Elastic waves propagating can be regulated and synthesized to realize rich non-Abelian phenomena and topological phases of solid media. Therefore, this work investigates the noncommutative mechanical responses and non-Abelian nodal braiding in three-dimensional elastic wave metamaterials with duality symmetry. The connection between dispersion relation and twisting angles is derived by lattice dynamics and next-nearest-neighbor interaction. Then, the duality transformation and self-dual conditions are revealed by geometric correspondence and the root-finding algorithm. Based on the semiclassical equations of motion and polarized pseudospin states, the spin phase evolution is obtained. The noncommutative propagation paths of the wave packet in real and reciprocal space are achieved as well. In addition, the quaternion charges and patch Euler classes can be calculated by loops encircling the nodal points and patches penetrating the nodal lines, respectively. They can characterize the non-Abelian topological properties of the nodal lines and predict topological phase transitions, i.e., nodal stability. The nontrivial braiding of nodal links in momentum and parameter space is revealed numerically by structural parameters. Experiments further support the bulk-edge correspondence and the presence of the gap node.

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• Mechanical & acoustical properties
• Mechanical metamaterials
• Topological materials