A three-dimensional actuated origami-inspired transformable metamaterial with multiple degrees of freedom

Abstract

Reconfigurable devices, whose shape can be drastically altered, are central to expandable shelters, deployable space structures, reversible encapsulation systems and medical tools and robots. All these applications require structures whose shape can be actively controlled, both for deployment and to conform to the surrounding environment. While most current reconfigurable designs are application specific, here we present a mechanical metamaterial with tunable shape, volume and stiffness. Our approach exploits a simple modular origami-like design consisting of rigid faces and hinges, which are connected to form a periodic structure consisting of extruded cubes. We show both analytically and experimentally that the transformable metamaterial has three degrees of freedom, which can be actively deformed into numerous specific shapes through embedded actuation. The proposed metamaterial can be used to realize transformable structures with arbitrary architectures, highlighting a robust strategy for the design of reconfigurable devices over a wide range of length scales.

Figure 1. Our work is inspired by snapology.

(a) Snapology is a type of modular, unit-based origami in which paper ribbons are folded, and ‘snapped' together to assemble extruded polyhedra, such as the extruded icosahedron shown. (b) Some of the geometries that can be made in this way, including the extruded icosahedron, are almost rigid. (c) In contrast, other geometries, including the extruded cube, have multiple degrees of freedom and can be easily deformed.

Figure 2. Analysis of the possible shapes of the extruded cube unit cell.

(a) The shape of the unit cell is found by extruding the edges in the direction normal to the faces of a rhombohedron, and can be fully described by the vectors p₁, p₂ and p₃ spanning the internal rhombohedron. (b) Regular tetrahedron containing all combinations of angles (γ₁, γ₂, γ₃) that are attainable. (c) State #1, #2, #3 and #4 are configurations that lie in the centre of the regular tetrahedron, on the centre of its faces, on the centre of its edges and on its vertices, respectively. (d) Contour plot showing the evolution of internal volume (v_int) of the unit cell as a function of γ₁, γ₂ and γ₃. (e) Contour plot showing the evolution of the strain energy (U) of the unit cell as a function of γ₁, γ₂ and γ₃. Note that the values of v_int and U are shown on the boundary of a sub-region of the regular tetrahedron, which, because of the symmetry of the unit cell, contains all possible configurations. Moreover, the orange and green lines on the unit cell indicate edges (hinges) whose energy are specified by the γ and φ angles, respectively.

Figure 3. Fabrication and deformation of a single extruded cube unit cell and the corresponding mechanical metamaterial.

(a) The unit cells were fabricated using three layers: two outer layers of polyethylene terephthalate (with thickness t=0.25 mm and 0.05 mm) and a layer of double-sided tape (t=0.05 mm) in the middle. The layers were cut in three steps to form flat building blocks with both flexible and rigid regions. (b) The extruded rhombi were formed by simply removing the building blocks from the layered sheet, folding them and sticking their ends together using the revealed adhesive tape. To form the unit cell, six cubes were attached together using the double-sided tape incorporated into the layered sheet. (c) State #1, #2, #3 and #4 can be realized by simply applying a compressive load. (d) A highly flexible mechanical metamaterial with a cubic microstructure was formed by connecting the outer edges of 64 identical unit cells. An external force can trigger a collective behaviour which shapes the cubic crystal into a number of different configurations. Scale bars, 3 cm.

Figure 4. Actuation of the unit cell and the corresponding mechanical metamaterial.

(a) To freely transform the entire unit cell, inflatable air pockets are placed on the hinges highlighted in orange (see the ‘Methods' section). (b) An internal pressure in the air pockets results in a moment in the hinges, causing the extruded rhombus to flatten. (c) Surfaces for which ∂U/∂γ_i=0. When moving between two states connected by a path that remains on one of these surfaces, the corresponding γ_i angle does not have to be actuated. (d) Configurations obtained by actuating the unit cell (with 3 actuators). (e) Improved actuation strategy to reach state #4. As expected state #4 does not fold completely flat, but instead deforms into the state with lowest strain energy for which φ₁=φ₂=φ₃=2π/3. (f) Actuation of the mechanical metamaterial (with 96 actuators). Note that all structures are actuated by connecting the air pockets to three separate syringes through transparent tubes. Scale bars, 3 cm.

Figure 5. Uniaxial compression of an extruded cube unit cell pre-folded into four different states.

(a) Snapshots of the loaded unit cell for states #1, #2 and #3. (b) The unit cell is folded into state #4 and then compressed by applying 10,000 N. Remarkably, the fully expanded state can be recovered after removal of the load. (c) Force-displacement curves under uniaxial compression of a single unit cell at different configurations. Note that the initial force at zero displacement indicates the force required to maintain the unit cell into its pre-deformed (folded) state. Scale bar, 4 cm.