Supplemental Material - Programmable Mechanical Metamaterials
Authors: Bastiaan Florijn, Corentin Coulais and Martin van Hecke
In the document we provide details for the 13 Movies accompanying the paper Programmable Mechanical Metamaterials.
Experiments: 5X5 biholar sheet
In Fig. 2 of the manuscript we have indicated the four datasets, (i)-(iv), which are representative for the four qualitatively different behaviors observed. In the four movies entitled ExpTrajectory i, ExpTrajectory ii, ExpTrajectory iii and ExpTrajectory iv we show the real space pictures of the deformed biholar sheets and force while the material is compressed and decompressed.
The experimental compression rate is 0.05 mm per second, and these movies are sped up by a factor eight. We note that while we carefully removed tiny slivers of rubber that are artefacts of our moulding process for the center hole, some of these slivers are visible in the other holes --- these do not affect the mechanics or measurements of the polarization.
The movie FEM trajectory iii shows snapshots of the finite elements simulation,for trajectory (iii) of Fig. 3 of the manuscript. The simulation is performed in Abaqus and uses neo-hookian material. The movie is in two parts; for total times less than 1, the material is confined horizontally up to the strain εx; for larger times, the material is compressed until it reaches the strain εy . The color represents the Von Mises stress.
The movies softmechanism trajectory X (where X is A, B, C, i, ii, iii, iv) present the real space configuration, geometric interpretation, force and polarization for our soft mechanism model for the seven trajectories discussed in Fig. 4 of the manuscript. While it may appear to be more intuitive to use yo as the control parameter, this leads to a rather complicated construction when multiple branches are present. Therefore we choose to use θ as our control paramater, and in each movie first increase θ (at a constant rate), starting from θlim, and then decrease θ, starting from θlim, where θlim is the critical angle of the underlying transcritical bifurcation. In this way, we guarantee that all possible states are found --- note that εy, Fy and Ωfollow directly from θ, and that even in cases when there are multiple solutions for a given εy, there is a unique θ for each solution.
Each of this movie has four panels.
In the top-left, we show snapshots of the mechanism, where the blue lines represent the springs. In the top-right, we show the trajectories of the various equilibria in the (x, y) space (see Fig. 4d). Here, the black dashed line represents the trajectory (xo(θ),yo(θ)), the pink curve is the M-curve which relates xi and yi of the mechanism, and the blue curve is the corresponding evolute or Σ-curve.
The black diamond represents the state of the mechanism (which is directly given by θ), the red circle has its center ar (xo(θ),yo(θ)) and is tangent to M, and the red dot depicts the instantaneous values of (xo(θ),yo(θ)) as θ is varied.
The bottom panels show the corresponding force curve Fy(εy) and the polarization curve Ω(εy).
Experiments: 9X11 biholar sheet
In Fig. 5b-d we discuss experiments on an inhomogeneously clamped biholar sheet and show 10 snapshots --- in Exp Multistable we show the full movie, illustrating the propagation of the polarization wave in full detail. As in Fig. 5c, the color of the ellipses codes for their polarization Ω.