  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.
