Frequency-dependent Effective Properties Research Articles

The non-reciprocal wave propagation in random piezoelectric composites with aligned semi-spheroid inclusions

The non-reciprocal wave propagation characteristics of 3D random piezoelectric composites with aligned semi-spheroid inclusions are theoretically predicted. The self-consistent equations about the frequency-dependent effective properties are established, in which the volume average of dynamic Green's operators in semi-spheroid domain is involved. Due to the non-centrosymmetric inclusions, the composites macroscopically exhibit non-reciprocity, i.e. the effective properties, wave speed, acoustic impedance, attenuation and resonance frequency under the two waves of identical form but incident in opposite directions are different. The numerical results show that when the aspect ratio of the inclusion is large, the differences of the effective wave speed and impedance in two opposite directions are significant, which provides a possible means for modulation of wave propagation in the 3D random composites.

Impedance tube method for characterization of one-dimensional electro-momentum coupled materials

Electro-momentum coupling is a macroscopically observable material response resulting from heterogeneous piezoelectric media with microscale asymmetries that produce unique cross-coupling between the bulk momentum of the material and the generated electric field. Recently, Pernas-Salomon et al. used a one-dimensional transmission line model to demonstrate that the electro-momentum coupling effect must be considered in order to retrieve physically meaningful effective properties of heterogeneous media with subwavelength asymmetries [Wave Motion 106, 102797, (2021)]. This work presents the specialization of their transmission line model to the classical impedance tube measurement technique in which the scattering coefficients and the spatial averages of the mechanical and electrical fields of a one-dimensional material can be measured in order to obtain estimates of the frequency-dependent effective properties of an electro-momentum coupled sample. We investigate an idealized analytical approximation and then a finite element model of the realistic impedance tube configurations in order to link the analytical model to realistic experimental conditions and geometry. We then provide preliminary test results extracted from an electro-momentum coupled unit cell in a water-filled impedance tube. [Research sponsored by the Defense Advance Research Project Agency and the Army Research Office and was accomplished under Grant No. W911NF-20-1-0349.]

Controlling the dispersion of metamaterials in three dimensions

We provide a platform to examine the effect of inclusion geometry on three-dimensional metamaterial crystals to tune frequency-dependent effective properties for control of leading order dispersive behaviour. The crystal is non-magnetic and made from all dielectric components. The design provides novel dispersive properties using subwavelength resonances controlled by the geometry of the media. We numerically calculate the effective tensors of the metamaterial to identify frequency intervals where the metamaterial exhibits band gaps as well as intervals of normal dispersion and double negative dispersion. The frequency intervals can be explicitly controlled by adjusting the geometry and placement of the dielectric inclusions within the period cell of the crystal.

Computational homogenisation of acoustic metafoams

Acoustic metafoams are novel materials recently proposed for low frequency sound attenuation. The design of their microstructure is based on the combination of standard acoustic foams with locally resonant acoustic metamaterials. This results in improved sound attenuation properties due to the interaction between viscothermal dissipation effects and the local resonance effects at the pore level. In this paper, the non-standard behaviour of such a metafoam with a complex two-phase microstructure is analysed through a multiscale approach. The macroscopic problem is described by general balance equations and at the microscopic scale a detailed representation of the microstructure is considered. The frequency dependent effective properties are used to explain the extraordinary acoustic performance. The homogenisation approach is also validated using direct numerical simulations, showing that the homogenisation technique is adequate in modelling both viscothermal dissipation and the local resonance effect within the metafoam microstructure.

Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation.

The small amplitude dynamic response of materials can be tuned by employing inhomogeneous materials capable of large deformation. However, soft materials generally exhibit viscoelastic behaviour, i.e. loss and frequency-dependent effective properties. This is the case for inhomogeneous materials even in the homogenization limit when propagating wavelengths are much longer than phase lengthscales, since soft materials can possess long relaxation times. These media, possessing rich frequency-dependent behaviour over a wide range of low frequencies, can be termed metamaterials in modern terminology. The sub-class that are periodic are frequently termed soft phononic crystals although their strong dynamic behaviour usually depends on wavelengths being of the same order as the microstructure. In this paper we describe how the effective loss and storage moduli associated with longitudinal waves in thin inhomogeneous rods are tuned by pre-stress. Phases are assumed to be quasi-linearly viscoelastic, thus exhibiting time-deformation separability in their constitutive response. We illustrate however that the effective incremental response of the inhomogeneous medium does not exhibit time-deformation separability. For a range of nonlinear materials it is shown that there is strong coupling between the frequency of the small amplitude longitudinal wave and initial large deformation.This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Hierarchically structured metamaterials with simultaneously negative mass density and Young’s modulus by using dynamic homogenization

We propose one-dimensional hierarchical metamaterials (HMMs) that have simultaneously negative mass density and Young’s modulus in the sense of effective properties. Frequency-dependent effective properties of the HMMs are evaluated by using a dynamic homogenization theory. By considering three necessary conditions for dynamic homogenizability, we verify that our proposed HMMs can have physically meaningful double negativity, whereas the conventional periodic structures cannot. In addition, HMMs yield the broader ranges of the frequency and the effective properties for double negativity due to the large number of geometrical parameters. By exploring all the possible effective properties of HMMs, we discuss whether we can design metamaterials with the opposite sign of material properties in nature, ultimately aiming at perfect impedance matching.

