Definition of effective properties

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Definition of effective properties

The main use of the programs described in this manual are to compute the effective properties of a multi-phase random composite. The effective properties of a composite material are defined simply in terms of the averages of various quantities over the system. First of all, for electric cases we have an applied field, maintained via the periodic boundary conditions, and in elastic cases, we have an applied strain, also maintained by the periodic boundary conditions. Theorems from composite theory assure us that the electric field average and the strain average must equal the applied quantities. In other words, if indicates an average over the entire system,

then if, in the electrical case,

is the applied field, the average of the microscopic field, which varies from pixel to pixel, is

while in the elastic case, a similar statement holds for the strain, where the applied strain is E_a :

In addition, an average can be performed over a single phase, denoted by

where V_n is the volume of phase n, and c_n is the volume fraction of phase n. The first integral is taken only over the volume of phase n, while the second integral is over the total volume, with

for phase n, and zero elsewhere.

There are two equivalent ways to define the effective quantities: an energy method or an average current/stress method. In the energy method, the total energy per unit volume is equated to the energy per unit volume of a uniform medium. The equation used is:

in the elastic case, and

in the electric case. Various applied fields and strains can be used to pick off various components of the effective quantities.

Another way to define the effective moduli or conductivity is through a stress average or a current density average. For the electric case, the average current density is given by

which then defines the effective conductivity, as the average field is just the applied field. For elastic cases, the equivalent expression for the average stress is

which then defines the effective moduli tensor. The average stress and current formulations are used in the programs, as one run can determine more of the effective properties than in the energy case, where there is only one number, the average energy, produced. The finite element programs do output the energy, however, so that this method can be used. Good review articles of composite theory include Refs. [14,15]. For the rest of this manual, the superscript e ƒ ƒ is dropped, so that properties like conductivity and elastic tensors without a phase label indicate effective properties.

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