Part-through cracks computation in an Euler-Bernoulli beam model

The reduction of the equations of a slender, brittle continuum to a one-dimensional theory and the possibility of accounting for the cracking of this body has received much attention recently. This contribution investigates the effect of cross-sectional geometry on damage localization in an Euler-Bernoulli beam. Two geometric fields characterizing the crack penetration depth on each side of the beam are introduced, from which the modulation of the bending and tensile parts of the strain energy by the crack depth is deduced. The geometry of the crack-tip envelope is obtained by energy minimization. We find that in the case of a Griffith-like dissipation potential, the characteristic size of the crack is a key element for predicting initiation at a finite external load. We can accurately predict the initiation position by regularizing the model in the spirit of Variational Brittle Fracture. Model predictions are compared against computational results obtained by the classical damage field theory.