A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains

Gazalet, Joseph; Dupont, Samuel; Kastelik, Jean‐Claude; Rolland, Quentin; Djafari‐Rouhani, Bahram

doi:10.1016/j.wavemoti.2012.12.010

Cited by 58 publications

(29 citation statements)

References 34 publications

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“…with an R-periodic amplitude ˜ u k (x ) ( Gazalet et al, 2013 ). In other words, to the expansion (2.2) , there corresponds a similar Blochwave expansion of the solution to (2.1) :…”

Section: Bloch-wave Expansionsmentioning

confidence: 93%

A generalized theory of elastodynamic homogenization for periodic media

Nassar

Auffray

2016

International Journal of Solids and Structures

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International audienceFor periodically inhomogeneous media, a generalized theory of elastody-namic homogenization is proposed so that even the long-wavelength and low-frequency asymptotic expansions of the resulting effective (or macro-scopic) motion equation can, approximately but simultaneously, capture all the acoustic and some of the optical branches of the microscopic dispersion curve. The key to constructing the generalized theory resides in incorporating new kinematical degrees of freedom in conjunction with rapidly oscillating body forces as microscopic and macroscopic loadings while satisfying an energetical consistency constraint reminiscent of Hill-Mandel lemma. By this constraint, an effective displacement field is naturally defined as the projection of a microscopic one onto the dual to the space of body forces. To illustrate these results, a two-phase string is studied in detail

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“…with an R-periodic amplitude ˜ u k (x ) ( Gazalet et al, 2013 ). In other words, to the expansion (2.2) , there corresponds a similar Blochwave expansion of the solution to (2.1) :…”

Section: Bloch-wave Expansionsmentioning

confidence: 93%

A generalized theory of elastodynamic homogenization for periodic media

Nassar

Auffray

2016

International Journal of Solids and Structures

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“…Let us detail the dynamic part of the identification processes, which is based on the computation of the dispersion curves for the unitary cell by using Bloch analysis (Dresselhaus et al, 2007;Farzbod and Leamy, 2011;Gazalet et al, 2013).…”

Section: General Identification Proceduresmentioning

confidence: 99%

On the validity range of strain-gradient elasticity: A mixed static-dynamic identification procedure

Rosi

Placidi

Auffray

2018

European Journal of Mechanics - A/Solids

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Wave propagation in architectured materials, or materials with microstructure, is known to be dependent on the ratio between the wavelength and a characteristic size of the microstructure. Indeed, when this ratio decreases (i.e. when the wavelength approaches this characteristic size) important quantities, such as phase and group velocity, deviate considerably from their low frequency/long wavelength values. This well-known phenomenon is called dispersion of waves. Objective of this work is to show that straingradient elasticity can be used to quantitatively describe the behaviour of a microstructured solid, and that the validity domain (in terms of frequency and wavelength) of this model is sufficiently large to be useful in practical applications. To this end, the parameters of the overall continuum are identified for a periodic architectured material, and the results of a transient problem are compared to those obtained from a finite element full field computation on the real geometry. The quality of the overall description using a strain-gradient elastic continuum is compared to the classical homogenization procedure that uses Cauchy continuum. The extended model of elasticity is shown to provide a good approximation of the real solution over a wider frequency range.

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“…(7) as the Floquet-Bloch relation. 1 Returning to our problem, there are several ways to proceed. We can connect the fields in adjacent regions (where u ′′ + k 2 u = 0; see Section 3.1) across the cells, or we can connect the fields in neighbouring periods of the periodic structure (Section 3.2).…”

Section: Periodic Media: Brief Reviewmentioning

confidence: 99%

“…That was done later by Bloch [11] in the context of the three-dimensional Schrödinger equation with a periodic potential. For further discussion, see [1].…”

Section: Connect Across One Cellmentioning

confidence: 99%

“…The wave bearing properties of periodic inhomogeneous media are important in many application areas in physics, engineering and applied mathematics and have been studied extensively [1]. The standard procedure is to consider unbounded inhomogeneous media, for which Floquet-Bloch conditions are imposed in order to determine their band-gap properties.…”

Section: Introductionmentioning

confidence: 99%

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One-dimensional reflection by a semi-infinite periodic row of scatterers

2015

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h i g h l i g h t s• A canonical one-dimensional scattering problem is solved exactly in 3 ways. • First approach: shift semi-infinite row by one period. • Second approach: solve for finite row and then take limit. • Third approach: solve semi-infinite problem (discrete Wiener-Hopf). a b s t r a c t Three methods are described in order to solve the canonical problem of the onedimensional reflection by a semi-infinite periodic row of identical scatterers. The exact reflection coefficient R is determined. The first method is associated with shifting the domain by a single period and subsequently considering two scatterers, one being a single scatterer and the second being the entire semi-infinite array. The second method determines the reflection coefficient R N associated with a finite array of N scatterers. The limit as N → ∞ is then taken. In general R N does not converge to R in this limit, although we summarize various arguments that can be made to ensure the correct limit is achieved. The third method considers direct approaches. In particular, for point masses, the governing inhomogeneous ordinary differential equation is solved using the discrete Wiener-Hopf technique.

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A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains

Cited by 58 publications

References 34 publications

A generalized theory of elastodynamic homogenization for periodic media

A generalized theory of elastodynamic homogenization for periodic media

On the validity range of strain-gradient elasticity: A mixed static-dynamic identification procedure

One-dimensional reflection by a semi-infinite periodic row of scatterers

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