Research Papers

Elasto-Dynamic Behavior of a Two-Dimensional Square Lattice With Entrained Fluid II: Microstructural and Homogenized Models

[+] 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 September 14, 2013; final manuscript received January 14, 2014; published online March 18, 2014. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 136(3), 031005 (Mar 18, 2014) (10 pages) Paper No: VIB-13-1317; doi: 10.1115/1.4026675 History: Received September 14, 2013; Revised January 14, 2014

This paper presents a detailed study of the pressure waves and effective mechanical properties of a closed-cell cellular solid with entrained fluid. Plane-harmonic-waves are analyzed in a periodic square with a finite-element model of a representative-volume element, which explicitly considers fluid-structure interactions, structural deformations, and the fluid dynamics of entrained fluid. The wall, cavity, and coupled-system resonance frequencies are identified as key parameters that describe the propagation characteristics. A tube-piston model based on computed microstructural deformations allows us to determine the effective stiffness tensor of an equivalent continuum at the macroscale. The analysis of dispersion surfaces indicates a single isotropic pressure mode for frequencies below resonance of the lattice walls, unlike Biot's theory which predicts two pressure modes. Shear modes are instead strongly anisotropic for all values of relative density ρ* describing both cellular ρ*<0.3 and porous solids ρ*0.3. The dependence of the pressure wave phase velocity on the relative density is analyzed for varying properties of the entrained fluid. Depending on the relative density and mass coupling of the solid and fluid phases, the microstructural deformations can be of three types: bending, through-the-thickness, and the combination of the two. For heavy and stiff entrained fluid, the bending regime is confined to extremely small values of relative density, whereas for light fluid such as a gas, deformations are of the bending-type for ρ*<0.1. Through-the-thickness deformations appear only for the heavy entrained fluid for large values of ρ*.

