Stable Phase of Ductile Fracture in Two and Three Dimensions, Final Stretch Model

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Abstract
Analysis given here is based on the final stretch concept employed in conjunction with a line plasticity model as suggested by the author in 1972 [6] and in this Journal in 1974 [8]. Its objective is to provide the equations governing quasi-static extension of a tensile crack contained in either partially or in a fully yielded specimen. Differential equations defining the apparent material resistance developed during the early stages of a ductile fracture process are derived from a requirement that the “essential work of fracture” dissipated in a small volume immediately ahead of the crack front, or equivalently, the “final stretch,” remains invariant in the process of ductile tear. The model suggests a certain near-tip distribution of displacements associated with a quasi-static Mode I crack such that the resulting strains are logarithmically singular at the crack tip. In contrast to the earlier work on this subject, here we impose no restrictions on the amount of plasticity which precedes the onset of crack growth and which accompanies spread of stable ductile fracture up to the point of global failure. The final results, which are illustrated in the diagrams of J-resistance curves, are analogous to the data obtained by other researchers on the basis of the incremental plasticity theory. Similarities between the present results and the solutions due to Hutchinson, Paris, and coworkers as well as the most recent data obtained by Shih and coworkers, are pointed out.

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