folding pattern. (Category 2) The lattices belonging to this category need to be sliced at the middle of their floors as well as the boundary of the floors. The unfolded state of these lattices is made of a backbone that comprises every other half-floor (A₁, A₂,…) and the remaining half-floors (B₁, B₂,…) that branch out of the backbone. Therefore, green and pink colors denote folding patterns that face each other after folding. Alternative arrangements of the half-floors are possible, as shown in the two alternative variants. Here, a truncated octahedron lattice is depicted as an example of this category of foldable lattices. (Category 3) The unfolding of category 3 lattices is similar to that of category 2 lattices (a similar notation and color code are used), except that the positioning of the half-floors is not orthogonal anymore. A rhombic dodecahedron lattice is shown as an example of the foldable lattices from this category.

Fig. 2. (Self-)folding of origami lattices.

(A) The folding sequences for a category 3 lattice (rhombic dodecahedron). Folding sequences of all other sample lattices of Fig. 1 can be seen in videos S1 to S3. (B) The time sequence of sequential self-folding in a three-story thick panel lattice. 3D-printed panels were hinged together using metal pins and a number of elastic rubber bands, and thus, the stored potential energy was used as a parameter for programming sequential self-folding. (C) The design of the self-folding lattice including the initial flat configuration and the final folded state. The hinges are designed to provide a confined space that locks the different floors after self-folding, thereby ensuring the integrity of the lattice and providing load-bearing capacity.

Fig. 3. Conventional and auxetic metallic origami lattices.

(A) An aluminum cubic lattice structure comprising three unit cells in each direction. (B) Multilayer assembly of a reentrant lattice structure (a variant of truncated octahedron). (C and D) Compression stress-strain results for cubic and reentrant structures. (E) The evolution of the average Poisson’s ratio with compressive strain, ε_yy, in the auxetic lattice. The evolution in the shape of the unit cells belonging to the ensemble that was used for calculating the Poisson’s ratio is depicted as well.

Fig. 4. Free-form ornamentation of origami lattices.

(A) EBID was used to deposit gasified platinum-based precursor on top of a polished flat origami truncated octahedron cut from a 200-μm pure titanium foil. Multiple 2D and hierarchical 3D patterns have been produced with feature sizes in the range of a few tens of nanometers. The surface ornaments were imaged using scanning electron microscopy (SEM). The dimensions of the ornaments were measured with atomic force microscopy (AFM; details in the Supplementary Materials). (B) A flat pure titanium foil was patterned using an ultrashort pulse laser. Three spots in the unfolded sheet were patterned using three different types of pattern design. The final configuration of folded lattice shows the arrangement of the three patterned spots. The profiles of the patterns were measured using 3D optical microscopy (details in the Supplementary Materials).