Effective Dynamic Properties of Microstructured Heterogeneous Materials

2012 ◽  
Author(s):  
Ankit Srivastava ◽  
Sia Nemat-Nasser
Keyword(s):  
Wave Propagation ◽  
Explicit Form ◽  
Constitutive Equations ◽  
Effective Properties ◽  
Dynamic Properties ◽  
Effective Property ◽  
Bloch Wave ◽  
Periodic Composites ◽  
Effective Dynamic ◽  
Dynamic Effective Properties

Central to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. While the theory of static effective property calculations goes back about 60 years, progress in the actual calculation of dynamic effective properties for periodic composites has been made only very recently. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites

2013 ◽  
Vol 1508 ◽  
Author(s):  
Ankit Srivastava ◽  
Sia Nemat-Nasser
Keyword(s):  
Wave Propagation ◽  
Dispersion Relation ◽  
Constitutive Relation ◽  
Effective Properties ◽  
Constitutive Relations ◽  
Bloch Wave ◽  
Ensemble Averaging ◽  
Periodic Composites ◽  
Effective Dynamic ◽  
Elastic Composites

ABSTRACTCentral to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. Recent advances in the theory of dynamic homogenization have established the coupled form of the constitutive relation (Willis constitutive relation). This coupled form of the constitutive relation naturally emerges from ensemble averaging of the dynamic fields and automatically satisfies the dispersion relation in the case of periodic composites. Its importance is also notable due to its invariance under transformational acoustics. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Applicability of Dynamic Homogenization for Acoustic Metamaterials

2014 ◽  
Author(s):  
Ankit Srivastava ◽  
Sia Nemat-Nasser
Keyword(s):  
Negative Refraction ◽  
Effective Properties ◽  
Dynamic Properties ◽  
Low Frequency ◽  
Bloch Wave ◽  
Acoustic Metamaterials ◽  
Infinite Domain ◽  
True Negative ◽  
Periodic Composite ◽  
Frequency Regime

Dynamic homogenization seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation in them. These properties can be used to predict and design for metamaterial behavior. However, there is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. By way of explicit examples we show that a comprehensive homogenization scheme proposed in earlier papers is applicable in a finite composite setting and in the low frequency regime. We also show that there exist good arguments for considering the second branch of a locally resonant composite a true negative branch. Furthermore, we note that infinite-domain homogenization is more applicable to finite cases of locally resonant metamaterial composites than it is to 2-phase composites. We also study the effect of the interface location on the applicability of homogenization. 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.

On a technique for deriving explicit transfer matrices of orthotropic layers

2015 ◽  
Vol 37 (4) ◽  
pp. 303-315 ◽  
Author(s):  
Pham Chi Vinh ◽  
Nguyen Thi Khanh Linh ◽  
Vu Thi Ngoc Anh
Keyword(s):  
Wave Propagation ◽  
Explicit Form ◽  
Half Space ◽  
Transfer Matrix ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Transfer Matrices ◽  
Orthotropic Layer ◽  
Wave Propagation Problem ◽  
Arbitrary Thickness

This paper presents  a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

scholarly journals “Fuzzy Band Gaps”: A Physically Motivated Indicator of Bloch Wave Evanescence in Phononic Systems

Crystals ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 66
Author(s):  
Connor D. Pierce ◽  
Kathryn H. Matlack
Keyword(s):  
Wave Propagation ◽  
Dispersion Relation ◽  
Quantitative Measure ◽  
Bloch Wave ◽  
Phononic Crystals ◽  
Wave Transmission ◽  
Band Gaps ◽  
Dissipative Systems ◽  
Frequency Range ◽  
Wave Modes

Phononic crystals (PCs) have been widely reported to exhibit band gaps, which for non-dissipative systems are well defined from the dispersion relation as a frequency range in which no propagating (i.e., non-decaying) wave modes exist. However, the notion of a band gap is less clear in dissipative systems, as all wave modes exhibit attenuation. Various measures have been proposed to quantify the “evanescence” of frequency ranges and/or wave propagation directions, but these measures are not based on measurable physical quantities. Furthermore, in finite systems created by truncating a PC, wave propagation is strongly attenuated but not completely forbidden, and a quantitative measure that predicts wave transmission in a finite PC from the infinite dispersion relation is elusive. In this paper, we propose an “evanescence indicator” for PCs with 1D periodicity that relates the decay component of the Bloch wavevector to the transmitted wave amplitude through a finite PC. When plotted over a frequency range of interest, this indicator reveals frequency regions of strongly attenuated wave propagation, which are dubbed “fuzzy band gaps” due to the smooth (rather than abrupt) transition between evanescent and propagating wave characteristics. The indicator is capable of identifying polarized fuzzy band gaps, including fuzzy band gaps which exists with respect to “hybrid” polarizations which consist of multiple simultaneous polarizations. We validate the indicator using simulations and experiments of wave transmission through highly viscoelastic and finite phononic crystals.

Consistent Hashin-Shtrikman Bounds on the Effective Properties of Periodic Composite Materials

1999 ◽  
Vol 66 (4) ◽  
pp. 858-866
Author(s):  
P. Bisegna ◽  
R. Luciano
Keyword(s):  
Composite Materials ◽  
Saddle Point ◽  
Variational Principles ◽  
Effective Properties ◽  
The Other ◽  
Constitutive Behavior ◽  
Homogenization Problem ◽  
Numerical Examples ◽  
Periodic Composites ◽  
Periodic Composite

In this paper the four classical Hashin-Shtrikman variational principles, applied to the homogenization problem for periodic composites with a nonlinear hyperelastic constitutive behavior, are analyzed. It is proved that two of them are indeed minimum principles while the other two are saddle point principles. As a consequence, every approximation of the former ones provide bounds on the effective properties of composite bodies, while approximations of the latter ones may supply inconsistent bounds, as it is shown by two numerical examples. Nevertheless, the approximations of the saddle point principles are expected to provide better estimates than the approximations of the minimum principles.

Locally resonant effective phononic crystals for subwavelength vibration control of torsional cylindrical waves

2021 ◽  
pp. 1-30
Author(s):  
Ignacio Arretche ◽  
Kathryn Matlack
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Effective Properties ◽  
Equations Of Motion ◽  
Phononic Crystal ◽  
Phononic Crystals ◽  
Unit Cell ◽  
Periodic Coefficients ◽  
Bloch Theorem ◽  
Plane Wave Propagation

Abstract Locally resonant materials allow for wave propagation control in the sub-wavelength regime. Even though these materials do not need periodicity, they are usually designed as periodic systems since this allows for the application of the Bloch theorem and analysis of the entire system based on a single unit cell. However, geometries that are invariant to translation result in equations of motion with periodic coefficients only if we assume plane wave propagation. When wave fronts are cylindrical or spherical, a system realized through tessellation of a unit cell does not result in periodic coefficients and the Bloch theorem cannot be applied. Therefore, most studies of periodic locally resonant systems are limited to plane wave propagation. In this paper, we address this limitation by introducing a locally resonant effective phononic crystal composed of a radially-varying matrix with attached torsional resonators. This material is not geometrically periodic but exhibits effective periodicity, i.e. its equations of motion are invariant to radial translations, allowing the Bloch theorem to be applied to radially propagating torsional waves. We show that this material can be analyzed under the already developed framework for metamaterials. To show the importance of using an effectively periodic system, we compare its behavior to a system that is not effectively periodic but has geometric periodicity. We show considerable differences in transmission as well as in the negative effective properties of these two systems. Locally resonant effective phononic crystals open possibilities for subwavelength elastic wave control in the near field of sources.

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