×

Converging bounds for the effective shear speed in 2D phononic crystals. (English) Zbl 1329.74121

Summary: Calculation of the effective quasistatic shear speed \(c\) in 2D solid phononic crystals is analyzed. The plane-wave expansion (PWE) and the monodromy-matrix (MM) methods are considered. For each method, the stepwise sequence of upper and lower bounds is obtained which monotonically converges to the exact value of \(c\). It is proved that the two-sided MM bounds of \(c\) are tighter and their convergence to \(c\) is uniformly faster than that of the PWE bounds. Examples of the PWE and MM bounds of effective speed versus concentration of high-contrast inclusions are demonstrated.

MSC:

74J05 Linear waves in solid mechanics
74Q20 Bounds on effective properties in solid mechanics
74E15 Crystalline structure
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Krokhin, A.A., Arriaga, J., Gumen, L.N.: Speed of sound in periodic elastic composites. Phys. Rev. Lett. 91, 264302 (2003)
[2] Kutsenko, A.A., Shuvalov, A.L., Norris, A.N., Poncelet, O.: Effective shear speed in two-dimensional phononic crystals. Phys. Rev. B 84, 064305 (2011) · Zbl 1228.74005
[3] Kutsenko, A.A., Shuvalov, A.L., Norris, A.N.: Evaluation of the effective speed of sound in phononic crystals by the monodromy matrix method. J. Acoust. Soc. Am. 130, 3553-3557 (2011)
[4] Nevard, J., Keller, J.B.: Reciprocal relations for effective conductivities of anisotropic media. J. Math. Phys. 26, 2761-2765 (1985) · Zbl 0582.73023
[5] Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994) · Zbl 0801.35001
[6] Pease, M.C. III: Methods of Matrix Algebra. Academic Press, New York (1965) · Zbl 0145.03701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.