Effective wave propagation in a prestressed nonlinear elastic composite bar

Abstract

The problem of determining the effective incremental response of nonlinearly elastic composite materials given some initial pre-stress, is of interest in numerous application areas. In particular the case when small amplitude elastic waves pass through a pre-stressed inhomogeneous structure is of great importance. Of specific interest is how the initial finite deformation affects the microstructure and thus the subsequent response of the structure. Modelling this effect is in general extremely difficult.In this article we consider the simplest problem of this type where the material is a one dimensional composite bar consisting of two distinct phases, periodically distributed. Neglecting lateral contractions, the initial deformation is thus piecewise homogeneous and we can therefore determine the incremental behaviour semi-analytically, given the constitutive behaviour (strain energy function) of the phases in question. We apply asymptotic homogenization theory in the deformed configuration in order to find the effective response of the deformed material in the low frequency limit where the wavelength of the propagating waves is much longer than the characteristic lengthscale of the microstructure. We close by considering the arbitrary frequency case and illustrate how the initial deformation affects the location of stop and pass bands of the material. Work is underway to confirm these results experimentally.

Exploiting Microstructural Instabilities in Solids and Structures: From Metamaterials to Structural Transitions

“…Therefore, predeformation allows for the controllable variation of elastic moduli and dispersion properties [221,[223][224][225]. Such tunability can be further enhanced by triggering mechanical instabilities along the loading path, since these result in dramatic geometric reconfigurations.…”

The effective wavenumber of a pre-stressed nonlinear microvoided composite

“…Nonlinear pre-stress can be extremely useful in practice, allowing us to tune materials in order to permit or restrict waves of specific frequency ranges to propagate through the structure. This property was described by Parnell [20] and discussed further in subsequent articles in different contexts [7], [1].…”

“…When the medium in question is inhomogeneous (for example a fibre-reinforced, or particulate composite material), where the host phase is nonlinear-elastic, pre-stress will almost always lead to non-homogeneous stretch distributions, except in very special cases (see e.g. [20]). Degtyar et al [5] analysed the case of stressed composites where residual stresses occur in the vicinity of inhomogeneities, in the context of linear elasticity and a self consistent scattering formulation of the effective wavenumber was used.…”

Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation

“…However, as discussed above, bistable or multistable building blocks in periodic solids and structures also promise interesting nonlinear dynamic effects, including solitary-wave propagation, which provides opportunities to focus acoustic signals in mechanical metamaterials [4,80,81]. Homogeneous solids undergoing finite elastic deformation [82] as well as periodic media experiencing nonlinear elastic instabilities [83] have been shown to exhibit acoustic band gaps that are controllable by the amount of nonlinear predeformation, yet the investigated waves again operate in the linear elastic regime. To date, only one example of a periodic elastic mechanical system has been reported that produces SineGordon solitons by allowing a kink propagation in the form of elastically connected rotating pendulums [84][85][86][87][88].…”