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


Feuillade, C., 1996, “The Attenuation and Dispersion of Sound in Water Containing Multiply Interacting Air Bubbles,” J. Acoust. Soc. Am., 99(6), p. 3412. [CrossRef]
Minnaert, M., 1933, “XVI. On Musical Air-Bubbles and the Sounds of Running Water,” Philos. Mag., 16(104), pp. 235–248 [CrossRef].
Stinson, M. R., and Champoux, Y., 1992, “Propagation of Sound and the Assignment of Shape Factors in Model Porous Materials Having Simple Pore Geometries,” J. Acoust. Soc. Am., 91(2), p. 685. [CrossRef]
Diebels, S., and Ehlers, W., 1996, “Dynamic Analysis of a Fully Saturated Porous Medium Accounting for Geometrical and Material Non-Linearities,” Int. J. Numer. Methods Eng., 39(1), pp. 81–97. [CrossRef]
Wang, X., and Lu, T. J., 1999, “Optimized Acoustic Properties of Cellular Solids,” J. Acoust. Soc. Am., 106(2), pp. 756–765. [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]
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]
Biot, M. A., and Willis, D. G., 1957, “The Elastic Coefficients of the Theory of Consolidation,” ASME J. Appl. Mech., 24(4), pp. 594–601.
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, 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(3), pp. 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]
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]
Gravade, M., Ouisse, M., Collet, M., Scarpa, F., Bianchi, M., and Ichchou, M., 2012, “Auxetic Transverse Isotropic Foams: From Experimental Efficiency to Model Correlation,” Acoustics 2012, Nantes, France, April 23–27.
Chekkal, I., Remillat, C., and Scarpa, F., 2012, “Acoustic Properties of Auxetic Foams,” High Performance Structures and Materials VI, WIT, Southampton, UK, pp. 119–130.
Lopatnikov, S., and Cheng, A.-D., 2004, “Macroscopic Lagrangian Formulation of Poroelasticity With Porosity Dynamics,” J. Mech. Phys. Solids, 52(12), pp. 2801–2839. [CrossRef]
Gibson, L., and Ashby, M., 1999, Cellular Solids: Structure and Properties (Cambridge Solid State Science Series), Cambridge University Press, Cambridge, UK.
Lu, T. J., Chen, F., and He, D., 2000, “Sound Absorption of Cellular Metals With Semi-Open Cells,” J. Acoust. Soc. Am., 108(4), pp. 1697–1709. [CrossRef] [PubMed]
Perrot, C., Chevillotte, F., and Panneton, R., 2008, “Dynamic Viscous Permeability of an Open-Cell Aluminum Foam: Computations Versus Experiments,” J. Appl. Phys., 103(2), p. 024909. [CrossRef]
Lind-Nordgren, E., and Göransson, P., 2010, “Optimising Open Porous Foam for Acoustical and Vibrational Performance,” J. Sound Vib., 329(7), pp. 753–767. [CrossRef]
Hoang, M. T., and Perrot, C., 2013, “Identifying Local Characteristic Lengths Governing Sound Wave Properties in Solid Foams,” J. Appl. Phys., 113(8), p. 084905. [CrossRef]
Evans, A. G., Hutchinson, J., and Ashby, M., 1998, “Multifunctionality of Cellular Metal Systems,” Prog. Mater. Sci., 43(3), pp. 171–221. [CrossRef]
Dempsey, B., Eisele, S., and McDowell, D., 2005, “Heat Sink Applications of Extruded Metal Honeycombs,” Int. J. Heat Mass Transfer, 48(3), pp. 527–535. [CrossRef]
Wang, A., and McDowell, D., 2005, “Yield Surfaces of Various Periodic Metal Honeycombs at Intermediate Relative Density,” Int. J. Plast., 21(2), pp. 285–320. [CrossRef]
Chevillotte, F., and Panneton, R., 2007, “Elastic Characterization of Closed Cell Foams From Impedance Tube Absorption Tests,” J. Acoust. Soc. Am., 122(5), pp. 2653–2660. [CrossRef] [PubMed]
Doutres, O., Atalla, N., and Dong, K., 2011, “Effect of the Microstructure Closed Pore Content on the Acoustic Behavior of Polyurethane Foams,” J. Appl. Phys., 110(6), p. 064901. [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]
Lee, C. Y., Leamy, M. J., and Nadler, J. H., 2009, “Acoustic Absorption Calculation in Irreducible Porous Media: A Unified Computational Approach,” J. Acoust. Soc. Am., 126(4), pp. 1862–1870. [CrossRef] [PubMed]
Stadler, M., and Schanz, M., 2010, “Acoustic Band Structures and Homogenization of Periodic Elastic Media,” PAMM, 10(1), pp. 427–428. [CrossRef]
Warner, M., Thiel, B., and Donald, A., 2000, “The Elasticity and Failure of Fluid-Filled Cellular Solids: Theory and Experiment,” Proc. Natl. Acad. Sci. (U.S.A.), 97(4), pp. 1370–1375. [CrossRef] [PubMed]
Roberts, A. P., and Garboczi, E. J., “Elastic Moduli of Model Random Three-Dimensional Closed-Cell Cellular Solids,” Acta Mater., 49(2), pp. 189–197. [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]
Kumar, R. S., and McDowell, D. L., “Generalized Continuum Modeling of 2-D Periodic Cellular Solids,” Int. J. Solids Struct., 41(26), pp. 7399–7422. [CrossRef]
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]
Spadoni, A., and Ruzzene, M., 2012, “Elasto-Static Micropolar Behavior of a Chiral Auxetic Lattice,” J. Mech. Phys. Solids, 60(1), pp. 156–171. [CrossRef]
Dorodnitsyn, V., and Spadoni, A., 2013, “Elasto-Dynamics of a 2D Square Lattice With Entrained Fluid I: Comparison With Biot's Theory,” ASME J. Vibr. Acoust., 136(2), p. 021024. [CrossRef]
Brillouin, L., 2003, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Dover, New York.
Martin, V., 2007, Éléments d'acoustique générale: de quelques lieux communs de l'acoustique a une premiere matrise des champs sonores, Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland.
Fahy, F. J., and Gardonio, P., 2007, Sound and Structural Vibration: Radiation, Transmission and Response, Elsevier, New York.
Graff, K., 1975, Wave Motion in ElasticSolids (Dover Books on Engineering Series), Dover, New York.
Wolfe, J. P., 2005, Imaging Phonons: Acoustic Wave Propagation in Solids, Cambridge University Press, Cambridge, UK.
Auld, B. A., 1973, Acoustic Fields and Waves in Solids, Vol. 1, Wiley, New York.


Grahic Jump Location
Fig. 1

Square lattice with walls of thickness h and cell length L with (a) ρ*< 30%, and (b) ρ*> 30%. The superposed unit cell in (a) has a thickness h. The RVE with the solid portion discretized (c) by either beam or 4-node plane elements, shown with appropriate degrees of freedom (DOF): ui, wi displacement DOFs for both types of elements, rotational DOF (φi) for beam elements only, and pressure pi for coupling and fluid elements. In both cases, coupling elements with both structural and pressure DOFs are employed to model fluid-structure interaction.

Grahic Jump Location
Fig. 2

Band structure for the RVE with L=100μm, ρ*=0.04 discretized with beam elements for the irreducible Brillouin zone with the high-symmetry points Γ, X, M. The 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. The solid lines are the solution to the FSI problem and the dashed lines are the solution to the structure-only case. The circled letters (a)–(f) denote the wavenumber combinations used to depict the deformed configurations in Fig. 3.

Grahic Jump Location
Fig. 3

Fluid-structure wavemodes corresponding to the wavenumber combinations indicated by the labels (a)–(f) in Fig. 2. The solid and dashed lines denote the deformed and initial configurations, respectively.

