A self-consistent approximation for crack bridging by elastic/perfectly plastic ligaments

摘要

In several contexts where the effectiveness of crack-bridging reinforcements degrades when the static or cyclic stress transmitted by the reinforcements exceeds a threshold value, the reinforcements can be idealized as elastic /perfectly plastic springs with the yield stress corresponding to the threshold stress for degradation. A self-consistent procedure is proposed for estimating the net crack tip stress intensity factor k_(tip) for a given applied stress, crack length and length of unbridged zone. Plausible analytical representations are postulated for the crack profile over the intervals where the crack-bridging springs respond elastically. The representations involve k_(tip) as a scaling factor, and lead to implicit formulae for k_(tip). When the applied stress exceeds the threshold for degradation, further simplifications yield explicit formulae for k_(tip). The best results obtained from these formulae agree closely over much of the parameter space with those obtained more laboriously by solving a non-linear integral equation. The approximations can be used to assess the influence of cyclic debonding on the effectiveness of externally bonded repairs, or of fiber reinforcement in metal laminates and metal-matrix composites. One of the approximations is an upper bound for k_(tip) and can therefore offer conservative estimates for the effectiveness of the bonded repair or fiber reinforcement.