Criteria for guaranteed breakdown in two-phase inhomogeneous bodies

TL;DR

This paper derives boundary-based lower bounds on maximum field strength in two-phase bodies to predict breakdown or nonlinearities, using analytical and numerical methods for various physical contexts.

Contribution

It introduces new boundary measurement bounds for breakdown prediction in two-phase materials, including optimal bounds and construction of special inclusion shapes.

Findings

Bounds are optimal for certain field configurations.

Numerical construction of $E_ ext{Omega}$-inclusions.

Applicable to conductivity and elasticity problems.

Abstract

Lower bounds are obtained on the maximum field strength in one or both phases in a body containing two-phases. These bounds only incorporate boundary data that can be obtained from measurements at the surface of the body, and thus may be useful for determining if breakdown has necessarily occurred in one of the phases, or that some other nonlinearities have occurred. It is assumed the response of the phases is linear up to the point of electric, dielectric, or elastic breakdown, or up to the point of the onset of nonlinearities. These bounds are calculated for conductivity, with one or two sets of boundary conditions, for complex conductivity (as appropriate at fixed frequency when the wavelength is much larger than the body, i.e., for quasistatics), and for two-dimensional elasticity. Sometimes the bounds are optimal when the field is constant in one of the phases, and using the algorithm of Kang, Kim, and Milton (2012) a wide variety of inclusion shapes having this property, for appropriately chosen bodies and appropriate boundary conditions, are numerically constructed. Such inclusions are known as $E_\Omega$-inclusions.