Grahic Jump Location
Fig. 4

1D tube-piston model applicable in the linear pressure mode regime

Grahic Jump Location
Fig. 5

Representation of the aligned tube-piston unit cells as a system of springs and masses in series with the effective stiffness and mass (keff, meff) for pressure wave propagation near the Γ point

Grahic Jump Location
Fig. 6

Band structure detail about the Γ point with ρ* = 0.04 for the (a) beam, (b) plane-element models [40], and (c) Superposition of Eq. (14) to the numerical band structure for high-frequency regimes. The circled label (1) denotes the FSI shear mode in the Γ-X direction.

Grahic Jump Location
Fig. 7

Case 1 (Table 1). Left (solid line) and right-hand (dashed line) sides (a) of Eq. (7), with the corresponding band structure (b) depicting the FSI solution with solid lines and the structure-only case with dashed lines. The ordinate is normalized with ω0.

Grahic Jump Location
Fig. 8

Case 2 (Table 1). Left (solid line) and right-hand (dashed line) sides of Eq. (7) for (a) β ≪ 1, and (b) β >> 1. (c) and (d) Corresponding band structures where the FSI solution is shown with solid lines and the structure-only case with dashed lines. The ordinates in (c) and (d) are normalized with ωfcc and ωfoc, respectively.

Grahic Jump Location
Fig. 9

Case 3 (Table 1). Left (solid line) and right-hand (dashed line) sides (a) of Eq. (7), with the corresponding band structure (b) depicting the FSI solution with solid black lines and the structure-only case with dashed lines. The ordinate is normalized with ωfcc.

Grahic Jump Location
Fig. 10

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 ρ*. Solid, dashed-dotted, and thick dashed lines correspond to the FE results based on the plane elements, beam elements, and Biot's theory for plane conditions, respectively. The dotted lines in panels (a) and (b) correspond to equivalent-continuum models for shear waves based on C44, which is defined in Sec. 3.2. The dotted lines in panels (c) and (d) correspond to the equivalent-continuum model for the pressure waves ((c) and (d)) given by Eq. (14). The thin dashed lines in panels (c) and (d) denote the pressure-wave velocity obtained by employing Eq. (8). All velocities correspond to the Γ-X direction.

Grahic Jump Location
Fig. 11

Static model of a square cavity enclosed by simply-supported beams of thickness h, with internal Pb and external pressure Pt

Grahic Jump Location
Fig. 12

Dispersion surfaces (a) for ρ* = 0.04 (case 1 from Table 1), and (b) for ρ* = 0.75 in the (kx,ky,ω)-domain, where |kxL|≤π and |kyL|≤π. Shear-mode surfaces are labeled with the circled numbers (1) and (3), while pressure mode surfaces are labeled with the circled numbers (2) and (4). The top view of each of the surfaces is provided on the right side of the figure.

Grahic Jump Location
Fig. 13

Dispersion surfaces corresponding to case 2 (Table 1) for (a) β ≪ 1, and (b) β >> 1 in the (kx,ky,ω)-domain, where |kxL|≤π and |kyL|≤π. Shear mode surfaces are labeled with the circled numbers (1) and (3), while pressure mode surfaces are labeled with the circled numbers (2) and (4). The top view of each of the surfaces is provided on the right side of the figure.

Grahic Jump Location
Fig. 14

Slowness curves (in s/m) for shear (left column) and pressure waves (right column). Case 1: α0Lf ≪ π/2, ρ* = 0.04 ((a) and (b)) and ρ* = 0.75 ((c) and (d)). Case 2: α0Lf >> π/2 with β ≪ 1 (ρ* = 0.08) (e) and β >> 1 (ρ* = 0.2) (f). All curves are evaluated for the constant frequency ω0 = 10 Hz. In (a) and (c), the solid and dashed lines denote, respectively, the shear slowness from the Christoffel matrix (see Eq. (26)) and from dispersion surfaces obtained with plane elements. In (b) and (d), the solid and dotted lines denote the pressure slowness from Eq. (26) and the plane elements, respectively. Panels (e) and (f) correspond to case 2 (Table 1) for β ≪ 1 and β >> 1, respectively, and depict the shear (anisotropic) and pressure slowness curves. Enlarged pressure slowness curves are superposed with insets in panels (e) and (f).

Grahic Jump Location
Fig. 15

Phase velocities for pressure waves for (a) entrained air, and (a) (b) water for 10-8≤ρ*≤1 in logarithmic scale. The dashed lines denote Biot's theory. Circled numbers label the logarithmically-linear regimes of cP denoted by solid lines. The dotted line in panel (b) is the approximation of the pressure phase velocity (see Eq. 31) given by the spring-mass model superposed in (b). Relative-density regimes and associated structural deformation mechanisms are indicated in panels (a) and (b).

Grahic Jump Location
Fig. 16

Fluid-structure (FSI) wave modes of a square unit cell with ρ* = 0.75 for (a) air, and (b) water, corresponding to the wavevector k along the x-axis with kxL = 0.03π, kyL = 0. The solid lines denote the initial configuration. The gray scale corresponds to the Euclidean norm of the nodal-displacement components v = ux2+uy2.




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