Elastic Properties of Depleted Networks and Continua

Abstract

Numerical simulations, effective medium theories and scaling arguments are used to examine the elastic properties of depleted networks and continua. The simplest model that embodies rotationally invariant forces is the central force only model. The numerical results for the central force model gives a universality class which is different from the conductivity problem on the same networks. The cental force threhold p_cen is much greater than the usual connectivty threshold p_c. Rotationally non invariant bond bending forces give the conductivity universality class, but a strong cross-over to the cenral force like behavior is observed near p_cen. Two dimensional bond percolation networks involving both central and rotationally invariant bond-bending forces were studied by numerical simulations and finite size scaling arguments. A critical exponent f (about 3.2), which is much higher than t (about 1.3), the conductivity exponent, is found. The effective exponent was found to depend on sample size L for small L. The scaling arguments based on the nodes-links-blobs picture can explain the sample size dependence of the effective exponent. Experimental data of Benguigui on elastic sheets with holes punched in them gives an elastic exponent which is in good agreement with simulations. The initial slope of the Young’s modulus vs. the fraction of holes was found to be in good ageement with the effective medium approximations (EMA). The EMA for elastic continua give classical exponents but predict that at p_c the ratio of the bulk and shear modulii approach a constant value which is independent of the modulii of the starting medium.