Every mechanism in 2d generates an exotic space of soft modes

Mechanical metamaterials can be designed with special deformation pathways (mechanisms) that allow them to change shape dramatically with little energetic cost. However, even in materials designed around a uniform mechanism, experiments reveal ubiquitous nonuniform deformations. Here, we show that the mechanism constitutes an approximate local symmetry of the material, and that in 2d each distinct mechanism generates a unique class of nonuniform deformations lower in energy than standard elastic modes. We introduce an illustrative new class of general mechanisms in lattices of corner-sharing quadrilaterals and use them to show that these modes are governed by a single second-order nonlinear differential equation which guarantees mechanical compatibility. This equation induces a bulk-boundary correspondence in which specifying the amplitude of the mode on the boundary fixes the deformation of the bulk, irrespective of additional microscopic details, which we then exploit to achieve on-demand target shapes. Our investigation shows that a single mechanism in 2d will generically give rise to an exotic space of nonlinear nonuniform soft modes, while in higher dimensions multiple mechanisms must be at play simultaneously to generate such nontrivial spaces of soft motions.