  The effective dynamic properties of specific periodic structures involving
rubber-like materials can be adjusted by pre-strain, thus facilitating the
design of custom acoustic filters. While nonlinear viscoelastic behaviour is
one of the main features of soft solids, it has been rarely incorporated in the
study of such phononic media. Here, we study the dynamic response of nonlinear
viscoelastic solids within a 'small-on-large' acoustoelasticity framework, that
is we consider the propagation of small amplitude waves superimposed on a large
static deformation. Incompressible soft solids whose behaviour is described by
the Fung-Simo quasi-linear viscoelasticy theory (QLV) are considered. We derive
the incremental equations using stress-like memory variables governed by linear
evolution equations. Thus, we show that wave dispersion follows a
strain-dependent generalised Maxwell rheology. Illustrations cover the
propagation of plane waves under homogeneous tensile strain in a QLV
Mooney-Rivlin solid. The acoustoelasticity theory is then applied to phononic
crystals involving a lattice of hollow cylinders, by making use of a dedicated
perturbation approach. In particular, results highlight the influence of
viscoelastic dissipation on the location of the first band gap. We show that
dissipation shifts the band gap frequencies, simultaneously increasing the band
gap width. These results are relevant to practical applications of soft
viscoelastic solids subject to static pre-stress.
