Dynamics of fracture propagation in the mesoscale: Theory

Abstract

A recent theoretical model (Blumenfeld, Phys. Rev. Lett. 76 (1996) 3703) is described for modes I and III crack propagation dynamics in noncrystalline materials on mesoscopic lengthscales. Fracture has been one of the longest standing problems in physics and materials science, and despite much eort, several fundamental issues have stubbornly resisted resolution:(i) Running cracks reach a steady-state velocity of roughly half the shear wave speed, while theoretical predictions based on energetics are twice as high. The discrepancy originates from dynamics, but a consistent dynamical model has been slow to emerge.(ii) There is little understanding of the mechanisms for crack initiation and arrest and the hysteresis between them. Lattice trapping, although relevant on the atomic scale, cannot explain this phenomenon on mesoscopic and macroscopic scales.(iii) Another intriguing phenomenon is appearance of velocity periodic oscillations in some materials and the relation between this and material properties.(iv) As a result of the above issues, there is currently no consensus on the form of the equations of motion that govern mesoscale fracture dynamics.Whether explicitly or implicitly, most traditional models use quasi-static and near-equilibrium concepts to analyse the dynamics of propagation. It is argued here that such approaches are bound to fail. Two reasons are responsible for this and consequently for the dire understanding of this problem: First, most fast fracture processes are usually restricted to post-mortem measurements of the already fractured system, while the process itself is too fast to capture. Only recently there emerged experiments where the dynamic process is continuously monitored. Second, it is strongly contended here that the fracture phenomenon is governed by dierent mechanisms on dierent length-scales, a crucial aspect that has not received sucient attention. In ideally brittle propagation, the crack is atomically sharp and therefore atomic potentials are important (5±10 # A). Anharmonicity plays a signi®cant role on this scale due to large local strains at the crack tip, which gives rise to a strong nonlinear behaviour. On large scales (>lm), continuum linear elasticity describes quite well the stress ®eld and the far-away elastic energetics. This is exactly because cracks propagate slower than the bulk speed of sound, which allows the bulk stress to relax to its static value in the frame of the moving crack. Ultimately, this is the reason why contour integral calculations of energy in¯ux into the crack tip are valid as long as the contours are taken well away from the tip. Between the atomic and the continuous scales there are at least two more relevant length-scales: One is that of the cohesive zone, which is the region where the continuous stress ®eld description breaks down due to the discreteness of the lattice. It is of the order of several lattice constants and about one order of magnitude above the atomic scale ($10±50 # A). The fourth length-scale, and the one we focus o...

The elastic displacement and stress in a material to promote reflection and transmission of an incoming wave

“…Evidence for geometric arrangements affecting the speed, reflection and refraction of a pulse have been brought out in studies involving periodic and nonperiodic arrangements of particles embedded in a material (Geng et al 2003;Kinra et al 1998Kinra et al , 1999Krasil'nikov 1999) and in a study involving oscillation of the speed of a crack tip (Blumenfeld 1998).…”