Phase transformation in helical structures

Martensitic phase transformation in crystalline solids is a first-order, solid to solid, diffusionless phase transformation. The structure exhibits changes in symmetry groups and structural parameters at the phase transformation temperature.

Using helical groups, we develop a diffusionless phase transformation theory for helical structures. We discovered a generic mechanism for phase transformation in nanotubes by forming compatible interfaces (Fig. 1(a)). Like the celebrated austenite-martensite interface in bulk martensitic materials, the conversion of one phase (green) to another requires twinning (red/blue) (Fig. 1(b)), but the mechanism of transformation is completely different from bulk martensites. In the case of phase transformations in crystals, this kind of twinning has led to materials with low hysteresis and dramatically improved resistance to transformational fatigue. Analogously, in helical structures, we predicted that a low-hysteresis helical phase transformation involving both twist and extension will also emerge by satisfying the derived compatibility/twinning condition, and can be applied to engineering applications such as artificial muscles and actuators.

Origami structures generated by isometry groups

Origami is an ancient art of paper folding. Designers use the isometric shape change induced by origami to achieve a desirable and highly reconfigurable 3D surface for applications in aerospace, medical devices, and architecture. The origami system is highly nonlinear and geometrically constrained. As a result, capturing the global properties of these structures–such as deformability and multistability with arbitrary folding lines–is a challenging task.

We tackle the global problem by harnessing the symmetry of origami. Applying the discrete helical groups to a four-fold origami as a unit cell, one can obtain helical Miura origami (HMO) (see Fig. 2(a)). The HMO is found to be multistable and can achieve stretch-twist coupling by phase transformation. Alternatively, using the isometry groups that generate quasicrystals, we can design highly deformable quasicrystalline origami (Fig. 2(b)). The study of quasicrystalline origami structures is in its infancy. From a broad perspective, the idea of using isometry groups bridges the design of origami with the structure of materials.

Deployable origami and kirigami structures

Deployable structures are widely used in aerospace engineering and medical devices such as foldable solar panels and medical stents. The structure is deployable when it is always geometrically compatible (no bend or stretch) upon varying the free parameter of the system continuously. Designing deployable structures usually requires solving the geometrical compatibility condition of the system.

Using group theory and kinematics, we formulate and solve the deployability condition for quadrilateral mesh origami. We obtain an efficient algorithm for designing all possible rigidly and flat-foldable quad-mesh origami. Based on the algorithm, we develop an inverse design framework for approximating arbitrary 3D surfaces using deployable origami. For example, in the top video, the designed origami pattern can be folded from a flat state to a vase-like state and then to a folded-flat state, without stretching or bending its panels.

We also solve the deployability condition for quadrilateral planar kirigami, in which the cuts form quadrilateral slits. The result reveals that the quadrilateral kirigami is deployable if and only if it has a compatible open state with parallelogram slits. The result holds for quadrilateral kirigami with different topologies; for example, the deployable kirigami has a hole in its initially compact state in the bottom video. We can use this result to design deployable kirigami approximating desired boundary shapes.

Geometrical evolution of liquid crystal elastomers

Liquid crystal elastomers (LCEs) are active materials that can have significant shape changes upon heat or illumination. A flat LCE sheet programmed with a 2D director field will contract along the director (blue lines in Fig. 3) and elongate perpendicularly (Fig. 3(a)). The deformation is therefore non-isometric. Non-isometric deformation induces non-trivial Gaussian curvature (GC). For example, in Fig. 3(b), the actuated twinned cones have positive GC concentrated at tips and negative GC concentrated along creases (red).

Using differential geometry, we quantify the GC generated by actuating the reference director pattern, with a focus on the GC concentrated along creases. We find that the main ingredients for the concentrated GC along creases are the first-order variations of directors and twinning angles at the reference curve. Analyzing the relationship between the GC and the mechanical response, we can program the actuated shape by programming the director pattern. For example, in Fig. 3(c), the crease with positive GC will curve downward, while the crease with negative GC will curve upward. We envision that the results are useful for designing active sheets, actuators, and soft robotics.