Navigating the landscape of nonlinear mechanical metamaterials for advanced programmability

We consider a flexible mechanical metamaterial comprising an elastomeric matrix with an embedded square array of circular holes. First, we use the deflated continuation technique of bifurcation analysis to explore its complex energy landscape, characterized by multiple bifurcations from which stable and unstable branches emanate. We then investigate how this landscape can be used to design materials with advanced programmability. We find that the response of the system can be constantly reprogrammed through local manipulation, moving it from one stable branch to another, and that small targeted imperfections can be harnessed to enhance such programmability.

Figure 1

(a), (b) Computed bifurcation diagrams for structures with an array of (a) $2\times 2$ and (b) $3\times 3$ holes. The color of the branches corresponds to the minimal eigenvalue ${\lambda }_{\text{min}}$, a measure of stability. The inset gray diagrams show the undeformed structures. The inset in panel (b) shows a zoomed-in region near the bifurcation. (c) Solutions corresponding to the labeled branches $h_{2\times 2}, h_{3\times 3}$, and $i$–$vii$ at ${\varepsilon }_a=-0.07$. Collections of branches are labeled based on having an identical mechanical response, and they are sublabeled with capital letters to differentiate between left and right symmetries.

Figure 2

(a) Experimentally measured stress-strain curve for the $2\times 2$ structure, overlaid on the numerical bifurcation diagram from Fig. 1 (colored lines). The solid black line shows the average response for four samples, and the gray envelope shows the standard deviation. The inset shows a deformed sample at ${\varepsilon }_a=-0.07$, in the sheared ${\mathit{ii}}_{\text{A}}$ mode. (b) Experimentally measured stress-strain curves for the $3\times 3$ structure. In 10 samples tested, 7 follow branch $\mathit{vi}$ and 3 follow branch vii; the averages and standard deviations for each set are plotted separately. The insets show deformed samples in both deformation modes at ${\varepsilon }_a=-0.07$. (c) Stress-strain curve for a single $2\times 2$ structure that is manually switched from branch ${\mathit{ii}}_{\text{A}}$ to ${\mathit{ii}}_{\text{B}}$ at ${\varepsilon }_S=-0.072$. The curve after the switch is shown in green. Insets show the structure before (${\varepsilon }_S^{+}$) and after (${\varepsilon }_S^{-}$) the switch. (d,e) Stress-strain curves for two differing $3\times 3$ structures that are manually switched from one branch to another at ${\varepsilon }_S=-0.06$. (f) Images of the $3\times 3$ structures before and after the switches.

Figure 3

Bifurcation diagrams for the perturbed structures using imperfection magnitudes of $\delta =5\times {10}^{-4}$ (top row) and $\delta =2\times {10}^{-3}$ (bottom row). The branches are colored according to the stability measure ${\lambda }_{\text{min}}$. The five columns correspond to perturbations based on the structures from branches ${\mathit{i}}_{\text{A}}, {\mathit{iv}}_{\text{C}}, v$, iii, $i$ in Fig. 1, which are plotted at the top of each column; the first four columns use perturbations based on stable branches, and the last column uses a perturbation based on an unstable branch. The images in panels (i) and (j) show the deformed structures at the end of two branches at ${\varepsilon }_a=-0.07$.

Figure 4

(a) Experimental stress-strain curve (black) for the $3\times 3$ structure with an $\mathit{iv}$ perturbation. For any applied ${\varepsilon }_a$ past the bifurcation point, all three previously predicted stable branches are realizable via manual switching. The stress-strain curves after switching are shown in green and include the noise introduced during the switching process (green vertical line). The inset images show the sample in the three different stable branches. (b) Bifurcation diagram for a $2\times 2$ structure with its central point (red dot) constrained to move vertically. Colored lines show the simulation results using the same color scheme as in Fig. 1. The solid line shows the experimental stress-strain curve, with the experimental snapshot showing the resulting polarized mode. All experimental images are captured at ${\varepsilon }_a=-0.07$.