Reduced model for fracture of geometrically exact planar beam: Non‐local variational formulation, ED‐FEM approximation and operator split solution

Abstract

SummaryIn this work, we present a novel reduced model for computing the solution to a bending fracture problem of the geometrically exact beam. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model. The non‐local variational formulation is proposed to deal with the softening phenomena characteristic of fracture. The fracture energy is introduced as the main fracture parameter, rather than (in general) ill‐defined characteristic length used for previous non‐local models. The discrete approximation is built in terms of the embedded‐discontinuity finite element method (ED‐FEM), which needs no initial crack for optimal performance nor detailed crack kinematics description, such as in the X‐FEM or the phase‐field approach. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best‐approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and stress) from local (internal) variables including the crack opening. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.