A new variational principle for cohesive fracture and elastoplasticity

Abstract

Variational methods for studying cohesive fracture and elastoplasticity have generally relied on minimizing an energy functional that is the sum of a stored elastic energy and a defect energy, corresponding to fracture or plasticity. The usual method for showing existence of minimizers is the Direct Method, whose success requires some properties of the defect energy that are not physically motivated, or in fact are contrary to physically desired properties. Here we introduce a new variational principle based on the idea of “necessity” of the defect, in the spirit of Garroni and Larsen (2009), reflecting the notion that these defects occur only if necessary in order for the elastic stress to be admissible, i.e., under the critical stress at which fracture or plasticity begins. The advantage is that the Direct Method only comes into play with a constraint on the defect set, which obviates some of the technical issues usually involved. The most significant advantage is that existence of global minimizers generally requires an infinite stress or strain threshold for plasticity or fracture, while our formulation is appropriate for finite thresholds. A further advantage is that the method produces local minimizers or locally stable states, rather than less physical global minimizers. General existence results will require new methods, but here we easily show existence in one dimension for both static and quasi-static solutions, even when global minima do not exist.