A nonlinear homogenization-based perspective on the soft modes and effective energies of some conformal metamaterials

There is a growing mechanics literature concerning the macroscopic properties of mechanism-based mechanical metamaterials. This amounts mathematically to a homogenization problem involving nonlinear elasticity. A key goal is to identify the “soft modes” of the metamaterial. We achieve this goal using methods from homogenization for some specific 2D examples – including discrete models of the Rotating Squares metamaterial and the Kagome metamaterial – whose soft modes are compressive conformal maps. The innovation behind this achievement is a new technique for bounding the effective energy from below, which takes advantage of the metamaterial’s structure and symmetry.