Percolation linear elasticity

Similarly, one can study the growth of the elastic constants (say the rigidity modulus) of a randomly formed elastic network, near the percolation point. The central force elastic problem (for networks formed out of linear springs only) belongs however to a different class of percolation problem, known as elastic percolation or central force percolation, and is discussed separately later (see Section 1.2.1(f)). 

In an earlier section, we have discussed the statistics of clusters of defects, produced for example in the random percolation processes. We have also discussed there how some of the linear responses, like the elastic moduli or... 

In the discrete lattice model, discussed above, each bond is identical, having identical threshold values for its failure. In the laboratory simulation experiments (discussed in the previous section) on metal foils to model such systems, holes of fixed size are punched on lattice sites and the bonds between these hole sites are cut randomly. If, however, the holes are punched at arbitrary points (unlike at the lattice sites as discussed before), one gets a Swiss-cheese model of continuum percolation. For linear responses like the elastic modulus Y or the conductivity E of such continuum disordered systems, there are considerable differences (Halperin et al 1985) and the corresponding exponent values for continuum percolation are higher compared to those of discrete lattice systems (see Section 1.2.1 (g)). We discuss here the corresponding difference (Chakrabarti et al 1988) for the fracture exponent Tf. It is seen that the fracture exponent Tf for continuum percolation is considerably higher than that Tf for lattice percolation Tf = Tf 4- (1 -h x)/2, where x = 3/2 and 5/2 in d = 2 and 3 respectively. 

Dividing the percolation system into cubes with linear dimension /, the infinite cluster system macroscopic elasticity can be defined as... 

Numerical studies [167] of planar (d = 2) elastic random percolation networks have shown that if their linear dimension L < 0.2c (c is the correlation length), then Poisson s ratio for the system is negative, and if L > 0.2J , Poisson s ratio is positive In this case, if L/c —> oo, the limiting value of Poisson s ratio is vp = 0.08 0.04 and is a universal constant that is, it does not depend on the relative values of the local elastic characteristics If... 

It is important to note that Eq. (2.7) should be used only in the vicinity of the hard phase percolation. Indeed, the meaning of the percolation exponent, 6, is to eiccomit for the fact that the percolated pathways are not linear but have some complicated morphology thus, the efficiency of the reinforcement ( fraction of elastically active hard phase elements ) is much less than 100%. 

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