9. Two-Dimensional Discrete Damage Models: Discrete Element Methods, Particle Models, and Fractal Theories

Discrete element methods (DEM) reviewed in this essay are limited to discontinuous models comprised of two-dimensional (2D) basic constitutive units such as circles, ellipses, or polygons given geometrical, structural, and contact properties that allow their assemblies to approximate phenomenological response of abstracted materials. The contacts are endowed with energy dissipation mechanisms and cohesive strength, which enables representation of damage-evolution phenomena such as crack nucleation and growth. By the way of dynamic interactions among particles, DEM are capable of coping with the complexity of fracture events in a simple and natural manner. On the other hand, the particle models are, in principle, offshoots of molecular dynamics (MD) adapted to simulation of materials at coarser scales. The role of atom is taken over by a continuum particle or quasi-particle, a basic constitutive unit that can represent, for example, a grain of ceramics, a concrete aggregate, and a particle of a composite. The computation domain is discretized into regular or random network of such particles generally interacting through nonlinear potentials within the realm of Newtonian dynamics. The model parameters should be identifiable with the macroscopic elastic, inelastic, and fracture properties of the material they aspire to represent, and the model should be structured in accordance with its morphology. Finally, a succinct survey of the percolation theory and fractal scaling laws of damage in discrete models is offered.