Research Papers

Elastodynamics of a Two-Dimensional Square Lattice With Entrained Fluid—Part I: Comparison With Biot's Theory

[+] Author and Article Information
Vladimir Dorodnitsyn

Institute of Mechanical Engineering,
École Polytecnique Fedérale de Lausanne,
Lausanne CH-1015, Switzerland

Alessandro Spadoni

Institute of Mechanical Engineering,
École Polytecnique Fedérale de Lausanne,
Lausanne CH-1015, Switzerland
e-mail: alex.spadoni@epfl.ch

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 11, 2013; final manuscript received December 9, 2013; published online February 5, 2014. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 136(2), 021024 (Feb 05, 2014) (10 pages) Paper No: VIB-13-1239; doi: 10.1115/1.4026349 History: Received July 11, 2013; Revised December 09, 2013

In the present paper, the performance of Biot's theory is investigated for wave propagation in cellular and porous solids with entrained fluid for configurations with well-known drained (no fluid) mechanical properties. Cellular solids differ from porous solids based on their relative density ρ*<0.3. The distinction is phenomenological and is based on the applicability of beam (or plate) theories to describe microstructural deformations. The wave propagation in a periodic square lattice is analyzed with a finite-element model, which explicitly considers fluid-structure interactions, structural deformations, and fluid-pressure variations. Bloch theorem is employed to enforce symmetry conditions of a representative volume element and obtain a relation between frequency and wavevector. It is found that the entrained fluid does not affect shear waves, beyond added-mass effects, so long as the wave spectrum is below the pores' natural frequency. One finds strong dispersion in cellular solids as a result of resonant scattering, in contrast to Bragg scattering dominant in porous media. Configurations with 0.0001ρ*1 are investigated. One finds that Biot's theory, derived from averaged microstructural quantities, well estimates the phase velocity of pressure and shear waves for cellular porous solids, except for the limit ρ*1. For frequencies below the first resonance of the lattice walls, only the fast-pressure mode of the two modes predicted by Biot's theory is found. It is also shown that homogenized models for shear waves based on microstructural deformations for drained conditions agree with Biot's theory.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Gibson, L. J., and Ashby, M. F., 1999, Cellular Solids: Structure and Properties (Cambridge Solid State Science Series), Cambridge University, Cambridge, UK.
Kumar, R. S., and McDowell, D. L., 2004, “Generalized Continuum Modeling of 2-D Periodic Cellular Solids,” Int. J. Solids Struct., 41(26), pp. 7399–7422. [CrossRef]
Spadoni, A., and Ruzzene, M., 2011, “Elasto-Static Micropolar Behavior of a Chiral Auxetic Lattice,” J. Mech. Phys. Solids, 60, pp. 156–171. [CrossRef]
Suiker, A. J., Metrikine, A. V., and De Borst, R., 2001, “Comparison of Wave Propagation Characteristics of the Cosserat Continuum Model and Corresponding Discrete Lattice Models,” Int. J. Solids Struct., 38(9), pp. 1563–1583. [CrossRef]
Eringen, A. C., 2001, Microcontinuum Field Theories: I. Foundations and Solids (Microcontinuum Field Theories), Springer, New York.
Martinsson, P. G., and Movchan, A. B., 2003, “Vibrations of Lattice Structures and Phononic Band Gaps,” Q. J. Mech. Appl. Math., 56(1), pp. 45–64. [CrossRef]
Phani, A. S., Woodhouse, J., and Fleck, N. A., 2006, “Wave Propagation in Two-Dimensional Periodic Lattices,” J. Acoust. Soc. Am., 119(4), pp. 1995–2005. [CrossRef] [PubMed]
Gonella, S., and Ruzzene, M., 2008, “Homogenization and Equivalent In-Plane Properties of Two-Dimensional Periodic Lattices,” Int. J. Solids Struct., 45(10), pp. 2897–2915. [CrossRef]
Mead, D. M., 1996, “Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964-1995,” J. Sound Vib., 190(3), pp. 495–524. [CrossRef]
Gonella, S., and Ruzzene, M., 2008, “Analysis of In-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices,” J. Sound Vib., 312(1), pp. 125–139. [CrossRef]
Spadoni, A., Ruzzene, M., Gonella, S., and Scarpa, F., 2009, “Phononic Properties of Hexagonal Chiral Lattices,” Wave Motion, 46(7), pp. 435–450. [CrossRef]
Casadei, F., and Rimoli, J. J., 2013, “Anisotropy-Induced Broadband Stress Wave Steering in Periodic Lattices,” Int. J. Solids Struct., 50, pp. 1402–1414. [CrossRef]
