Wave propagation through elastic granular and granular auxetic materials

The wave propagation through a material with a granular micro‐structure is investigated. The simplest such medium is the granular medium of equal spheres. The theory demonstrates that for long wavelengths the micro‐structured material exhibits modes of wave motion that are also found for a traditional medium (that is, a medium that satisfies simple continuum stress equilibrium laws and no consideration for particle spin and moment of inertia of the particles is accounted for). However, an extra shear wave and rotational oscillatory particle motion are also found. The limit of a traditional material is obtained by letting tangential contact stiffnesses vanish. Towards higher wave numbers the micro‐structure reveals itself in the shear wave speeds by displaying strongly anomalous behaviour compared to a traditional isotropic, elastic medium. An expansion in terms of structural sums (these are sums over the contact interaction weighed with a string of branch vectors) is explored. The lowest order non‐trivial terms (second order structural sums) yield basic behaviour, but a higher order expansion (including fourth order sums) shows a more pronounced anomaly for wave numbers approaching the inverse particle diameter. Yet higher order terms capture more information than can possibly be present in a granular medium and it is senseless to examine these. The theory is relevant to granular materials and granular auxetics. In the latter case the tangential contact stiffness needs to exceed the normal stiffness, which is not possible for ordinary solid contacts, but can be achieved with special arrangements.