Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements

TL;DR

This paper extends universal bounds on electrical and elastic responses of two-phase bodies to arbitrary shapes, enabling improved volume fraction estimation from boundary measurements, with applications to cavities and multiphase materials.

Contribution

It develops improved bounds for two-phase bodies of arbitrary shape and extends their application to multiphase and anisotropic materials, enhancing volume fraction estimation methods.

Findings

Bounds can be improved and extended to arbitrary shapes.

New methods for estimating volume fractions from boundary measurements.

Effective bounds for cavities using simple pressure measurements.

Abstract

Universal bounds on the electrical and elastic response of two-phase (and multiphase) ellipsoidal or parallelopipedic bodies have been obtained by Nemat-Nasser and Hori. Here we show how their bounds can be improved and extended to bodies of arbitrary shape. Although our analysis is for two-phase bodies with isotropic phases it can easily be extended to multiphase bodies with anisotropic constituents. Our two-phase bounds can be used in an inverse fashion to bound the volume fractions occupied by the phases, and for electrical conductivity reduce to those of Capdeboscq and Vogelius when the volume fraction is asymptotically small. Other volume fraction bounds derived here utilize information obtained from thermal, magnetic, dielectric or elastic responses. One bound on the volume fraction can be obtained by simply immersing the body in a water filled cylinder with a piston at one end and measuring the change in water pressure when the piston is displaced by a known small amount. This bound may be particularly effective for estimating the volume of cavities in a body. We also obtain new bounds utilizing just one pair of (voltage, flux) electrical measurements at the boundary of the body.