Taylor expansions and bounds for the effective conductivity and the effective elastic moduli of multicomponent composites and polycrystals

We describe a construction, based on variational inequalities, which gives a hierarchy of upper and lower bounds (of odd orders 2k+1), on the various effective moduli of random multiphase materials and polycrystals. The bounds of order 2k+1 on a given effective modulus can be explicitly evaluated if a truncated Taylor expansion of the given modulus is known to order 2k+1. Our approach is motivated by prior investigations of Beran and other authors. Our calculations do not involve Green functions or n-point correlation functions, and they are very simple. We thus rederive known and obtain new sequences of bounds on the different effective moduli. We also describe a method that, for cell materials (i.e. materials in which cells of smaller and smaller length scales cover all space, with material properties assigned at random), permits one to calculate the truncated Taylor expansions that are needed for the explicit evaluation of the bounds. In connection with this, we show that the first coefficient in the low volume fraction expansion of any effective modulus of a cell material, coincides with the corresponding low volume fraction coefficient for an array of cells randomly distributed in a matrix.