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.

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.

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.

On Eshelby's S-tensor under various magneto-electro-elastic constitutive settings, and its application to multiferroic composites

2016 ◽  
Vol 01 (03n04) ◽  
pp. 1640002 ◽  
Author(s):  
Yang Wang ◽  
George J. Weng
Keyword(s):  
Effective Properties ◽  
General Procedure ◽  
Constitutive Relations ◽  
Magnetoelectric Coupling ◽  
Coupling Coefficients ◽  
Multilayer Composites ◽  
Inclusion Shape ◽  
Multiferroic Composite ◽  
Tensor Formula ◽  
Elastic Composites

The magneto-electro-elastic Eshelby S-tensor is the key to the study of linear effective properties of magneto-electro-elastic composites. There are eight different ways to write the constitutive relations, and each is associated with a specific kind of boundary condition and Eshelby S-tensor. In this work, we provide a general procedure to convert the magneto-electro-elastic Eshelby S-tensor from one system to another. As an application, we use it to calculate the magnetoelectric coupling coefficients of a piezoelectric–piezomagnetic multiferroic composite under stress-and strain-prescribed boundary conditions. We demonstrate that the calculated results are significantly different. In particular, it is shown that, under an applied stress, the magnetoelectric coupling coefficient [Formula: see text], is much stronger than that under an applied strain, while for [Formula: see text], the values are positive under a prescribed stress but negative under a prescribed strain. The effects of inclusion shape, volume concentration and geometrical exchange, are also examined. For ready applications, the explicit forms of S-tensor, [Formula: see text] and [Formula: see text], of 1-3 fibrous and 2-2 multilayer composites are also provided at the end.

Heat Conduction in Composites: Homogenization and Macroscopic Behavior

1997 ◽  
Vol 50 (6) ◽  
pp. 327-356 ◽  
Author(s):  
Piotr Furman˜ski
Keyword(s):  
Heat Conduction ◽  
Effective Properties ◽  
Constitutive Relations ◽  
Local Heat ◽  
Temperature Fields ◽  
Heat Sources ◽  
Ensemble Averaging ◽  
Macroscopic Behavior ◽  
Classical Behavior ◽  
Homogeneous Media

The review article discusses methods of macroscopic averaging of heat conduction in composite materials that lead to models of homogenized, macroscopic behavior of these media. It is shown that essentially two continuum models are in use: 1) the effective medium and 2) the mixture. The ensemble averaging technique allows one to derive the constitutive relations for both models assuming Fourier-like conduction on the microstructure level of a composite. These constitutive relations contain effective, macroscopic properties of the composite material which can be forecast when properties of individual constituents, the form of thermal interaction at constituent interfaces, amount of each material and its distribution are known. For weakly varying mean temperature fields, thermal behavior of the composite is essentially the same as homogeneous media but, for stronger variation, a non-classical behavior is observed. This non-classical behavior can be associated either with space nonlocality and memory phenomena or with wall effects and, in some cases, with influence of local heat sources on the effective properties. Most of these effects are not well known and need further detailed studies. The article includes 158 references.

Effective Properties of Discrete Random Composites Wherein the Host and Inclusion Phases Obey Different Constitutive Relations

1991 ◽  
Vol 253 ◽  
Author(s):  
V V. Varadan ◽  
R. T. Apparao ◽  
V. K. Varadan
Keyword(s):  
Wave Propagation ◽  
Effective Properties ◽  
Constitutive Relations ◽  
Effective Medium ◽  
Acoustic Wave Propagation ◽  
Piezoelectric Composites ◽  
Inclusion Phase ◽  
Random Composites ◽  
Polarization Study ◽  
Effective Medium Theories

ABSTRACTIn studying the effective medium theories, polarization is hardly given a consideration in deciding the effective properties of a composite where the host and inclusion phases follow different constitutive equations. A significant conclusion of this paper is that eventhough the composite has discrete inclusions, with the inclusion phase obeying different constitutive properties than the host, the effective medium shows a preference for the inclusion behavior rather than the host which is continuous. As an example, results on polarization study are detailed for the specific case of chiral composites. Application of similar principles is presently explored in more complex problems like the elastic wave propagation through piezoelectric composites and the acoustic wave propagation through sediments.

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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