Metamaterials are of great interest on account of their tunable material proper-ties, arising from an internal microstructure, which can often be chosen to give a desired behaviour. The behaviours of acoustic metamaterials are often derived via analogy with electromagnetic metamaterials, and typically described using Willis relations. Here an elastic metamaterial consisting of a series of elastic plates interspaced by fluid is considered. It is found that the effective constitutive equations contain higher derivative terms than would normally be present in the Willis (or elastic) constitutive equations for an anisotropic material. This leads to behaviour that is highly dispersive due to an “emergent scale” associated with bending on the constitutive equations for an anisotropic material. This leads to behaviour that is highly dispersive due to an “emergent scale” associated with bending on the constituent plates. The existence of this scale is due to the boundary conditions on the plate-fluid interfaces and appears to be a purely elastic phenomena. The metamaterial in this case has no resonant inclusions; nevertheless, the ef-fective behaviour leads to broadband dispersion and is different from anisotropic elasticity. The dependence of the dispersion curves on the material properties of the constituents is examined and the connection with fractional derivative theories and rotons considered.