Stability Theory of Cohesive Crack Model

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Abstract
A mathematical formulation is proposed to describe equilibrium and stability behavior of the cohesive‐crack model (mode 1 cracking). Under the condition that no unloading occurs, total potential energy of the system can be defined, including energy dissipation in the process zone. Displacement and crack propagation can be formally solved from the potential energy. The properties of stable crack propagation are discussed using second‐order potential‐energy variation. The slope of load‐deflection curve is determined as a function of the second‐order variations of potential energy. Particularly, the second‐order variation of the potential energy with respect to displacement is found to be critical at incipient crack instability, a feature very useful in calculating critical quantities such as peak load. Based on the proposed stability theory, an analytic model is derived for three‐point beams to demonstrate the validity and usefulness of the formulation. The predictions of the analytical model compared excellently with numerical‐simulation results. Comparison with experimental data is not attempted, since the cohesive crack model was already validated experimentally by Hillerborg (1976) and Hillerborg et al. (1983) and many other authors.

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