On the Duality of Complex Geometry and Material Heterogeneity in Elastodynamics: From Topographic Reliefs to Mechanical Metamaterials

From cell membranes to mountains, and from arches to buildings, we are surrounded by countless curved systems possessing both beauty and function.
Here, we introduce a novel perspective for understanding wave propagation in domains with curved boundaries using coordinate transformation techniques mapping the curved geometry in the stress-free reference domain into a gradually varying anisotropic impedance distribution in a new stress-free domain with flat boundaries. Such transformation reveals areas of possible amplification and localization of elastodynamic disturbance without solving the elastodynamic equations and sheds lights into long-standing observations of ground motion amplification at steep crests and valleys during earthquakes. We also show that with increased surface waviness, the contrast in the impedance in the mapped domain increases leading to emergence of band gaps and evolution of local resonant effects. These results provide a new pathway for designing homogeneous Metamaterials exploiting space curvature rather than material heterogeneities lowering the barrier towards their realization using conventional manufacturing techniques. It also provides methodology for transforming anisotropic material properties into geometric curvature and vice versa.