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

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 wellknown drained (no fluid) mechanical properties. Cellular solids differ from porous solids based on their relative density دپ*<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 finiteelement model, which explicitly considers fluidstructure interactions, structural deformations, and fluidpressure 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 addedmass 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≤دپ*≤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 دپ*→1. For frequencies below the first resonance of the lattice walls, only the fastpressure 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.