Harmonic waves in one-, two- and three-dimensional composites: Bounds for eigenfrequencies

Nemat‐Nasser, Sia; Fu, F. C. L.; Minagawa, S.

doi:10.1016/0020-7683(75)90034-7

Cited by 43 publications

(38 citation statements)

References 8 publications

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“…Their close correspondence indicates that our micro-structurally-based method may be used to calculate the dispersion relation for more complex 2-and 3-dimensional cases, where exact solutions are not available, as we shall report elsewhere. Here we also use our approach to homogenize a 4-layered composite and compare the results with those obtained by the field integration of the stress and displacement mode-shapes obtained using the mixed variational formulation of Nemat-Nasser et al [15,16,17]. It is shown that the micro-structurally-based method gives homogenized results which converge to the field integrationbased homogenization results that are based on either the exact solution or the mixed variational formulation.…”

Section: Introductionmentioning

confidence: 97%

Overall dynamic constitutive relations of layered elastic composites

Nemat‐Nasser

Srivastava

2011

Journal of the Mechanics and Physics of Solids

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A method for homogenization of a heterogeneous (finite or periodic) elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations. It is shown that when the spatial variation of the field variables is restricted by a Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress modeshapes and, necessarily, the dispersion relations. It also gives as a matter of course point-wise solution of the elasto-dynamic field equations, to any desired degree of accuracy. The resulting overall dynamic constitutive relations however, are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micro-mechanical modeling of a representative unit cell of the composite proposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].We show that, for a micro-structured elastic composite, the overall effective massdensity and compliance (stiffness) are always real-valued and positive, whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are however couple and the coupling constitutive parameters are always each others complex conjugates for any heterogeneous elastic unit cell, such that the overall energy-density is always real and positive. In this paper, we have sought to separate the overall constitutive relations which should depend only on the composition and structure of the unit cell, from the overall field equations which should hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations.It is shown, by way of an example of a bi-layered composite, that dispersion curves obtained by our method accurately produce the exact results of Rytov [3]. The method is also used to calculate the effective parameters for a 2-layered composite and the results are compared with those of homogenization based on the field integration of the exact solution (Willis [4], and Nemat-Nasser et al. [5]), and certain relevant issues are clarified. Finally the method is used to homogenize both a symmetric and a non-symmetric 4-layered composite and the results for the symmetric case are compared with those reported by Nemat-Nasser et al. as the exact solution. Thus, this method provides a powerful solution and homogenization tool to use in many cases where the unit cell contains inclusions of complex geometry.

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Section: Introductionmentioning

confidence: 97%

Overall dynamic constitutive relations of layered elastic composites

Nemat‐Nasser

Srivastava

2011

Journal of the Mechanics and Physics of Solids

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“…Furthermore, it is of interest to investigate the convergence behaviors of the three variational principles under different compliance and density variations. Nemat-Nasser et al 25 in 1975 proved that the mixed quotient, in general, converges faster than the other quotients. Babuska and Osborn 30 presented in 1978 their theoretical analysis on the convergence rates of the three quotients, and related them to the function spaces of the density and compliance functions.…”

Section: Study Of Convergence Ratesmentioning

confidence: 96%

“…Convergence rates of three variational principles, the displacement Rayleigh quotient, where the displacement field is varied, the stress Rayleigh quotient, where the stress field is varied, and the mixed quotient, 17,[24][25][26] where both the displacement and stress fields are varied, are considered in this paper. The mixed quotient was proposed by Nemat-Nasser 24 in 1972, which was derived from the works of Hellinger, 27 Reissner, 28 Hu and Washizu.…”

Section: Introductionmentioning

confidence: 99%

Variational methods in phononics

Srivastava

2015

SPIE Proceedings

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Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the eigenvalue problems resulting from the displacement Rayleigh quotient, the stress Rayleigh quotient and the mixed quotient. The convergence rates of the three quotients are found to be related to the continuity and differentiability of the density and compliance variation over the unit cell. In general, the mixed quotient converges faster than both the displacement Rayleigh and the stress Rayleigh quotients, however, there exist special cases where either the displacement Rayleigh or the stress Rayleigh quotient shows the exact same convergence as the mixed-method. We show that all methods converge faster for smoother material property variations, but when density variation is rough, the difference between the mixed quotient and stress Rayleigh quotient is higher and similarly, when compliance variation is rough, the difference between the mixed quotient and displacement Rayleigh quotient is higher. Since eigenvalue problems such as those considered in this paper tend to be highly computationally intensive, it is expected that these results will lead to fast and efficient algorithms in the areas of phononics and photonics.

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“…Plane wave expansion method [5], multiple scattering technique [6], and new-quotient method [7] have been used successfully to calculate the band structure of PCs. Furthermore, there have been efforts to calculate the frequency-dependent effective elastodynamic properties of PCs [8].…”

Section: Introductionmentioning

confidence: 99%

Design optimization of layered periodic composites for desired elastodynamic response

Sadeghi

Nemat‐Nasser

2015

SPIE Proceedings

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In this work, we present a systematic method to design layered periodic composites (PCs) for a prescribed elastodynamic response. Our focus is on optimization problems with equality and/or inequality constraints. Constrained optimization problems are reduced to unconstrained problems, and a genetic algorithm is used to find the optimal design. Symmetric 3-phase layered PCs are considered and the thickness of each phase is chosen as a design parameter. Three cases are presented for illustration purposes: (1) the design of an acoustic filter for maximum bandwidth, (2) the design of an acoustic filter for maximum attenuation, and (3) the design of a PC for maximum attenuation and minimum reflection of acoustic waves.

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Harmonic waves in one-, two- and three-dimensional composites: Bounds for eigenfrequencies

Cited by 43 publications

References 8 publications

Overall dynamic constitutive relations of layered elastic composites

Overall dynamic constitutive relations of layered elastic composites

Variational methods in phononics

Design optimization of layered periodic composites for desired elastodynamic response

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