Two-dimensional Phononic Crystals with Acoustic-Band Negative Refraction

A two-dimensional phononic crystal (PC) can exhibit longitudinal-mode negative energy refraction on its lowest (acoustical) frequency pass band. The effective elastodynamic properties of a typical PC are calculated and it is observed that the components of the effective density tensor can achieve negative values at certain low frequencies on the acoustical branches for the longitudinal-mode pass-band, and that negative refraction may be accompanied by either positive or negative effective density. Furthermore, such a PC has a high anisotropy ratio at certain low frequencies, offering potential for application to acoustic cloaking where effective material anisotropy is essential.Comment: in Proceedings of the ASME 2016 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Stowe, VT (2016

Refraction Characteristics of Phononic Crystals

The 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 inclusions. The band-structure, group velocity, and energy-flux vector are calculated using a powerful variational method which accurately and efficiently yields all the field quantities over multiple frequency pass-bands. Equifrequency contours and energy-flux vectors are calculated as functions of the wave-vector. 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 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. complete reflection with no energy transmission, depending on the frequency, and direction and the wave length of the plane-wave which is incident from the homogeneous solid to the interface. By comparing our results with those obtained using the Rayleigh quotient and, for the layered case, with the exact solutions, the remarkable accuracy and the convergence rate of the present solution method are demonstrated. MatLab codes with comments will be provided

Bounds on Effective Dynamic Properties of Elastic Composites

We present general, computable, improvable, and rigorous bounds for the total energy of a finite heterogeneous volume element or a periodically distributed unit cell of an elastic composite of any known distribution of inhomogeneities of any geometry and elasticity, undergoing a harmonic motion at a fixed frequency or supporting a single-frequency Bloch-form elastic wave of a given wave-vector. These bounds are rigorously valid for \emph{any consistent boundary conditions} that produce in the finite sample or in the unit cell, either a common average strain or a common average momentum. No other restrictions are imposed. We do not assume statistical homogeneity or isotropy. Our approach is based on the Hashin-Shtrikman (1962) bounds in elastostatics, which have been shown to provide strict bounds for the overall elastic moduli commonly defined (or actually measured) using uniform boundary tractions and/or linear boundary displacements; i.e., boundary data corresponding to the overall uniform stress and/or uniform strain conditions. Here we present strict bounds for the dynamic frequency-dependent constitutive parameters of the composite and give explicit expressions for a direct calculation of these bounds

Overall Dynamic Constitutive Relations of Micro-structured Elastic Composites

A method for homogenization of a heterogeneous (finite or periodic) elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations. It is shown that when the spatial variation of the field variables is restricted by a Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress modeshapes and, necessarily, the dispersion relations. It also gives as a matter of course point-wise solution of the elasto-dynamic field equations, to any desired degree of accuracy. The resulting overall dynamic constitutive relations however, are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micro-mechanical modeling of a representative unit cell of the composite proposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].Comment: 23 pages, 6 figures, submitted to JMP

On the Limit and Applicability of Dynamic Homogenization

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction

Mixed-variational formulation for phononic band-structure calculation of arbitrary unit cells

This paper presents phononic band-structure calculation results obtained using a mixed variational formulation for 1-, and 2-dimensional unit cells. The formulation itself is presented in a form which is equally applicable to 3-dimensiomal cases. It has been established that the mixed-variational formulation presented in this paper shows faster convergence with considerably greater accuracy than variational principles based purely on the displacement field, especially for problems involving unit cells with discontinuous constituent properties. However, the application of this formulation has been limited to fairly simple unit cells. In this paper we have extended the scope of the formulation by employing numerical integration techniques making it applicable for the evaluation of the phononic band-structure of unit cells displaying arbitrary complexity in their Bravais structure and in the shape, size, number, and anisotropicity of their micro-constituents. The approach is demonstrated through specific examplesComment: arXiv admin note: substantial text overlap with arXiv:1310.638