A numerical method for fracture of rods

We consider fracture problems for one-dimensional bodies, such as rods, through energy minimization. For an elastic-brittle body β occupying in the reference configuration the domain Ω, the energy depends on both some unknown closed crack set K and a displacement field u, smooth on Ω ∖ K. On Ω ∖ K a bulk energy density and on K an interface energy density are defined. As a result of this assumption the total potential energy is a nonconvex functional. We propose a new numerical method for the search of minimal states of rods by minimizing the energy both on K and u. To seek a minimizer, a nonlinear gradient iterative procedure is employed. During the process of minimization both the displacement field u and the set K evolve toward local minima. To illustrate our approach, some examples are reported.