The complex material behavior associated with multiple pop-in phenomena has been treated with some degree of success. For pop-in length, the analysis considers an effective load reduction at the crack front due to plastic deformation. The effective stress intensity factor is decreased by the plastic deformation while it is increased by the length of pop-in, l. Due to these competing effects, the stress intensity, K, first increases and then decreases with increasing l. The point at which the effective K reduces back to the initial K defines l. For a second or third pop-in, the stress intensity increment, ΔK, between pop-ins is interpreted in terms of a changing fracture criterion (plane strain to plane stress) and the crack-tip blunting that occurs with increasing stress intensity. Although semi-quantitative in nature, these analyses do predict the experimentally observed increase in l and decrease in ΔK associated with increasing K. The limits of these models provide additional insight into fracture behavior. (a) As l → 0: A geometrical lower bound for pop-in behavior is hypothesized; (b) As ΔK → 0: The critical stress intensity factor at any particular thickness is given thus allowing establishment of the fracture-mode transition curve.
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