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# A reprogrammable mechanical metamaterial with stable memory

## Abstract

Metamaterials are designed to realize exotic physical properties through the geometric arrangement of their underlying structural layout1,2. Traditional mechanical metamaterials achieve functionalities such as a target Poisson’s ratio3 or shape transformation4,5,6 through unit-cell optimization7,8,9, often with spatial heterogeneity10,11,12. These functionalities are programmed into the layout of the metamaterial in a way that cannot be altered. Although recent efforts have produced means of tuning such properties post-fabrication13,14,15,16,17,18,19, they have not demonstrated mechanical reprogrammability analogous to that of digital devices, such as hard disk drives, in which each unit can be written to or read from in real time as required. Here we overcome this challenge by using a design framework for a tileable mechanical metamaterial with stable memory at the unit-cell level. Our design comprises an array of physical binary elements (m-bits), analogous to digital bits, with clearly delineated writing and reading phases. Each m-bit can be independently and reversibly switched between two stable states (acting as memory) using magnetic actuation to move between the equilibria of a bistable shell20,21,22,23,24,25. Under deformation, each state is associated with a distinctly different mechanical response that is fully elastic and can be reversibly cycled until the system is reprogrammed. Encoding a set of binary instructions onto the tiled array yields markedly different mechanical properties; specifically, the stiffness and strength can be made to range over an order of magnitude. We expect that the stable memory and on-demand reprogrammability of mechanical properties in this design paradigm will facilitate the development of advanced forms of mechanical metamaterials.

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## Data availability

The CAD models of the moulds are available for reproduction. Source data are provided with this paper.

## Code availability

The codes that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We acknowledge funding from École Polytechnique Fédérale de Lausanne.

## Author information

Authors

### Contributions

T.C., M.P. and P.M.R. conceived the research problem. T.C. fabricated the samples, conducted experiments, performed the finite-element simulations and analysed the data. T.C., M.P. and P.M.R. interpreted the results, and wrote and edited the manuscript.

### Corresponding authors

Correspondence to Mark Pauly or Pedro M. Reis.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Corentin Coulais and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Characterizations of silicone materials.

a, Dimensions of the silicone cast dog-bone specimens. The gauge length is indicated. All dimensions are in millimetres. b, Stress–strain curves obtained from uniaxial testing of VPS32, VPS16 and the MRE (the VPS8-NdFeB composite). The corresponding Ogden hyperelastic model results using the parameters listed in c are overlaid as solid lines. c, Table listing the material parameters obtained from fitting the experimental data to the Ogden hyperelastic model for each of the silicone materials used in the fabrication of an m-bit. Source data

### Extended Data Fig. 2 Fabrication sequence of the m-bit.

a, Fabrication of the bistable shell, starting with a two-part 3D-printed mould for the shell and then an additional mould for the magnetic cap. b, c, Fabrication of the curved columns and the top lid.

### Extended Data Fig. 3 Programming and testing stage of the m-bits.

a, Photograph of the experimental apparatus. The m-bit specimens were attached to the xy stage (white) that moved atop of the transparent, acrylic plate (with an engraved grid for alignment). b, Alignment of the electromagnetic coils and the 6 × 6 m-bit array at the start of programming. c, The vertical position of the coils and the array is shown. The loading arm of the static testing machine passes through the top electromagnetic coil. We note that the top coil is mounted independently of the base plate and does not come into contact with the specimens or the loading plate. d, Controlled motion of the array in relation to the electromagnetic coils.

### Extended Data Fig. 4 Magnetic field generated by the Maxwell coil configuration.

a, Geometry and dimensions of the electromagnetic coils. Both top and side views are shown. The coordinates z and r are the vertical and the radial axes, respectively. b, Magnetic field probed horizontally across the diameter. c, Magnetic field probed concentrically across the height of the electromagnetic coil pair. d, Gradient of the field along the centre line. Solid lines represent the analytical solutions from equation (6) and (7) for the field and the field gradient, respectively. Source data

### Extended Data Fig. 5 Parametric study of the bistable shell.

a, A planar Von Mises bistable truss structure. b, Comparison between geometrical design and results from FE simulations. c, Conic geometry of the axisymmetric silicone bistable shell. d, e, A connector is cast at the apex concentric to the shell to enable displacement-controlled experiments. f, The geometrical quantities that define the shell geometry, t, g and α, are explored parametrically. g, h, i, Maximum bistable force fb,1 (g), ratio between two extrema forces fb,2/ fb,1 (h) and the distance between the two stable states dEq (i) as a function of slenderness (t/r) and of the inclination angle α. The boxed quantities in the axes correspond to the complete force-displacement plots in j and k. The parameter pair indicated with the red dot is used for the fabricated specimen. j, k, l, Loading responses obtained by varying the inclination angle α (j), the thickness of the shell t (k) and the thickness of the hinge of the shell g (l). Source data

