Locally resonant band gaps in periodic beam lattices by tuning connectivity

Lattice structures have long fascinated physicists and engineers not only because of their outstanding functionalities, but also for their ability to control the propagation of elastic waves. While the study of the relation between the connectivity of these systems and their static properties has a long history that goes back to Maxwell, rules that connect the dynamic response to the network topology have not been established. Here, we demonstrate that by tuning the average connectivity of a beam network $\left(\overline{z}\right)$, locally resonant band gaps can be generated in the structures without embedding additional resonating units. In particular, a critical threshold for $\overline{z}$ is identified, far from which the band gap size is purely dictated by the global lattice topology. By contrast, near this critical value, the detailed local geometry of the lattice also has strong effects. Moreover, in stark contrast to the static case, we find that the nature of the joints is irrelevant to the dynamic response of the lattices. Our results not only shed new light on the rich dynamic properties of periodic lattices, but also outline a new strategy to manipulate mechanical waves in elastic systems.

Figure 1

Band structures of triangular periodic beam lattices. (a) Dispersion relation of the triangular lattice with welded joints. (b) Dispersion relation of the triangular lattice with pin joints. The lattice and unit cells with both welded and pin joints are shown as insets. Band gaps are shown as gray-shaded areas and the flat bands at their lower edge are highlighted in red. The Bloch modes of the flat band (the third mode) at high symmetry points of the Brillouin zone $(G,X$, and $M$) [25] are shown in (c) and (d) for the case of welded and pin joints, respectively. A plot presenting both band structures normalized by the same factor is included in the Supplemental Material [25].

Figure 2

Band structure of periodic lattices with $\overline{z}=3$ and welded joints. (a) Hexagonal lattice $\left(\alpha =2\pi /3\right)$; (b) topologically equivalent lattice with $\alpha =\pi /2$; (c) topologically equivalent lattice with $\alpha =5\pi /12$; and (d) topologically equivalent lattice with $\alpha =\pi /3$. The lattice structures and unit cells used in the calculations are shown as insets. Note that for $\alpha =\pi /3$ the arrangement of the beams is the same as for the triangular lattice, but the connectivity is still $\overline{z}=3$. In fact, although for clarity the green-colored joints are drawn separately in the unit cell of (d), they are positioned at the same spatial location. No locally resonant band gap is found for any of the configurations. Results for the same lattices with pin joints are provided in the Supplemental Material [25].

Figure 3

Dynamic response of periodic lattices with $3<\overline{z}<6$. (a) Representative unit cells constructed either starting from the triangular lattice and removing a number of randomly chosen nodes or starting from the hexagonal lattice and filling a number of randomly chosen hexagons with six triangles. For two unit cells with $\overline{z}=4.615$ the band structures are also shown. (b) Relation between normalized average band gap size $\Delta \omega /{\omega }_{\mathrm{welded}}$ and coordination number $\overline{z}$. The band gap size is computed as the difference between the upper and lower edges of the gap. The error bar at each data point indicates the band gap range of frequencies spanned by all different configurations characterized by the same value of $\overline{z}$.

Figure 4

Normalized band gap width $\Delta \omega /{\omega }_{*}$ for rhombic lattices that are topologically equivalent to the square lattice. The lattice structures and joint types used in the calculations are shown as insets. The results for lattices with both welded and pin joints are reported. Note that ${\omega }_{*}={\omega }_{\mathrm{welded}}$ and ${\omega }_{*}={\omega }_{\mathrm{pin}}$ for lattices with welded and pin joints, respectively.