Ruzzene, M., Scarpa, F., and Soranna, F., 2003, “Wave Beaming Effects in Two-Dimensional Cellular Structures,” Smart Mater. Struct., 12(3), pp. 363–372. [CrossRef]
Liu, X. N., Hu, G. K., Sun, C. T., and Huang, G. L., 2011, “Wave Propagation Characterization and Design of Two-Dimensional Elastic Chiral Metacomposite,” J. Sound Vib., 330(11), pp. 2536–2553. [CrossRef]
Xu, Y. L., Tian, X. G., and Chen, C. Q., 2012, “Band Structures of Two Dimensional Solid/Air Hierarchical Phononic Crystals,” Physica B: Condens. Matter, 407(12), pp. 1995–2001. [CrossRef]
Biot, M. A., 1956, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range,” J. Acoust. Soc. Am., 28(2), pp. 168–178. [CrossRef]
Allard, J., and Atalla, N., 2009, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, 2nd ed., Wiley, New York.
Carcione, J., 2007, Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media (Handbook of Geophysical Exploration: Seismic Exploration), Elsevier Science, New York.
Auriault, J. L., and Sanchez-Palencia, E., 1977, “Etude du Comportement Macroscopique d'un Milieu Poreux Saturé Déformable,” J. Méc., 16(4), pp. 575–603.
Burridge, R., and Keller, J. B., 1981, “Poroelasticity Equations Derived From Microstructure,” J. Acoust. Soc. Am., 70(4), pp. 1140–1146. [CrossRef]
Thompson, M., and Willis, J. R., 1991, “A Reformulation of the Equations of Anisotropic Poroelasticity,” ASME J. Appl. Mech., 58, p. 612–616. [CrossRef]
Cheng, A. H.-D., 1997, “Material Coefficients of Anisotropic Poroelasticity,” Int. J. Rock Mech. Min. Sci., 34(2), pp. 199–205. [CrossRef]
Dormieux, L., Molinari, A., and Kondo, D., 2002, “Micromechanical Approach to the Behavior of Poroelastic Materials,” J. Mech. Phys. Solids, 50(10), pp. 2203–2231. [CrossRef]
Chekkal, I., Remillat, C., and Scarpa, F., 2012, “Acoustic Properties of Auxetic Foams,” High Performance Structures and Materials VI, WIT Press, Ashurst, UK, pp. 119–130.
Gueven, I., Kurzeja, P., Luding, S., and Steeb, H., 2012, “Experimental Evaluation of Phase Velocities and Tortuosity in Fluid Saturated Highly Porous Media,” PAMM, 12(1), pp. 401–402. [CrossRef]
Chevillotte, F., Perrot, C., and Panneton, R., 2010, “Microstructure Based Model for Sound Absorption Predictions of Perforated Closed-Cell Metallic Foams,” J. Acoust. Soc. Am., 128(4), pp. 1766–1776. [CrossRef] [PubMed]
Perrot, C., Chevillotte, F., and Panneton, R., 2008, “Bottom-Up Approach for Microstructure Optimization of Sound Absorbing Materials,” J. Acoust. Soc. Am., 124(2), pp. 940–948 [CrossRef]. [PubMed]
Cook, R., 2001, Concepts and Applications of Finite Element Analysis, Wiley, New York.
Kittel, C., 2004, Introduction to Solid State Physics, Wiley, New York.
Brillouin, L., 2003, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices (Dover Phoenix Editions), Dover, New York.
Bažant, Z. P., and Christensen, M., 1972, “Analogy Between Micropolar Continuum and Grid Frameworks Under Initial Stress,” Int. J. Solids Struct., 8(3), pp. 327–346. [CrossRef]
Biot, M. A., 1956, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range,” J. Acoust. Soc. Am., 28(2), pp. 179–191. [CrossRef]
Brutsaert, W., 1964, “The Propagation of Elastic Waves in Unconsolidated Unsaturated Granular Mediums,” J. Geophys. Res., 69(2), pp. 243–257. [CrossRef]
Santos, J. E., Douglas, Jr., J., Corberó, J., and Lovera, O. M., 1990, “A Model for Wave Propagation in a Porous Medium Saturated by a Two-Phase Fluid,” J. Acoust. Soc. Am., 87(4), pp. 1439–1448. [CrossRef]
Carcione, J. M., Cavallini, F., Santos, J. E., Ravazzoli, C. L., and Gauzellino, P. M., 2004, “Wave Propagation in Partially Saturated Porous Media: Simulation of a Second Slow Wave,” Wave Motion, 39(3), pp. 227–240. [CrossRef]
BiotM. A., and Willis, D. G., 1957, “The Elastic Coefficients of the Theory of Consolidation,” ASME J. Appl. Mech., 24(4), pp. 594–601.
Stadler, M., and Schanz, M., 2010, “Acoustic Band Structures and Homogenization of Periodic Elastic Media,” PAMM, 10(1), pp. 427–428. [CrossRef]
Yavari, B., and Bedford, A., 1991, “Comparison of Numerical Calculations of Two Biot Coefficients With Analytical Solutions,” J. Acoust. Soc. Am., 90(2), pp. 985–990. [CrossRef]


Grahic Jump Location
Fig. 1

Square lattice with walls thickness h and cell length L with (a) ρ*<0.3 and (b) ρ*>0.3. The superposed unit cell in (a) has thickness h/2.

