Determining the volume fraction in 2-phase composites and bodies using time varying applied fields

TL;DR

This paper presents a method to determine the volume fractions of two phases in composites or bodies by using specially designed time-varying boundary conditions, enabling exact measurements from boundary data.

Contribution

It introduces a novel approach to extract phase volume fractions from boundary measurements using quasistatic, time-varying fields, independent of microstructure details.

Findings

Boundary measurements at a specific time yield phase volume fractions.

Time variations can be tailored to determine volume fractions at any time.

Boundary conditions can be designed to retrieve responses at specific frequencies.

Abstract

A body $\Theta$ containing two phases, which may form a periodic composite with microstructure much smaller that the body, or which may have structure on a length scale comparable to the body, is subjected to slowly time varying boundary conditions that would produce an approximate uniform field in $\Theta$ were it filled with homogeneous material. Here slowly time varying means that the wavelengths and attenuation lengths of waves at the frequencies associated with the time variation are much larger than the size of $\Theta$, so that we can make a quasistatic approximation. At least one of the two phase does not have an instantaneous response but rather depends on fields at prior times. The fields may be those associated with electricity, magnetism, fluid flow in porous media, or antiplane elasticity. We find, subject to these approximations, that the time variation of the boundary conditions can be designed so boundary measurements at a specific time $t=t_0$ exactly yield the volume fractions of the phases, independent of the detailed geometric configuration of the phases. Moreover, for specially tailored time variations, the volume fraction can be exactly determined frommeasurements at any time $t$, not just at the specific time $t=t_0$. We also show how time varying boundary conditions, not oscillating at the single frequency $\omega_0$, can be designed to exactly retrieve the response at $\omega_0$.