Smooth Triaxial Weaving with Naturally Curved Ribbons

Triaxial weaving is a handicraft technique that has long been used to create curved structures using initially straight and flat ribbons. Weavers typically introduce discrete topological defects to produce nonzero Gaussian curvature, albeit with faceted surfaces. We demonstrate that, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously, which is not feasible using traditional techniques. Further, we reveal that the shape of the physical unit cells is dictated solely by the in-plane geometry of the ribbons, not elasticity. Finally, we leverage the geometry-driven nature of triaxial weaving to design a set of ribbon profiles to weave smooth spherical, ellipsoidal, and toroidal structures.

Figure 1

Representative family of triaxially woven unit cells for different numbers of ribbons (rows, $n=\{5,6,7\}$) and for different values of the in-plane curvature in their middle segment (columns, $k_2=\{-0.033,-0.013,0,0.013,0.033\}{\mathrm{mm}}^{-1}$). All ribbons have three segments, each of arc length ${\ell }_1={\ell }_2={\ell }_3=15\mathrm{mm}$; only the middle one is curved ($k_2\ne 0{\mathrm{mm}}^{-1}$), while those in the two extremities are naturally flat ($k_1=k_3=0{\mathrm{mm}}^{-1}$). Tomographic $\mu \mathrm{CT}$ scans (gray images) are juxtaposed on FEM simulations, color coded by the distance between their respective centerlines locations $e/\ell$. See the associated video in the Supplemental Material [23].

Figure 2

Unit cells woven with straight ribbons. (a) Photograph of a unit cell with $n=5$ straight ribbons. (b) Experimental data of the framed centerlines of the cell in (a) extracted from $\mu \mathrm{CT}$ ([23], Secs. 1.1–3). (c) FEM-computed version of (b). (d) Average interior angles of the $n$-gon $⟨{\theta }^{\circ }⟩$ and average integrated geodesic curvatures $⟨{\kappa }_g^{\circ }⟩$ vs $n$. The horizontal dashed lines at $⟨{\theta }^{\circ }⟩=2\pi /3$ and $⟨{\kappa }_g^{\circ }⟩=0$ are drawn to aid visual comparison with the geometric prediction. (e) Integrated Gaussian curvature of the unit cells with straight ribbons ${\mathcal{K}}_n^{\circ }$, computed through Eq. (1), vs $q^{\circ }=6-n$. The solid line is the prediction from Eq. (2). Inset: FEM-computed unit cells with $q^{\circ }=\{-3,-2,\dots ,3\}$, or equivalently $n=\{9,8,\dots ,3\}$.

Figure 3

Weaving with curved ribbons. (a) Schematics of the ribbons (top) and the planar representation of a typical unit cell. The ribbons have three distinct curved segments of arc length ${\ell }_j$ and normalized curvature ${\kappa }_j$ ($j=1$, 2, 3; color coded as red, green, and blue, respectively). (b) Integrated Gaussian curvature of the unit cells vs $q^{*}=6-n(1+{\kappa }^{*})$, with ${\kappa }^{*}$ from Eq. (3). Inset: unit cells with $n=6$, ${\kappa }_1={\kappa }_3=0$, and from left to right ${\kappa }_2=\{0.5,0.2,0,-0.2,-0.5\}$. (c1),(c2) Photographs of spherical weaves with (c1) straight and (c2) curved ribbons. The weaves consist of 12 pentagonal (blue shaded region) and 20 hexagonal (orange shaded region) unit cells. (d1),(d2) Reconstructed $\mu \mathrm{CT}$ images of the weaves in (c1) and (c2), respectively. The color bar indicates the normalized voxelwise radial deviation $\delta /R$ from a sphere of radius $R=42\mathrm{mm}$.

Figure 4

Nonspherical weaves with initially curved ribbons, reconstructed by photogrammetry. The color bar indicates the normalized deviation from a target surface $e/w$, where $w$ is the ribbon width. (a1)–(a3) Ellipsoidal weaves of aspect ratios, $a/b=\{0.75,1.25,1.5\}$, respectively. (b) Toroidal weave of inner radius $r_i=35\mathrm{mm}$ and outer radius $r_o=105\mathrm{mm}$. The planar geometries of the underlying curved ribbons for each of these weaves are provided in the Supplemental Material ([23], Secs. 3.2–3.3)