Rigidity percolation model of polymer fracture

A theory of the fracture of polymers with network microstructure was developed that was based on the vector, or rigidity percolation (RP) model of Kantor and Webman, in which the modulus, E, is related to the lattice bond fraction p, via E similar to [p - p(c)](tau). The Hamiltonian for the lattice was replaced by the strain energy density function of the bulk polymer, U = sigma(2)/2E, where sigma is the applied stress and p was expressed in terms of the lattice perfection via the bond density nu, with the entanglement molecular weight, nu = rho/M-e and appropriate measures of crosslink density for rubber, thermosets, and carbon nanotubes. The stored mechanical energy, U, was released by the random fracture of nuD(o)[p - p(c)] over stressed hot bonds of energy D-o approximate to 330 kJ/mol. The polymer fractured critically when p approached the percolation threshold p(c), and the net solution was obtained as sigma = (2EnuD(o) [p - p(c)])(1/2) with a fracture energy, G(1c) similar to [p - p(c)]. The fracture strength of amorphous and semicrystalline polymers in the bulk was well described by, sigma = [ED(o)rho/16 M-c](1/2), or sigma approximate to 4.6 GPa/M-e(1/2). Fracture by disentanglement was found to occur in a finite molecular weight range, M-c < M < M-*, where M-*/M-c approximate to 8, such that the critical draw ratio, lambda(c) = (M/M-c)(1/2), gave the molecular weight dependence of the fracture as G(1c) similar to [(M/M-c)(1/2) - 1](2). The critical entanglement molecular weight, M-c, is related to the percolation threshold, p(c), via M-c = M-e/(1 - p(c)). Fracture by bond rupture was in accord with Flory's suggestion, G/G(*) = [1 - M-c/M], where G(*) is the maximum fracture energy. Fracture of an ideal rubber with p = 1 was determined not to occur without strain hardening at lambda > 4, such that the maximum stress, sigma = E (lambda - 1/lambda) = 3.75E. The fracture proper-ties of rubber were found to behave as sigma similar to nu, sigma similar to E, and G(1c) similar to nu. For highly crosslinked thermosets, it was predicted that sigma similar to (Enu)(1/2), sigma similar to (X - X-c)(1/2), and G(1c) similar to nu(1/2), where X is the degree of reaction of the crosslinking groups and X-c defines the gelation point. When applied to carbon nanotubes (SWNT and MWNT) of diameter d and hexagonal bond density nu = j/b(2), the nominal stress as a function of diameter is sigma(d) = [16 EDo(p - p(c)) j/b](1/2)/d approximate to 211/d (GPa.nm) and the critical force, F-c(d) approximate to 166 d (nN/nm), in which j = 1.15, b = 0.142 nm, E approximate to 1 Tpa, and D-o 518 kJ/mol. For polymer interfaces with 1 chains per unit area of length L and width X similar to L-1/2, G(1c) is then similar to [p - p(c)], where p similar to SigmaL/X. The results predicted by the RP fracture model were in good agreement with a considerable body of fracture data for linear polymers, rubbers, thermosets, and carbon nanotubes. (C) 2004 Wiley Periodicals, Inc.