两相体的电和弹性响应的普遍界限及其对边界测量的体积分数界限的应用
Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements
摘要：
Nemat-Nasser和Hori已经获得了两相（和多相）椭球体或平行六面体的电和弹性响应的通用界限。在这里，我们将展示如何改进它们的边界，并将其扩展到任意形状的物体。虽然我们的分析是两相体各向同性相，它可以很容易地扩展到多相体各向异性成分。我们的两相边界可以用在一个相反的方式绑定的相所占的体积分数，并为电导率减少到Capdeboscq和Vogelius的体积分数时，是渐近小。其他体积分数的界限，这里利用从热，磁，介电或弹性响应获得的信息。体积分数的一个界限可以通过简单地将主体浸入在一端具有活塞的充水圆柱体中并测量当活塞位移已知的小量时水压的变化来获得。该界限对于估计身体中的空腔的体积可能特别有效。我们还获得了新的边界，利用只有一对（电压，通量）的电测量在身体的边界。
英文摘要：
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.
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