An energy based analysis of the pull-out problem

We are focusing on the fibre pull-out problem by considering a cylindric elastic reinforced beam characterised by the small ratio ε of its radius to its length, clamped at one end while at the other one the fibre is stretched. If one applies the fracture Griffith criterion with the 3D theory, the debonding process never starts, that indicates the well-known inability of Griffith theory to predicts the initiation of cracking in a sound structure. On the other hand, the Griffith energy approach works and accounts for a complete debonding process when the mechanical fields are suitably approximate as in the shear-lag model. We show here that this approximate solution is nothing but the global minimiser of the first order approximation of the total energy arising from the asymptotic beam theory in which the ratio ε is considered as a small parameter. So we follow (Francfort and Marigo, 1998) and replace Griffith criterion by the principle of least energy in the full 3D context. In particular, we show that, from a more accurate approximation of the elastic energy which takes into account the boundary layers effects, the initiation of the debonding corresponds to a crack, the length of which is of the order of ε, appearing brutally when the load reaches a critical value. Numerical values of parameters are obtained from computations using the finite element method.