Elastodynamic gauge transformation for controlling vibration mode shapes in Willis networks

Designing mechanical metamaterial networks to precisely control vibration mode shapes at targeted frequencies has shown great potential for applications in fluid and particle manipulation, nondestructive structural health monitoring and evaluation, as well as the protection of sensitive and fragile structures. However, the inverse design of metamaterial networks to simultaneously satisfy vibration mode shape and frequency requirements remains elusive. Here, we suggest a rapid analytical approach, called elastodynamic gauge transformation, to exploit design guidelines on microscopic geometry and local material properties to construct metamaterial networks to achieve sought-after vibration mode shapes at target frequencies. We operate a displacement gauge in the transformation over the Lagrangian of metamaterial networks and build the relationships between the desired displacement field and microscopic design. This approach is exemplified through discrete string, beam, and linear spring networks. We find that transformed string networks keep the same network geometry as those before the transformation, but each of the strings should be grounded and display Willis coupling. Further, beam networks can also conserve their virtual microscopic geometry, but additional grounded torsional springs and their associated rotational Willis coupling must be included. Nevertheless, linear spring networks no longer maintain their virtual network geometry after the transformation, where both the directions of the linear springs and the shapes of the masses need to be properly adjusted. We also interpret the transformed discrete string, beam, and linear spring networks at the continuum limit by performing elastodynamic gauge transformation over homogenized membrane, plate, and Cauchy elasticity models. Numerical simulations are performed to validate the elastodynamic gauge transformation for string, beam, and linear spring networks. Elastodynamic gauge transformation offers a new approach for the rapid design of metamaterial structures to achieve desired vibration mode shapes at target frequencies.