A one-dimensional variational formulation for quasibrittle fracture

Besides efficient techniques allowing for the finite-element modeling of propagating displacement discontinuities, the numerical simulation of fracture processes in quasibrittle materials requires the definition of criteria for crack initiation and propagation.Among several alternatives proposed in the literature, the possibility to characterize energetically the discontinuous solution has recently attracted special interest.In this work, the initiation and propagation of cohesive cracks in an inhomogeneous elastic bar, subject to an axial body force is considered.The incremental finite-step problem for the evolving discontinuity is formulated accounting for progressive damage in the cohesive interface.For assigned loading conditions, it is shown that the equilibrium of the system and the position where the crack actually forms can be obtained from the minimality conditions of an energy functional including the bulk elastic energy and the crack surface energy.The subsequent step-by-step propagation of the cohesive crack is also obtained from the minimality conditions of an energy functional defined for each step.The issue of the algorithmic selection of the energetically more convenient solution is briefly discussed.