Improving the evaluation of static bifurcations in locally damaged parabolic arches

Even local small cracks may induce instability and failure or operativity loss before the crisis for the intact material is actually attained. Thus, damage models and the possibility of their identification have become trendy subjects in structural mechanics. We previously studied buckling and post-buckling of parabolic arches with a local transverse small crack by two perturbations of the finite field equations and boundary conditions. One perturbation describes non-trivial fundamental paths adjacent to the undamaged initial shape, the second describes the germ of a bifurcated path in the vicinity of a critical point. We think the arch consisiting of two undamaged parts, connected by lumped elasticities quantified by notions of linear fracture mechanics, at the cracked cross-section. In this paper we highlight that in a previous investigation of ours on the same subject some physically meaningful terms were wrong in the perturbed equations. This implies that the values of the incremental external load considered in the applications of that investigation need to be amended. Thus, here we present perturbed equations that are complete and, by the same procedure, we find more reliable numerical results for the critical loads and a more refined description of the germ of the post-buckling path, besides providing clear physical interpretation for seemingly paradoxical results.