Controlling Displacement Fields in Polar Willis Solids via Gauge Transformations

Rigid-body displacement and deformation constitute the total displacement field of a solid. Harnessing the former calls for well-organized kinematic elements, and controlling the latter allows for creation of shape-morphing materials. A solid capable of simultaneously controlling both rigid-body displacement and deformation remains unknown. Here, we exploit gauge transformations to show how the total displacement field in elastostatic polar Willis solids can be harnessed at will and how those solids can be realized in the form of lattice metamaterials. The transformation method we develop leverages a displacement gauge in linear transformation elasticity, giving rise to polarity and Willis coupling such that the resulting solids not only break minor symmetries of the stiffness tensor, but display cross coupling between stress and displacement. We realize those solids using a combination of tailored geometries, grounded springs, and a set of coupled gears and numerically demonstrate a range of satisfactory, and peculiar, displacement control functions. Our work provides an analytical framework for the inverse design of grounded polar Willis metamaterials to achieve arbitrary displacement control functions by design.