Longitudinal elastic wave control by pre-deforming semi-linear materials

TL;DR

This paper demonstrates that pre-deforming semi-linear hyper-elastic materials allows broadband control of longitudinal elastic waves through transformation methods, with potential applications in designing novel elastic wave devices.

Contribution

It proves that the governing equations for longitudinal waves in semi-linear materials remain form-invariant under symmetric pre-deformation, enabling wave control without principal stretch constraints.

Findings

Longitudinal wave control achieved via pre-deformation.

Band structure shifts observed in pre-deformed materials.

Spring lattice mimics semi-linear material behavior.

Abstract

An incremental wave superimposed on a pre-deformed hyper-elastic material perceives an elastic media with the instantaneous modulus of the current material. This offers a new route with a broadband feature to control elastic waves by purposely creating finite deformation field. This study proves that the governing equation of a semi-linear material under a symmetric pre-deformation condition maintains the form invariance for longitudinal wave, so the longitudinal wave control can be made by transformation method without the constraint condition on principle stretches, but this is not the case for shear waves. Therefore pre-deforming a semi-linear material provides a potential method for treating longitudinal and shear waves differently. Examples with elastic wave control and band structure shift through pre-deforming a semi-linear material are provided to illustrate this finding. Finally, a one-dimensional spring lattice is proposed to mimic a semi-linear material, and the dispersion relation for longitudinal waves in a sandwich structure with such spring lattice is shown to be invariant during elongation, confirming the result found based on a homogeneous semi-linear material. These results may stimulate researches on designing new hyper-elastic microstructures as well as designing new devices based on pre-deformed hyper-elastic materials.