Some of the most interesting refraction prop- erties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or ellipt...Some of the most interesting refraction prop- erties 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 multi- inclusions. The corresponding band structure, group velocity, and energy-flux vector are calculated using a powerful mixed variational method that accurately and efficiently yields all the field 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-flux vectors are read- ily calculated as functions of the wave-vector components. By superimposing the energy-flux 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 con- taining 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 homoge- neous solid can display： （1） negative refraction with negative phase velocity refraction; （2） negative refraction with pos- itive phase velocity refraction; （3） positive refraction with negative phase velocity refraction; （4） positive refraction with positive phase velocity refraction; or even （5） completereflection with no energy transmission, depending on the fre- quency, 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, analyti- cal expressions are given for the key calculation quantities. Expressions for displacement, velocity, linear momentum, strain, and s展开更多
文摘Some of the most interesting refraction prop- erties 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 multi- inclusions. The corresponding band structure, group velocity, and energy-flux vector are calculated using a powerful mixed variational method that accurately and efficiently yields all the field 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-flux vectors are read- ily calculated as functions of the wave-vector components. By superimposing the energy-flux 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 con- taining 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 homoge- neous solid can display： （1） negative refraction with negative phase velocity refraction; （2） negative refraction with pos- itive phase velocity refraction; （3） positive refraction with negative phase velocity refraction; （4） positive refraction with positive phase velocity refraction; or even （5） completereflection with no energy transmission, depending on the fre- quency, 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, analyti- cal expressions are given for the key calculation quantities. Expressions for displacement, velocity, linear momentum, strain, and s
