Geometrical-parameter Bounds on the Effective Moduli of Composites
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چکیده
We study bounds on the effective conductivity and elastic moduli of two-phase isotropic composites that depend on geometrical parameters that take into account up to three-point statistical information concerning the composite microstructure. We summarize existing bounds, apply a special fractional linear transformation to simplify their functional forms, and describe two approaches to improve such bounds. These approaches allow us to get new and improved geometrical-parameter bounds on the elastic moduli of two-dimensional composites. Applications of the bounds for effective-medium geometries as well as random arrays of aligned fibers in a matrix are discussed. 1. GEOMETRICAL PARAMETERS AND BOUNDS ON THE EFFECTIVE MODULI It is well known that effective properties of random two-phase composite materials generally depend upon an infinite set of correlation functions that statistically characterize the microstructure (see review by Torquato (1991) for references). An example of such a correlation function is the so-called n-point probability function S,, defined by the relation where Z(x) is the characteristic function of one of the phases, say phase 1, i.e. Z(x) = 1, if xEphase 1, 0, otherwise. (1.2) The angular brackets in (1.1) denote an ensemble average. For statistically homogeneous media and under the ergodic hypothesis, one can equate ensemble and volume averages. In particular, the one-point probability function S, is the probability of finding a point in phase 1, which is equal toyi, the volume fraction of phase 1, i.e. S, =f, = l-f2 = (Z(x)). (1.3) For statistically isotropic media, the quantity S,(v) is the probability of finding the
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تاریخ انتشار 1995