Piezoelectric gauge transformation for inverse design of polar Willis transducers

Piezoelectric lattices with delicately designed microscopic geometry are powerful building blocks to construct integrated sensors and actuators with versatile, yet unconventional, responses absent from bulk materials. However, the inverse design of the microscopic geometry to achieve a sought-after electromechanical response remains elusive. Here, we suggest an analytical approach, called piezoelectric gauge transformation, to design piezoelectric lattice transducers that can deform to an arbitrary desired displacement field when a voltage is applied. We first develop continuum piezoelectric gauge transformation and find that the transformed piezoelectric material displays piezoelectric polarity and Willis coupling in the sense that the applied electric field generates asymmetric stress and body force, and both rigid body rotation and translation induce electric charges. To design this polar and Willis-type piezoelectric material, we develop discrete piezoelectric gauge transformation and propose feasible lattice design guidelines. Numerical simulations are performed to validate the piezoelectric gauge transformation and demonstrate a range of appealing displacement control functions. The study presents a complete theoretical framework for the inverse design of lattice transducers to achieve arbitrary desired actuated displacement fields, beneficial to the development of soft actuators, robotics, and other piezoelectric devices.