Conformal symmetry & elastodynamics of dilational metamaterials

Two-dimensional mechanical metamaterials can be imbued with a low-energy uniform dilational mode via precise geometry utilizing rigid plates connected by thinner ligaments that bend easily, so-called planar kirigami. Recent work has shown that such structures exhibit static conformal deformations at low energies, able to accommodate a large range of loading conditions by primarily activating the local mechanism. In this talk, I examine the role that conformal symmetry plays in the dynamics of these dilational metamaterials. This symmetry ensures the existence of conformal boundary modes operating at frequencies set by the small bulk modulus, rather than the large shear modulus. Additionally, even the bulk modes are subject to potentially infinite conformal symmetries, leading to conserved quantities that generalise the concept of momentum for dilational materials. The strength of these effects is controlled by finite cell size and the small ratio of bulk to shear modulus, which itself is determined by the geometric design and material properties of the system. Using simulations of rotating-square lattices, we explore how the inherent geometric non-linearity, viscoelasticity, and driving terms influence these phenomena under realistic conditions as can be achieved experimentally.