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions \begin{document}$u^\varepsilon : \Omega_\varepsilon \to \mathbb{R}$\end{document} to a Helmholtz equation in the limit \begin{document}$\varepsilon \to 0$\end{document} with the help of two-scale convergence. The domain \begin{document}$\Omega_\varepsilon $\end{document} is obtained by removing from an open set \begin{document}$\Omega\subset \mathbb{R}^n$\end{document} in a periodic fashion a large number (order \begin{document}$\varepsilon ^{-n}$\end{document} ) of small resonators (order \begin{document}$\varepsilon $\end{document} ). The special properties of the meta-material are obtained through sub-scale structures in the perforations.

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions $u^\varepsilon: \Omega_\varepsilon \rightarrow \mathbb{R}$ to a Helmholtz equation in the limit $\varepsilon\rightarrow 0$ with the help of two-scale convergence. The domain $\Omega_\varepsilon$ is obtained by removing from an open set $\Omega\subset \mathbb{R}^n$ in a periodic fashion a large number (order $\varepsilon^{-n}$) of small resonators (order $\varepsilon$). The special properties of the meta-material are obtained through sub-scale structures in the perforations.

Viscoelectroelastic closed form models for frequency and time dependent effective properties of reinforced viscoelectroelastic composites

In this paper, closed form models of time and frequency dependent effective properties of reinforced viscoelectroelastic composites are elaborated. Based on the correspondence principle of linear viscoelectroelasticity and the Mori-Tanaka micromechanical mean field approach, the effective properties are derived in the Laplace-Carson domain and then inverted numerically to the time domain. New analytical models for decoupled and coupled effective properties of transversely isotropic viscoelectroelastic materials are elaborated using compact matrix formulations. Explicit relationships are presented for different effective viscoelectroelastic moduli of piezoelectric materials with various inclusions shapes. Various viscoelastic time dependent models for the phases can be used. Only few Eshelby tensor coefficients are required to be analytically or numerically computed. Results based on the elaborated effective viscoelectroelastic models are presented for various types of piezoelectric inclusions and compared to the available predictions. These models are suitable for designers to predict composite viscoelectroelastic moduli of heterogeneous materials with optimized active and passive characteristics.

Mathematical modeling of the overall time-dependent behavior of non-ageing viscoelastic reinforced composites

New mathematical and numerical formulations are developed for effective time-dependent non ageing viscoelastic behavior of linear viscoelastic composites. The modeling is based on the dynamic Green’s functions, integral equations, and Volterra product. The time concentration tensor is derived through numerical solution of the integral equation and the Mori–Tanaka micromechanical model. The developed modeling gives explicit expressions of effective properties through convolution products in the Stieltjes space. A numerical procedure is developed allowing the calculation of the tensoriel convolution product and its inverse. Based on Carson–Laplace transform, the frequency dependent effective properties are also obtained and converted to the time domain. For viscoelastic Prony laws and inclusions with various shapes, numerical results are presented and compared with the approach based on the Laplace transform and the correspondence principle.

Micromechanical effective medium modeling of metamaterials of the Willis form

The unique behavior of acoustic metamaterials (AMM) results from deeply sub-wavelength structures with hidden degrees of freedom rather than the inherent material properties of their constituents. This distinguishes AMM from classical composite or cellular materials and also complicates attempts to model their overall response. This is especially true when sub-wavelength structures yield anisotropic effective material response, a key feature of AMM devices designed using transformation acoustics. Further, previous work has shown that the dynamic response of heterogeneous materials must include coupling between the overall strain and momentum fields [Milton and Willis, Proc. R. Soc. A 463, 855–880, (2007)]. A micromechanical homogenization model of the overall Willis constitutive equations is presented to address these difficulties. The model yields a low-volume-fraction estimate of anisotropic and frequency-dependent effective properties in the long-wavelength limit. The model employs volume averages of the dyadic Green’s function calculating the particle displacement resulting from a unit force source. This Green’s function is shown to be analogous to one that determines the particle velocity in a fluid resulting from a unit dipole moment. The predicted effective properties for isotropic materials with spherical inclusions fall within the Hashin-Shtrikman bounds and agree with self-consistent estimates. [Work supported by ONR.]

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. Through arguments of calculated reflected energy we note that infinite-domain homogenization is much more applicable to finite cases of the 3-phase composite than it is to the 2-phase composite. In fact, the applicability of dynamic homogenization extends to most of the first branch (negligible reflection) for the 3-phase composite. This is in contrast with a periodic composite without local resonance where the approximation of homogenization worsens with increasing frequency over the first branch and is demonstrably bad on the second branch. 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.

Effective Dynamic Constitutive Relations for 3-D Periodic 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.

Negative effective dynamic mass-density and stiffness: Micro-architecture and phononic transport in periodic composites

We report the results of the calculation of negative effective density and negative effective compliance for a layered composite. We show that the frequency-dependent effective properties remain positive for cases which lack the possibility of localized resonances (a 2-phase composite) whereas they may become negative for cases where there exists a possibility of local resonance below the length-scale of the wavelength (a 3-phase composite). We also show that the introduction of damping in the system considerably affects the effective properties in the frequency region close to the resonance. It is envisaged that this demonstration of doubly negative material characteristics for 1-D wave propagation would pave the way for the design and synthesis of doubly negative material response for full 3-D elastic wave propagation.