Abstract
The small amplitude dynamic response of materials can be tuned by employing inhomogeneous materials capable of large deformation. However, soft materials generally exhibit viscoelastic behaviour, i.e. loss and frequency-dependent effective properties. This is the case for inhomogeneous materials even in the homogenization limit when propagating wavelengths are much longer than phase lengthscales, since soft materials can possess long relaxation times. These media, possessing rich frequency-dependent behaviour over a wide range of low frequencies, can be termed metamaterials in modern terminology. The sub-class that are periodic are frequently termed soft phononic crystals although their strong dynamic behaviour usually depends on wavelengths being of the same order as the microstructure. In this paper we describe how the effective loss and storage moduli associated with longitudinal waves in thin inhomogeneous rods are tuned by pre-stress. Phases are assumed to be quasi-linearly viscoelastic, thus exhibiting time-deformation separability in their constitutive response. We illustrate however that the effective incremental response of the inhomogeneous medium does not exhibit time-deformation separability. For a range of nonlinear materials it is shown that there is strong coupling between the frequency of the small amplitude longitudinal wave and initial large deformation. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.