Quantitative analysis most probable failure current and distibution

2 Quantitative analysis most probable failure current and distibution (a) Lattice percolation 

A quantitative analysis of the failure process was made by Duxbury et al (1987) by modelling the system by a lattice and we shall present their results. The simplest lattices were taken a square lattice in two dimensions and a simple cubic lattice in three dimensions, in which the bonds are all equal resistors to begin with. Each resistor can stand a current up to io. If i io, the resistor is fused and becomes a perfect insulator. It is believed that the results are not dependent on the type of the lattice as it was proven in the case of percolation. The size of the lattice is L in two dimensions it 

Dilute limit the most probable failure current The smallest defect which can be made is to remove one resistor. If it is one of the resistors perpendicular to the current flow (horizontal resistors), nothing happens since in the perfect lattice only resistors parallel to the current flow carry current. The simplest defect is the absence of one of the parallel resistors (vertical resistors) in the sample. For the time being, we suppose that the new defect is far from the boundaries of the sample. The problem of defects near the boundaries was investigated by Li and Duxbury (1987). 

The enhancement coefficient was calculated in two dimensions, and it is equal to 4/tt. This means that for p — 1, if is decreased by the factor 7t/4. For a perfect sample in d = 2 

We shall now determine what is the probability of occurrence of a dangerous defect as defined in the preceding section. In the present case of a lattice, it consists in n neighbouring resistors belonging to the same plane perpendicular to the current flow. In d = 2, it is merely a line of n removed bonds and in d = 3, it is an ensemble of n removed bonds forming a hole with approximately the shape of a disc. The current in the immediately adjacent parallel resistor will be in accordance with the enhancement factor in an elliptic defect 

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