The application of fractal and quantum geometry to brittle fracture

This paper examines three apparently disparate views of fracture in brittle materials with the purpose of showing the interrelationship of the fractur e process at different length scales. Quantitative fractographic analysis o f brittle fracture surfaces shows that there are characteristic markings on the surfaces that are self-similar and scale invariant, implying that frac tal analysis is a reasonable approach to analyzing these surfaces. The frac tal dimension is directly proportional to the fracture energy, gamma, durin g fracture for many brittle materials, i.e., gamma = 1/2Ea(0)D* where E is the elastic modulus, a(0) a structural parameter and D* is the fractal dime nsional increment. Analysis of previous results of molecular dynamics model ing shows that the fractal nature of the fracture surface is consistent wit h the predicted surface produced during simulated fracture. The fractal dim ensional increment, D*, of the simulated fracture surface in a silica glass and silicon single crystal over a single order of magnitude matched well w ith those values measured on fracture surfaces of beams fractured in flexur e for the same materials at length scales 1000-100000 times larger. Finally , a ring contraction mechanism, modeled using semi-empirical quantum mechan ical molecular orbital methods, is shown to be a likely step in the fractur e of silica tetrahedra along the crack front. The geometry of the structure formed at the atomic scale from these ring contractions can be related to a(0) and leads to the basis for propagation into a fractal structure. Thus, we have identified a reasonable atomic level foundation for the structural parameter, a(0). Based on a comparison of atomic and molecular modeling wi th macroscopically measured values of D* and a(0), are suggest that the fra cture process is a percolation of a series of ring contractions along the c rack front, which result in the observed fracture surfaces for several brit tle materials. (C) 1999 Elsevier Science B.V. All rights reserved.