Grahic Jump Location
Fig. 2

Discretized RVE: beam and four-node plane elements are shown with appropriate degrees of freedom. In both cases, coupling elements with both structural and pressure degrees of freedom enforce fluid-structure interaction.

Grahic Jump Location
Fig. 3

Transformation (a) from direct (i1,i2) to reciprocal basis (i1*,i2*) with first and irreducible Brillouin zones with symmetry points Γ = (0,0),X = (π/L,0),M = (π/L,π/L). RVE (b) with associated master (light gray) and slave (dark gray) degrees of freedom.

Grahic Jump Location
Fig. 4

Band structure for RVE with h/L=0.02,ρ* = 0.02 discretized with beam elements for the irreducible Brillouin zone with high-symmetry points Γ, X, M. Left ordinate is normalized by the first natural frequency of a clamped-clamped beam, ω0; the second ordinate is normalized by the first natural frequency of the fluid cavity alone, ωc. Solid lines are the solution to the FSI problem; dashed lines are the solution to the structure-only case. Circled letters (a)–(i) denote wavenumber combinations used to depict deformed configurations in Fig. 4.

Grahic Jump Location
Fig. 5

Fluid-structure (FSI) wavemodes corresponding to the wavenumber combinations indicated by the labels (a)-(i) in Fig. 4. Solid lines denote the deformed configuration, while dashed lines denote the initial configuration.

Grahic Jump Location
Fig. 6

Band structure (a) for the same parameters as Fig. 4, computed with plane elements. Solid lines are the solution to the FSI configuration; dashed lines are the solution to the structure-only configurations. FSI solution obtained with beam elements is superposed with dotted lines. Superposition of FSI band structures (b) with entrained air (1a) and water (1w) with detailed view of first shear wavemode. Homogenized phase velocities denoted as cSair and cSw for air and water, respectively.

Grahic Jump Location
Fig. 7

Band structure (beam elements) for the FSI configuration (a) for the same parameters as Fig. 4 in solid lines with the first four natural frequencies of a clamped-clamped beam superposed as dashed lines. The ordinate is normalized by ω0. Thin dashed lines are the solution to the structure-only case. Band structure (beam elements) and mode shapes for the FSI configuration (b) for entrained fluid with Bf = 1.42×10-7 Pa and ρf = 1.2 kg/m3 with ω∧ = ω/ωc on the right ordinate and ω¯=ω/ω0 on the left. Solid lines are shear modes, while dashed lines denote the fast-pressure modes.

Grahic Jump Location
Fig. 8

Examples of the unit cell model with different walls thickness: (a) h/L = 0.16, (b) h/L = 0.2, (c) h/L = 0.5, and (d) h/L = 1. Solid elements are shown in light gray, fluid elements in white, and coupled elements in dark gray.

Grahic Jump Location
Fig. 9

Band structure for a porous medium with entrained air for (a) h = 0.16L, (b) h = 0.2L, (c) h = 0.5L, and (d) h = L, corresponding to the RVEs of Fig. 9. Solid lines represent solution of the FSI configuration; dashed lines in panels (a)-(c) are the solution to the structure-only case. Shear and longitudinal (pressure) wavemodes are denoted as S and L, respectively. Dashed lines in (d) denote the phase velocities of Eq. (17).

Grahic Jump Location
Fig. 10

Band structures for a square lattice with entrained air with h/L = 0.02 ((a) and (d)), h/L = 0.4 ((b) and (e)), and h/L = 0.8 ((c) and (f)). Panels (a)–(c) highlight pressure modes, while panels (d)–(f) highlight shear modes. Thin dashed lines denote shear and pressure modes from Biot's theory. Thick dashed lines denote the shear-wave velocity of Eq. (20). The insets in panels (a)–(c) show the RVE for each configuration.

Grahic Jump Location
Fig. 11

Phase velocities for shear ((a) and (b)) and pressure ((c) and (d)) waves for entrained air ((a) and (c)) and water ((b) and (d)) for increasing relative density. Solid and dashed-dotted lines correspond to the RVE model discretized with plane and beam elements, respectively. Dashed lines denote Biot's theory for plane conditions. Dotted lines in panels (a) and (b) correspond to the homogenized model for shear in drained conditions [2]. All velocities correspond to wavenumbers in the Γ-X direction.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In