Fig. 3: The torsional deformation and mechanical characterization of triangular ring metamaterials.

a The prototype snapshots of initial stable state and the final deformed state of designed triangular ring metamaterials. The ring metamaterials in initial state from top to bottom are constructed by C_I, C_II, C_III, and R functional groups. The geometry of triangular ring metamaterials is fixed by 3D-printed plastic structural supports, see detail in Supplementary Fig. 27. The outer iron ring and the cords arranged at the evenly divided position are used to apply the external force at outer wall of ring metamaterials. The rendered images in the middle column show the force applied form during deformation. The green arrow indicates functional groups that can be transformed to each other. b Radial stretch R_i/R_i0 varied as a function of sweeping angle ratio ΔΘ/ΔΘ_max. R_i and ΔΘ are, respectively, the inner tangent circle radius of the inner cavity and sweep angle under torsional deformation. The solid lines represent theoretical results, and the discrete points denote experimental results. Source data are provided as a Source Data file. c Semi-experimental torque–radial strain results of triangular ring metamaterials. The abscissa ΔR_i/R_i0 represents the ratio of the variation of the inner radius, as characteristic size, of the metamaterial to its initial inner radius and denotes the difference between deformed radial stretch and initial radial stretch. The negative value corresponds to the torsional contraction deformation, and the positive value corresponds to the torsional expansion deformation. d Comparison of equivalent stiffness modulus between triangular rings and those of quadrilateral rings. The equivalent stiffness modulus is calculated by Supplementary Eq. (44).