Focusing and Mode Separation of Elastic Vector Solitons in a 2D Soft Mechanical Metamaterial

Soft mechanical metamaterials can support a rich set of dynamic responses, which, to date, have received relatively little attention. Here, we report experimental, numerical, and analytical results describing the behavior of an anisotropic two-dimensional flexible mechanical metamaterial when subjected to impact loading. We not only observe the propagation of elastic vector solitons with three components—two translational and one rotational—that are coupled together, but also very rich direction-dependent behaviors such as the formation of sound bullets and the separation of pulses into different solitary modes.

Figure 1

(a) Schematic of the system. (b) Definition of velocity of squares. (c)–(f) Contour plots of parallel velocity $v_{\parallel }$ and $t\text{−}d$ contour plots of velocity along indicated direction for impact angles of $\varphi =0°$, 15°, 30°, and 45°. The magenta squares are those to which the displacement is applied. The full time evolution is available in Movie S1 of the Supplemental Material [29].

Figure 2

Numerical results for a circular model with 60 squares along its diameter. (a) Contour plots of $v_{\parallel }$ at $t=12\mathrm{ms}$ for all four impact angles. (b) Spatial-temporal map of $v_{\parallel }$ for all considered impact angles. The magenta squares in (a) are those to which the displacement is applied. The full time evolution is available in Movie S2 of the Supplemental Material [29].

Figure 3

(a) Schematic highlighting the input signal ${\mathbf{A}}_{\mathrm{in}}$ and the translational amplitude of the excited soliton $\mathbf{A}$. (b) Translational amplitude $\mathbf{A}$ of the solitons excited by impacts of amplitude $A_{\mathrm{in}}=7\mathrm{mm}$ for $\varphi \in [0,45°]$. The triangular and circular markers correspond to the velocities extracted from our experimental and numerical results. (c)–(f) Translational amplitude and associated deformation for all solitary modes. Note that the displacements and rotations are five times amplified, excited by input signals with $A_{\mathrm{in}}=7\mathrm{mm}$ and (c) $\varphi =0°$, (d) $\varphi =45°$, (e) $\varphi =15°$, and (f) $\varphi =30°$.

Figure 4

(a) Schematic highlighting the full width at half maximum, $W_{\perp }$. (b),(c) Time evolution of $W_{\perp }$ as extracted from (b) experiments and (c) simulations for $\varphi =0°$ (red markers) and $\varphi =45°$ (blue markers).