Modular metamaterials composed of foldable obelisk-like units with reprogrammable mechanical behaviors based on multistability

A new type of modular metamaterials with reprogrammable mechanical properties is proposed based on the multistability in decoupled units. This metamaterial consists of periodically arranged foldable obelisk-like (FO) units, and each unit has three interchangeable states: two different soft states and a stiff state. Therefore, such metamaterial can possess various mechanical properties with different state combinations of units. Both theoretical and experimental investigations are conducted to understand the multistability in one unit and the reprogrammed mechanical properties in a two-dimensional tessellation. Additionally, we investigate the inverse question that whether the identical force response can be generated with different geometrical design of the metamaterial and propose a way to build 3D metamaterials with intended architectures. This work establishes general principles for designing mechanical metamaterials with independently transformable modules, and opens new avenues for various potential applications such as: self-locking materials, impact mitigation and stiffness transformation materials.


FO unit mechanics
When (plane angles ( , ) and lengths ( , ) are fixed, we define the energy of torsional spring BJ as ([3] in main text references) BJ ( BJ , 0 ; ) = 0.5 BJ ( − 0 ) 2 (S7) The energy of virtual spring AB is defined as But in practice, AB needs to be offset with a small angle * as a calibration: AB * ( AB , * ; ) = AB ( AB ; + * ) Thus, the total energy of the unit is where ℎ = 2BE ̅̅̅̅ sin ( 2 )
From the force-displacement curves (the raw data), stiffness of pattern "x" obtained from the gray unit is 1.07N/mm, and that from the orange unit is 1.49 N/mm, so the average is 1.28N/mm; stiffness of pattern "1" is 0.013N/mm; stiffness of pattern "0" is 0.026N/mm. The stiffness is obtained by the slope. The force 2×2 is derived by the total energy ∑ 4 =1 and the displacement as

Force model of the × tessellation
When there is no force applied, ℎ denotes the initial height: ℎ = ℎ 0 . The energies of a unit in pattern "1" and "0" are different, because the parameters ( BJ , 0 , AB , * ) in pattern "1" and "0" are different as shown in Section 4. (b) Vibration (dynamic) experiment for the 1D bar with 6 units. In the experiment setup, the bottom of the bar is bonded on the shaker, and the top of the bar is bonded with a mass piece and an acceleration sensor. The bar in patterns "1x1x1x" (blue) and "x1x1x1" (brown) have the similar vibration response, but there is a frequency shift (∆ 1 ≈ 21.62Hz) between them and pattern "xxxxxx" (black, upper plot). The interval between the two peaks of pattern "0xx0xx", ∆ 2 ≈ 12.86Hz (purple), changes to ∆ 3 ≈ 37.42Hz (green) when the bar is set in pattern "100100"

Other examples of reprogrammable static and dynamic properties
(lower plot). The photographs of the five patterns are shown below the experiment setup. The vibration responses of other patterns are shown in Fig. S6.
First, we intend to design a 1D bending bar with multiple units and experimentally demonstrate its reprogrammable stiffness. By fixing its right end as a cantilever beam, we pressed the free left end and obtained four different force responses with four patterns, as shown in Fig. S5a. It is found that when fixing the pattern for No.1 and No.3 unit (counting from left), pattern "x" for No.2 and No.4 unit yield large force response (the force of "x1x1" (orange) < the force of "xxxx" (blue), and "1111" (red) < "1x1x" (black)). When fixing the pattern for No.2 and No.4 unit, pattern "1" for No.1 and No.3 unit yield large force ("x1x1" (orange) < "1111" (red), and "xxxx" (blue) < "1x1x" (black)).
Although the structural patterns "x1x1" and "1x1x" are symmetric, their force responses are quite different. This is because No.1 and No.3 unit in pattern "1x1x" need larger forces when current pattern "1" switches to pattern "x". Since three stable states can be found for one unit ("1", "x", "0" ), there are in total 3 ( is the unit number) mechanical properties of the 1D bar. For the 1D bending bar, the stress-free heights ℎ 0 of the patterns in orange, red, blue, and black frame (Fig. S5a) are 76 mm, 78 mm, 85 mm and 87 mm, respectively.
Further, we made and then tested the vibration behaviors of the 1D bar with 12 geometrical patterns (here we choose five typical patterns). Interestingly, we find that the 1D bar has the dynamic reprogrammability which can be potentially used for a dynamic low-frequency wave filter. In Fig.   S5b (upper), there is a resonant frequency shift between some patterns (e.g., between "x1x1x1" (brown) and "xxxxxx" (black), ∆ 1 ≈ 21.62Hz) while nearly no change can be found between the "swapped" patterns (e.g., between "x1x1x1" (brown) and "1x1x1x" (blue)). In Fig. S5b (lower), the frequency interval between the two peaks can also be tuned in a programmatic fashion (e.g., for pattern "0xx0xx" (purple) the interval is ∆ 2 ≈ 12.86Hz; for pattern "100100" (green) the interval changes to ∆ 3 ≈ 37.42Hz). Each experimental result is robust as it repeats in 20 experiments.
In the dynamic study, the amplitude ratio was defined by the acceleration ratio, i.e., = out in , where denote the amplitude and acceleration, respectively. The subscripts "out" and "in" denote the output and input, respectively. Here, out comes from the acceleration sensor, and in = 0.0002m/s 2 in all vibration tests. The excitation signal is harmonic sinusoidal wave. In Fig. S5b, the normalized amplitude is defined as / max , where max is the maximum value in each test.
The total mass of the mass piece and the acceleration sensor is about 4.0g. The mass of the 1D bar is about 6.87g. The results were averaged by 20 vibration tests. For the tested samples, = = 60°, = 20mm and = 36mm (see main text Figure 1a for the definition of geometrical parameters).
The overall vibration responses are shown in Fig. S6.

Inverse design for gradient tessellations
Based on the Sections FO unit geometry and FO unit mechanics, by fixing the mechanics parameters ( BJ , 0 , AB , * ) using the fitting results and unfixing the geometrical parameters For the target tessellation with 3 × 3 units in Fig. 3a, the total energy is t ( ) = 9 (30°; ) Thus the force can be defined as where ℎ = 4BE ̅̅̅̅ sin ( 2 ) Fig. S7. Compression model of 3 × 3 tessellation.

Construction of 3D metamaterials
First, we construct a 3D cube with × × voxels, and then assign FO units only into the voxels with s( , , ) > 0 to form a TPMS-based structure (where ( , , ) is the order number of a voxel in X, Y, and Z direction). In the main text, the four TPMSs ( where the factor = 4 π⁄ , = 6 π⁄ .

Volume measurement
The fine sands with the average particle size about 38μm were used to measure the volume of the totally deployed 3D FO units. First, we measured the density of the sands as = 1.29g/ml, and then measured the mass of the sands that were loaded into the given unit. Finally, the full volume was calculated by / . The experimental results of the paper-made structures are slightly different from the numerical predictions. Fig. S8 shows that the calculated values are lower than the measured values. This is because when pouring the sands into a unit, all the facets were bended outward.
The volume is calculated by where S ∆GIH denotes the area of triangle GIH when the unit is totally unfolded (i.e., I, J and H are collinear). indicate the maximum and minimum results. Each structure was measured five times.

Force response of a unit fabricated using plastic sheets
Fig. S10. Force-displacement curve of a unit fabricated using plastic sheets. We find similar negative stiffness after the compression of 2 mm.