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

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.Comment: 5 figures, 27 page

Bounds on the volume fraction of 2-phase, 2-dimensional elastic bodies and on (stress, strain) pairs in composites

Bounds are obtained on the volume fraction in a two-dimensional body containing two elastically isotropic materials with known bulk and shear moduli. These bounds use information about the average stress and strain fields, energy, determinant of the stress, and determinant of the displacement gradient, which can be determined from measurements of the traction and displacement at the boundary. The bounds are sharp if in each phase certain displacement field components are constant. The inequalities we obtain also directly give bounds on the possible (average stress, average strain) pairs in a two-phase, two-dimensional, periodic or statistically homogeneous compositeComment: 16 pages, 2 figures, Submitted to Comptes Rendus Mecaniqu

Structural information of composites from complex-valued measured bulk properties

This paper is concerned with the estimation of the volume fraction and the anisotropy of a two-component composite from measured bulk properties. An algorithm that takes into account that measurements have errors is developed. This algorithm is used to study data from experimental measurements with an unknown microstructure. The dependence on the microstructure is quantified in terms of a measure in the representation formula introduced by D. Bergman. We use composites with known microstructures to illustrate the dependence on the underlying measure and show how errors in the measurements affect the estimations of the structural parameters

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

In two phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and to simplify the analysis Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored.Comment: 36 pages, 7 figure