  Hierarchical structures with constituents over multiple length scales are
found in various natural materials like bones, shells, spider silk and others,
all of which display enhanced quasi-static mechanical properties, such as high
specific strength, stiffness and toughness. At the same time, the role of
hierarchy on the dynamic behaviour of metamaterials remains largely unexplored.
This study assesses the effect of bio-inspired hierarchical organization as
well as of viscoelasticity on the wave attenuation properties of continuous
mechanical metamaterials. We consider single-phase metamaterials formed by
self-similar unit cells with different hierarchical levels and types of
hierarchy. Results highlight a number of advantages through the introduction of
structural hierarchy. Band gaps relative to the corresponding non-hierarchical
structures are mostly preserved, while additional "hierarchically-induced" band
gaps appear. Additionally, the hierarchical configuration allows the tuning of
the band gap frequencies of regular metamaterial to lower frequencies, with a
simultaneous significant reduction of the global structural weight. We show
that even small viscoelastic effects, not treated in the current literature,
are essential in determining this behaviour. The approach we propose allows the
addition of hierarchical elements to existing metamaterial configurations, with
the corresponding improvement of the wave damping properties, thus providing
indications for the design of structures for practical applications.
