A discrete nonlocal damage mechanics approach

In this paper the authors develop the governing equations for a finite element model of micro-cracking based on a novel approach, eschewing differential equations and continuum mechanics. Instead of first stating the continuum balance laws and constitutive relations followed by discretization, the body is discretized first and then the equations of equilibrium are directly stated for the discretized body. It is shown that, as a result, the balance laws and constitutive relations can be entirely stated in terms of edge forces and lengths rather than strains. Furthermore, by ensuring that microcracks always propagate along the dual mesh which represents the possible fracture microplanes in the element, the need for creating additional nodes, gap elements or cohesive zones is avoided. Finally, the notion of the survival probability of a fracture microplane is introduced and the transition probability evolution is described by using probabilistic notions from population models. Thus the resulting governing equations can be solved by a conventional elastic predictor, followed by a nonlocal fracture corrector, making this convenient to augment conventional elements with fracture abilities.