Refraction characteristics of phononic crystals
摘要
Some of the most interesting refraction properties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or elliptical multiinclusions.Thecorrespondingbandstructure,groupvelocity,and energy–flu vector are calculated using a powerful mixed variational method that accurately and efficientl yields all the fiel quantities over multiple frequency pass-bands. The background matrix and the inclusions can be anisotropic,each having distinct elastic moduli and mass densities.Equifrequency contours and energy–flu vectors are readily calculated as functions of the wave-vector components.By superimposing the energy–flu vectors on equifrequency contours in the plane of the wave-vector components, and supplementing this with a three-dimensional graph of the corresponding frequency surface, a wealth of information is extracted essentially at a glance. This way it is shown that a composite with even a simple square unit cell containing a central circular inclusion can display negative or positive energy and phase velocity refractions, or simply performs a harmonic vibration(standing wave), depending on the frequency and the wave-vector. Moreover, that the same composite when interfaced with a suitable homogeneous solid can display:(1) negative refraction with negative phase velocity refraction;(2) negative refraction with positive phase velocity refraction;(3) positive refraction with negative phase velocity refraction;(4) positive refraction with positive phase velocity refraction; or even(5) completereflectio with no energy transmission, depending on the frequency, and direction and the wavelength of the plane-wave that is incident from the homogeneous solid to the interface.For elliptical and rectangular inclusion geometries, analytical expressions are given for the key calculation quantities.Expressions for displacement, velocity, linear momentum,strain, and stress components, a