Reduced enhanced elastic continua as acoustic metamaterials

Abstract

We consider a linear complex elastic material whose body point motion is described by two vectorial generalised co-ordinates of any dimension. We suppose that the medium obeys Lagrange equations. Elastic energy depends on both of these vectorial generalised co-ordinates but does not depend on the gradient of one of them (“special” co-ordinate). Such a complex medium can be interpreted as a “bearing continuum” (corresponding to the non-special co-ordinate), whose body points are enhanced by body points of “continuum of dynamic absorbers” (corresponding to the special co-ordinate), the latter not connected directly between them. We obtain that under some restrictions for the strain energy bulk plane harmonic waves in this infinite medium have a band gap (or band gaps), the medium being thus a single negative acoustic metamaterial, and for some cases decreasing part(s) of dispersion curve, thus being a double negative acoustic metamaterial in this zone. We consider some examples of such continua with rotational degrees of freedom.