46 research outputs found
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