### Extended Data Fig. 6 Parametric study of the curved columns.

a, Representative example of the geometry of the curved columns. The initial amplitude Aini and the column thickness tc are used as the parameters. b, Initial stiffness as a function of tc and Aini, obtained from the finite-element simulations. The entire force-displacement curves for the boxed quantities are plotted in c and d. c, d, Stress–strain curves of the columns with increasing thickness (c) and with increasing amplitude (d). e, The initial amplitude, Aini, of the curved column is optimized so that a vertical displacement of Δd = lini − lf results in a final amplitude of Af. The arc length of the curve must be preserved, given that we assume that no axial deformation occurs. f, The first two iterations of the optimization. g, The square error between the arc lengths of the initial curve and the final curve is shown for different intermediate displacements. Source data

### Extended Data Fig. 7 Measuring the spatially heterogeneous magnetic fields of an m-bit array.

a, A Hall sensor traverses above each m-bit and records the magnitude of the generated magnetic field. The unfiltered value is reported in real time. Four different patterns are used for demonstration purposes: b, all m-bits in the OFF state; c, column-wise alternating; d, checkerboard; and e, smiley face. The photographs are extracted from Supplementary Video 7. Source data

### Extended Data Fig. 8 Time-evolution of the state change of an m-bit.

a, Three programming cycles of one m-bit, showing the current control (top) and the resulting position of the magnetic cap (bottom). b, c, Detailed plots (over 2 s) of the state changes from OFF to ON (b) and from ON to OFF (c). d, Snapshots of the slow-motion videos at critical stages of the snap-through event. Source data

### Extended Data Fig. 9 Experimental mechanical testing of square assemblies of m-bits.

a, Experimental force-displacement plots of square arrays of sizes from 1 × 1 to 6 × 6 in either the all-ON (ξ = 1) or the all-OFF (ξ = 0) state. b, Force displacement for the different arrays normalized by the respective number of m-bits. Source data

### Extended Data Fig. 10 Programmable strain energy density of the m-bit array.

Experimental data showing a nearly linear increase in the strain energy density $$\hat{U}$$ of an 6 × 6 array as a function of the programming ratio (in the range 0 ≤ ξ ≤ 1, that is, as an increasing number of m-bits are switched from the OFF to the ON state). Source data

## Supplementary information

### Video 1

SI Video 1. Mechanical testing of an m-bit in both ON (left panel) and OFF (right panel) states. For either case, the same m-bit is compressed cyclically at the same speed (1 mm s-1, both upward and downward) and down to the same maximum displacement (8 mm). Two distinct behaviours are observed, resulting in significantly different mechanical responses. In both cases, the deformation is elastic (reversible). In particular, the bistable shell remains in its current equilibrium state (ON or OFF) throughout the entire deformation. The videos are recorded at 60 Hz using a Nikon D850 digital camera with a Micro-Nikkor AF-S VR 105mm f / 2.8G IF-ED lens. The side view is shown and the playback is in real time.

### Video 2

X-ray computed Tomography scans of the bistable elastic shell at different critical displacements: (I) initial, (II) highest force, (III) inflection point, (IV) lowest force, and (V) second equilibrium. The conditions for these points, I-V, are specified in the caption of Fig. 3 of the main text. The reconstructed 3D geometries are shown from one corner of the bistable shell to the mid plane; the animation shows different cross-sectional planes for each of the static I-V configurations. The apex of the shell is held by the connector shown in the Extended Data Fig. 5d, e and excluded from the reconstruction due to the connector’s lower density (which is below threshold for the X-ray imaging). The results from the FE simulation results are superimposed at the mid plane for each of these five stages, I-V, and accurately match the experimental reconstruction.

### Video 3

Programming of a single m-bit. Two cameras are used to simultaneously capture both the side view of the sample being programmed (left panel), as well as the output of the power supply (right panel). The power supply is current-controlled and is set to 10 A when switched on. The m-bit is stable in both states when the power is switched off. The playback is sped up by a factor of two from real time.

### Video 4

Cyclic programming of a single m-bit over ≈100 cycles, under the same experimental conditions as in video S3. The playback is sped up by a factor of 25 from real time. In the experimental results presented in Extended Data Fig. 8, we tested up to 1000 consecutive cycles.

### Video 5

Programming and testing of a 6×6 m-bit array. The entire automated programming and testing sequence is shown. The cameras capture simultaneously: a view of the experimental setup, the current input into the EM coils (output from the power supply), and the force displacement measurements obtained the load testing machine. Three programming patterns are shown, namely ξ=1.0, 0.5, and 0.0. For the intermediate programming ratio (ξ=0.5), a checker board pattern was used. The sample and the apparatus were inspected at the end of each step, even if these pauses are now apparent in the video. The playback is sped up by a factor of 5 from real time.

### Video 6

Slow motion videos of three programming cycles. The cycle number is indicated on the top-right corner of the video. To provide accurate motion tracking, a white dot was placed at the top of the magnetic cap. A slow-motion camera (MotionBLITZ EoSens® mini, Mikrotron) was used to image the snapping behaviour. The playback is slowed down by a factor of 0.02 from real time.

### Video 7

Real-time recording and display of magnitude field measurements over four test patterns. A Hall sensor attached to a x-y stage is used to obtain the magnitude field magnitude over each m-bit. A 6-fold increase is measured from the ON state compared to the OFF, allowing for simple classification.

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Chen, T., Pauly, M. & Reis, P.M. A reprogrammable mechanical metamaterial with stable memory. Nature 589, 386–390 (2021). https://doi.org/10.1038/s41586-020-03123-5

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