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Research Papers

# The Dynamic Compressive Response of Open-Cell Foam Impregnated With a Newtonian Fluid

[+] Author and Article Information
M. A. Dawson

Department of Mechanical Engineering,  Massachusetts Institute of Technology, Cambridge, MA 02139dawson@mit.edu

G. H. McKinley

Department of Mechanical Engineering,  Massachusetts Institute of Technology, Cambridge, MA 02139

L. J. Gibson

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

J. Appl. Mech 75(4), 041015 (May 14, 2008) (11 pages) doi:10.1115/1.2912940 History: Received May 18, 2007; Revised February 14, 2008; Published May 14, 2008

## Abstract

This analysis considers the flow of a highly viscous Newtonian fluid in a reticulated, elastomeric foam undergoing dynamic compression. A comprehensive model for the additional contribution of viscous Newtonian flow to the dynamic response of a reticulated, fluid-filled, elastomeric foam under dynamic loading is developed. For highly viscous Newtonian fluids, the flow in the reticulated foam is assumed to be dominated by viscous forces for nearly all achievable strain rates; Darcy’s law is assumed to govern the flow. The model is applicable for strains up to the densified strain for all grades of low-density, open-cell, elastomeric foam. Low-density, reticulated foam is known to deform linear elastically and uniformly up to the elastic buckling strain. For strains greater than the elastic buckling strain but less than the densified strain, the foam exhibits bimodal behavior with both linear-elastic and densified regimes. The model presented in this analysis is applicable for all strains up to the densified strain. In the bimodal regime, the model is developed by formulating a boundary value problem for the appropriate Laplace problem that is obtained directly from Darcy’s law. The resulting analytical model is more tractable than previous models. The model is compared with experimental results for the stress-strain response of low-density polyurethane foam filled with glycerol under dynamic compression. The model describes the data for foam grades varying from $70ppito90ppi$ and strain rates varying from $2.5×10−3to101s−1$ well. The full model can also be well approximated by a simpler model, based on the lubrication approximation, which is applicable to analyses where the dimension of the foam in the direction of fluid flow (radial) is much greater than the dimension of the foam in the direction of loading (axial). The boundary value model is found to rapidly converge to the lubrication model in the limit of increasing aspect ratio given by the ratio of the radius $R$, to the height $h$, of the foam specimen with negligible error for aspect ratios greater than $R∕h∼4$.

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## Figures

Figure 1

One-regime model of fluid-filled cylindrical foam with strain less than the elastic buckling strain, ε<εel*. Velocity of fluid (solid arrow). Relative velocity of fluid with respect to the velocity of foam (dotted arrow).

Figure 2

(a) Bimodal regime model of fluid-filled cylindrical foam compressed beyond elastic buckling strain, εel*<ε<εd. Velocity of fluid. (solid arrow) (b). Top symmetric half of bimodal regime model of fluid-filled cylindrical foam compressed beyond elastic buckling strain, εel*<ε<εd, in the reference frame of the densified regime. Relative velocity of fluid with respect to the velocity of foam (dotted arrow).

Figure 3

The fraction of the flux into the linear-elastic regime (α) as a function of strain in the bimodal model

Figure 4

Stress-strain response of the 90ppi foam under a quasistatic load rate of ε̇=1×10−2s−1. (i) Strain corresponding to deviation from linear-elastic regime. (ii) Strain corresponding to peak stress before the plateau region.

Figure 5

Stress plotted against strain for 70ppi foam. Experimental data (◆). Contribution to the stress response of fluid model given by Eqs. 41,42 (---), the solid model given by Eqs. 1,3 (–⋅–), and the total model (––).

Figure 6

Stress plotted against strain rate for 70ppi foam at ε=0.60. Experimental data (◼), the contribution to the stress response of fluid model given by Eqs. 41,42 (—) and solid model given by Eqs. 1,3 (---).

Figure 7

(a) Stress plotted against strain rate for 70ppi foam. Experimental data at 0.60 strain (◼), and 0.30 strain (●), respectively. Model given by combining Eqs. 2,3 with Eq. 42 at ε=0.30 (—) and ε=0.60 (---). (b) Stress plotted against strain rate for 80ppi foam. Experimental data at 0.60 strain (◼), and 0.30 strain (●), respectively. Model given by combining Eqs. 2,3 with Eq. 42 at ε=0.30 (—) and ε=0.60 (---). (c) Stress plotted against strain rate for 90ppi foam. Experimental data at 0.60 strain (◼), and 0.30 strain (●), respectively. Model given by combining Eqs. 2,3 with Eq. 42 at ε=0.30 (—) and ε=0.60 (---).

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