Abstract
The effect of nonlinear elastic pre-stress on coupled compressional and vertically polarised shear elastic wave propagation in a two-dimensional periodic structure is investigated. The medium consists of cylindrical annuli embedded on a periodic lattice in a uniform host material. An identical inhomogeneous deformation is imposed in each annulus and the theory of small-on-large is used to find the incremental wave equations governing subsequent small-amplitude elastic waves. The plane-wave-expansion method is employed in order to determine the permissable eigenfrequencies. It is found that the application of pre-stress has a significant effect on the band structure, allowing stop bands to be controlled. The sensitivity of the choice of constitutive behaviour is studied and it is shown that the fundamental shear wave mode is largely unchanged for the class of strain energy functions considered here, whereas the compressional mode is considerably more sensitive to this choice.
Highlights
Improvements in engineering and technology rely greatly on advanced, complex materials, which often possess intricate microstructure, permitting macroscopic behaviour that is not present in naturally occurring materials
In this paper the theory of small-on-large has been employed to incorporate pre-stress into a phononic crystal made up of a compressible elastic material. The effect of this pre-stress on the band structure was investigated for two different strain energy functions
For the coupled pressure and shear mode we have shown that applying pre-stress can have the effect of switching on and off stop bands, giving a mechanism for real time tuning
Summary
Improvements in engineering and technology rely greatly on advanced, complex materials, which often possess intricate microstructure, permitting macroscopic behaviour that is not present in naturally occurring materials. Incompressible hyperelastic annular cylinders were embedded periodically in an elastic medium and their deformation ensured that band gaps could be switched on and off. Further it was shown, based on the theory of hyperelastic cloaking [32,33,34,35] that some materials permit deformation in a manner such that in a normal setting from a geometric viewpoint they would possess band gaps but the prestress prohibits this; these materials were termed phononic cloaks. The same geometry as considered in [31] is employed, as depicted in Fig. 1 but here the annular cylinders are permitted to be compressible in order to accommodate the propagation of compressional waves
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