A new integral equation approach to elastodynamic homogenization

Parnell, William J.; Abrahams, I. David

doi:10.1098/rspa.2007.0254

Cited by 11 publications

(3 citation statements)

References 30 publications

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“…29 The static effective density is also recovered by application of a recently proposed scheme of homogenization applied to the SH problem: the so-called integral equation method proposed by Parnell and Abrahams. 30 This method also recovers Eq. ͑35͒ at leading order, with successive lattice corrections for materials where scatterers are positioned on a periodic lattice.…”

Section: ͑34͒mentioning

confidence: 67%

Estimating the dynamic effective mass density of random composites

Martin

Maurel

Parnell

2010

The Journal of the Acoustical Society of America

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The effective mass density of an inhomogeneous medium is discussed. Random configurations of circular cylindrical scatterers are considered, in various physical contexts: fluid cylinders in another fluid, elastic cylinders in a fluid or in another solid, and movable rigid cylinders in a fluid. In each case, time-harmonic waves are scattered, and an expression for the effective wavenumber due to Linton and Martin [J. Acoust. Soc. Am. 117, 3413-3423 (2005)] is used to derive the effective density in the low frequency limit, correct to second order in the area fraction occupied by the scatterers. Expressions are recovered that agree with either the Ament formula or the effective static mass density, depending upon the physical context.

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Section: ͑34͒mentioning

confidence: 67%

Estimating the dynamic effective mass density of random composites

Martin

Maurel

Parnell

2010

The Journal of the Acoustical Society of America

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“…In this regime, effective elastic moduli, viscosities and other mechanical properties can be derived by using homogenization methods and micromechanical techniques [11][12][13]. In particular when the microstructure is distributed periodically, closed-form solutions can often be found [14][15][16][17]. The field of metamaterials is related to, but rather distinct from, this homogenization scenario in the sense that a metamaterial can give rise to a frequency-dependent response even in this low-frequency (homogenization) limit, usually being associated with an induced resonance of the microstructure [18,19].…”

Section: Introductionmentioning

confidence: 99%

Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation

Parnell

Pascalis

2019

Phil. Trans. R. Soc. A.

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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’.

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“…The key novelty of the proposed scheme is the form of solutions that are derived as shall be illustrated below. The method to be discussed extends the work in [ 18 ] (referred to as PA below), where a new homogenization scheme was devised based on the integral equation form of the governing equation, considering antiplane wave propagation in the low-frequency limit, so that an equation of the form ( 1.2 ) was derived, and the leading-order result was determined. The work of PA was itself inspired by the method introduced in [ 19 ], in which expressions for the effective elastic properties of three-dimensional random particulate media were determined, but restricted to the dilute-dispersion limit [ 20 ].…”

Section: Introductionmentioning

confidence: 99%

An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems

et al. 2017

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In Parnell & Abrahams (2008 Proc. R. Soc. A 464, 1461–1482. (doi:10.1098/rspa.2007.025410.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.

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A new integral equation approach to elastodynamic homogenization

Cited by 11 publications

References 30 publications

Estimating the dynamic effective mass density of random composites

Estimating the dynamic effective mass density of random composites

Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation

An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems

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