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Research Papers

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

[+] 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 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

## Abstract

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 $ρ*$.

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## Figures

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.

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.

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.

Fig. 4

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

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

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.

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.

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.

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.

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.

Fig. 11

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

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.

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.

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).

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).

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.

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