Singly connected bonds

Fig. 11. Filler network chain according the L-N-B-model i.e., a set of N singly connected bonds under an applied force at the two ends of the chain (after [90])...

The density distribution function of the number of singly connected bonds,... 

It is interesting to note that the (L-N-B)-model leads to similar expressions for the moduli like the VTG-model apart from the first summand of Eq. (38). However, contrary to the semi-empirical weighting functions W(y6) of the VTG-model, the corresponding density distribution function/la(y) in the (L-N-B)-model is related to the morphological structure of the filler network, i.e., the distribution of singly connected bonds in a percolation network. Unfortunately, this distribution function is not known, exactly. Therefore, a simple exponential... 

The above described lack of smoothness at y = ya is essential. It refers to the characteristic power law distribution functions of cluster sizes in percolation, indicating that the most frequent number Lx of singly connected bonds is unity. This leads to a spontaneous fast decline of G when y exceeds the value yapp> since all L-N-B-chains with Lx=1 break simultaneously at this amplitude. Experimental results show that a smooth transition of G with varying strain amplitude appears that cannot be described by a power law distribution function or the assumed exponential type of/lfl (y). 

Fig. 1.2. Portion of a random bond percolating cluster backbone, connecting the points A and B. Here, the thick black lines represent the singly connected bonds or red bonds which, if cut, will disconnect the connection between A and B. The bonds in the blob portions are indicated by dotted lines. The dangling bonds are indicated by thin black lines (cf. StauflPer and Aharony 1992).

In conclusion, the two criteria, namely that the current in a link or that the temperature of a singly-connected bond reaches a specific value at the failure, give the same results. It was in fact expected, since the two quantities ATgc and ii, are related by the relation (2.43). We shall see below the experimental results giving support for the second approach and making the fuse model a realistic one. 

In eqn (2.49), the currents that will give the main contribution are those through the singly-connected bonds, and with this assumption B becomes... 

These expressions help us to understand why B diverges at pc- Besides the terms p — Pc and 6Tgc also diverges at pc- This is because, as one approaches pc, the number of Unks decreases and therefore the current through a link increases, giving rise to the increase in the temperature in the singly-connected bonds. We now define a new exponent for the variations of D ... 

In fact, this is a lower bound for the exponent x since we included all the singly-connected bonds in a link. An upper bound is obtained if one takes equal to one, when only one of the singly-connected bonds... 

If all the strain energy is supported by the number of singly connected bonds only (see Section 1.2.1), then one gets an overestimate of the elastic energy per bond, which comes out to of the order of El/Lc 2 (d+Te/i/-i/i/) When one equates this energy with the threshold energy of the bond, one gets the lower bound ... 

For d > 6, the links of the super-lattice network are indeed made up of singly connected bonds (see Section 1.2.1), and therefore both the bounds become equal and equalities in (3.14) become exact. The above bounds are valid in general for any elastic network, provided the exponents like Te, I/, de, etc. are appropriate for the network under consideration. For... 

In fact, if we denote by cjy (< af) the stress at which the irreversible plastic deformation occurs and if cry (Ap) y, then one can estimate Ty in the lattice cases, using the following picture due to Bergman (1986). As one approaches pc, the singly connected bonds in the links are strained more... 

Figure 3 Schematic sketch of a two-dimensional percolation cluster structural elements are single connecting bonds, dead (or dangling) ends, and loops (blobs) on all scales (this is only an illustration and not a real generated cluster)...

The cluster is dominated by the immense fluctuations in connectivity and density and it can be shown, by simulations, that many structural elements like single connecting bonds, dangling ends, loops and blobs are critical quantities, i.e, their number diverges with a certain exponent at the thres-hold. " This is another indication that one cannot expect validity of the mean field exponents as the Flory-Stockmayer theory calculates. 

---