Recovery of inclusion separations in strongly heterogeneous composites from effective property measurements

Abstract

An effective property of a composite material consisting of inclusions within a host matrix depends on the geometry and connectedness of the inclusions. This dependence may be quite strong if the constituents have highly contrasting properties. Here, we consider the inverse problem of using effective property data to obtain information on the geometry of the microstructure. While previous work has been devoted to recovering the volume fractions of the constituents, our focus is on their connectedness—a key feature in critical behaviour and phase transitions. We solve exactly a reduced inverse spectral problem by bounding the volume fraction of the constituents, an inclusion separation parameter and the spectral gap of a self-adjoint operator that depends on the geometry of the composite. We present a new algorithm based on the Möbius transformation structure of the forward bounds whose output is a set of algebraic curves in parameter space bounding regions of admissible parameter values. These results advance the development of techniques for characterizing the microstructure of composite materials. As an example, we obtain inverse bounds on the volume fraction and separation of the brine inclusions in sea ice from measurements of its effective complex permittivity.