Effective medium theory of elastic waves in random networks of rods

TL;DR

This paper develops an effective medium theory for elastic waves in random rod networks, deriving wavelength-dependent elastic properties and wave dispersion relations, with implications for designing low-density, high-stiffness materials.

Contribution

It introduces a novel effective medium approach for elastic wave propagation in random rod networks, including wavelength-dependent moduli and dispersion relations, applicable to porous and composite materials.

Findings

Waves are dispersive with velocities decreasing at higher wave vectors.

Derived static elastic moduli and dispersion relations for ultrasonic waves.

Potential for creating low-density materials with high stiffness-to-weight ratios.

Abstract

We formulate an effective medium (mean field) theory of a material consisting of randomly distributed nodes connected by straight slender rods, hinged at the nodes. Defining novel wavelength-dependent effective elastic moduli, we calculate both the static moduli and the dispersion relations of ultrasonic longitudinal and transverse elastic waves. At finite wave vector $k$ the waves are dispersive, with phase and group velocities decreasing with increasing wave vector. These results are directly applicable to networks with empty pore space. They also describe the solid matrix in two-component (Biot) theories of fluid-filled porous media. We suggest the possibility of low density materials with higher ratios of stiffness and strength to density than those of foams, aerogels or trabecular bone.