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Research Papers

# Analysis of Phononic Bandgap Structures With Dissipation

[+] Author and Article Information
Jakob S. Jensen

e-mail: jsj@mek.dtu.dk
Department of Mechanical Engineering,
Technical University of Denmark,
Building 404, 2800 Denmark

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 14, 2012; final manuscript received February 12, 2013; published online June 6, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 135(4), 041015 (Jun 06, 2013) (8 pages) Paper No: VIB-12-1149; doi: 10.1115/1.4023901 History: Received May 14, 2012; Revised February 12, 2013

## Abstract

We study wave propagation in periodic materials with dissipation using two different formulations. An $ω(k)$-formulation yields complex frequency solutions for nonvanishing dissipation whereas a $k(ω)$-formulation leads to complex wave numbers. For small (realistic) levels of material dissipation and longer wavelengths, we show that the two formulations produce nearly identical results in terms of propagation constant and wave decay. We use the $k(ω)$-formulation to compute loss factors with dissipative bandgap materials for steady-state wave propagation and create simplified diagrams that unify the spatial loss factor from dissipative and bandgap effects. Additionally, we demonstrate the applicability of the $k(ω)$-formulation for the computation of the band diagram for viscoelastic composites and compare the computed loss factors for low frequency wave propagation to existing results based on quasi-static homogenization theory.

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## Figures

Fig. 1

(a) Unit cell of periodic material consisting of two phases with different density and elasticity modulus. The cell side is 1 cm. (b) Illustration of the irreducible Brillouin zone and propagation directions specified by the linear relation between the two wave vector components ky = akx.

Fig. 2

Band diagram considering the boundary of the irreducible Brillouin zone for the four materials in Table 1. Equation (21) has been solved for 120 points along each edge for the first six eigenvalues.

Fig. 3

Band diagram for the nondissipative material ND

Fig. 4

Each band diagram is created by solving Eq. (21) for a specific value of a, which is shown above the corresponding plot. Material D1 with δ = 0.1.

Fig. 5

Each band diagram is created by solving Eq. (21) for a specific value of a, which is shown above the corresponding plot. Material D2 with δ = 0.1.

Fig. 6

Each band diagram is created by solving Eq. (21) for a specific value of a, which is shown above the corresponding plot. Material D3 with δ = 0.5.

Fig. 7

Illustration of how spatial and temporal decay is related through Eq. (23). Top row shows solutions from Γ to X, while bottom row shows solutions from Γ to M, for (a) material D1 with δ = 0.1, (b) material D2 with δ = 0.1, and (c) material D3 with δ = 0.5.

Fig. 8

Each simple band diagram is created by solving Eq. (21) for a specific value of a, which is shown above the corresponding plot. The considered material is D2.

Fig. 9

Simple band diagram for material D2 created by (a) solving Eq. (21) for a range of a values from 0 to 1, or (b) traversing only the border of the irreducible Brillouin zone

Fig. 10

Unit cell of material optimized for high loss factor, from Ref. [26]. The cell side is 1 cm.

Fig. 11

Ratio according to Eq. (26) for the material in Fig. 10 with constant phase properties. (a) Top and (b) bottom border of the irreducible Brillouin zone.

Fig. 12

Ratio according to Eq. (26) for the material in Fig. 10 with frequency dependent phase properties. (a) Top and (b) bottom border of the irreducible Brillouin zone.

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