The fuse problem

This is a difficult question which has not yet received a complete answer. But we can discuss qualitatively the failure process. We consider the following experiment. We increase slowly the applied voltage until a local failure takes place. At this point we stop increasing the voltage, and one can have different possibilities. We shall mention only two of them, (i) After the first failure, the whole sample fails by a succession of local failures until complete failure. This is a cascade effect, (ii) After the first failure, nothing happens. One has to increase the voltage for the appearance of new local failures until complete failure. 

By analogy with the mechanical fracture (discussed in the next chapter), the case (i) corresponds to a brittle failure and the case (ii) to a ductile failure. Although there is no definite answer, it is believed that in the case of percolation disorder, the electrical failures are mostly brittle-like failures the voltage (or the current) for the first failure is often the voltage (or the current) for the failure of the whole sample, especially for disorder concentrations near the percolation threshold. We shall see later a different type of disorder which can give ductile failure. 

We close this introduction with a final remark about the modelling of the failure. In a real situation, failure takes place in solid samples which are, by nature, continuous in space. However, many studies (numerical and experimental) have been made on lattices. In all these studies, it is an implicit assumption that one can replace a continuous solid by a lattice. For example, a conducting solid can be described by a lattice in which the bonds between sites are identical resistors. It is a very common practice in percolation type models of disordered solids. We stress that this transformation (continuous solid to lattice) defines a particular length scale the length of the unit cell of the lattice. This implies that defects appear by discrete steps and this does not correspond always to real situations. We shall see later how to remove this limitation. 

If this transformation (continuous solid to lattice) does not present a particular problem in electricity, the elastic case is more subtle and needs to be approached with some caution. We shall return later to this point. 

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In the present work, we shall not discuss the exact nature of the failure, i.e. its microscopic mechanism. In the fuse problem, the mechanism of the failure is very well-known (it is merely the Joule effect), but in the dielectric problem the mechanism is much more complicated (O Dwyer 1973). The reason is that we intend to attack the problem from a point of view which is of tremendous importance for statistical analysis. If the sample is perfectly homogeneous the failure will take place in all the portions of the sample. In the first case the current density is uniform in the sample and in the second, the electric field is the same everywhere. If the threshold value is reached, the failure will be general and the sample will explode. In fact, this never happens. The failure always begins as a local event and progressively becomes general. This is because there are weak points in the system. The failure always begins at these weak points. The existence of weak points is due to the fact that solids are never homogeneous. This means that the... 

Fig. 2.2. Failure paths, (a) In the dielectric problem, the failure path gives the possibility for a current to flow from one electrode to the other, (b) In the fuse problem, the failure path is made of insulating elements and it prevents the current across the sample.

One can ask, what exactly is Vb or If First, we have to imagine how the sample looks after the failure. In the dielectric problem, after the failure has taken place, there is a conducting path composed of conducting portions which connects the two electrodes (Fig. 2.2a). In the fuse problem, after the failure, the current cannot flow since there is now an insulating path more or less perpendicular to the current direction (Fig. 2.2b). It is also possible that other parts of the sample, which do not belong to the failure... 

For the fuse problems with various other criteria of failure, for example breakdown due to local Joule heating in a random thermal fuse model, see Sornette (1987) and Sornette and Vanneste (1992) and also Section 2.2.6. 

Other kinds of disorder distribution of the failure threshold Until now, we supposed that the disorder was of the random percolation kind. We describe here another kind of disorder in the case of the resistor lattice. Only the two-dimensional case has been studied. The disorder comes from the fact that the resistors do not have the same failure current or failure voltage. Although in the fuse problem the current is the relevant quantity, Kahng et al (1988) developed this model considering only a distribution of threshold voltage. 

As in the fuse problem, we begin with the explanation of how the presence of defects can increase the local breakdown field. A defect is a local change in the properties of the sample. In an insulator, the defects are conducting parts of the sample. We consider again a spherical defect (circular in two dimensions) and we draw the equipotential surfaces or lines (in two dimensions). In a pure sample, these surfaces or lines are parallel to the electrodes (Fig. 2.12a) but in a sample with one defect they show distortions near it. For a two-dimensional sample, the new equipotential lines are shown in Fig. 2.12(b). One sees that in the vicinity of the defect there is an increase of the field. The sample will break at an applied voltage smaller than the one which is needed to break a pure sample. This is the enhancement effect identical to that of the fuse problem and consequently the curve Vb(p) will exhibit an infinite slope when p goes to zero. 

The dielectric breakdown problem can be solved very easily from the solution of the fuse problem in two dimensions using the concept of duality. This concept is largely used in the case of composite materials and in percolation for problems in d = 2 or with cylindrical symmetry (Mendelson 1975, Bowman and Stroud 1989). Here we follow the derivation of Bowman and Stroud. 

Fig. 2.13. Comparison of constant current lines in the fuse problem and the equipotential lines in the dielectric problem in two dimensions. The two figures are identical but rotated by 90°.

In the present case, one introduces conducting defects in the form of circles in two dimensions or spheres in three dimensions with randomly positioned centres having the possibility to overlap. The size of one defect is the characteristic length of the problem, as mentioned in the case of the fuse problem. 

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