Cusp catastrophe interpretation of fracture instability

Abstract

A cohesive crack model is applied to analyse slow crack growth in elastic-softening materials. The shape of the structural load-displacement response is changed substantially by varying the size-scale while keeping the geometrical shape of the structure unchanged. The softening branch becomes steeper when the sizescale increases. A critical size-scale exists for which the softening slope is infinite. In such a case the load carrying capacity drastically decreases for relatively small displacement increments. Then, for size-scales larger than the critical one, the softening slope becomes positive and part of the load-displacement path becomes virtual if the loading process is displacement-controlled. In such a case, the loading capacity will present a discontinuity with a negative jump. The size-scale transition from ductile to brittle behaviour is governed by a nondimensional brittleness number S E which is a function of material properties and structure size-scale. A truly brittle failure occurs only with relatively low fracture toughnesses G ic, high tensile strengths σ u, and/or large structure size-scales b, i.e. when S E = G ic σ ub → 0. On the other hand, if the loading process is controlled by a monotonically increasing function of time (e.g. the crack mouth opening displacement), the snap-back instability in the load-displacement curve can be captured experimentally. When the post-peak behaviour is kept under control up to the complete structure separation, the area delimited by the load-displacement curve and the displacement-axis represents the product of G ic and the initial ligament area. Finally, it is verified that. for S E → 0, the maximum load for catastrophic failure is provided by the simple LEFM condition : K 1 = K IC = G icE (plane stress), and that there is no slow crack growth prior to instability.