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Research Papers

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

[+] Author and Article Information

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

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

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

## Abstract

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 $0.0001≤ρ*≤1$ 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.

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## Figures

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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