Quasi-static Fracture Evolution with Cohesive Energy

Li, Yiqing

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Quasi-static Fracture Evolution with Cohesive Energy

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The last fifteen years have seen much success in the analysis of quasi-static evolution for Griffith fracture, which is the mathematically natural starting point for studying fracture. At the same time, attempts have been made to show existence for similar models based on cohesive fracture rather than Griffith. These models are generally viewed as physically more realistic than Griffith, in that they are better models for crack nucleation. These attempts at existence proofs have been unsuccessful without very strong additional assumptions, for example, specifying the crack path a priori. The main purpose of this thesis is to characterize as well as possible the mathematical difficulties in cohesive fracture, and to make progress toward an existence result without the prescribed crack path assumption. So far, the most powerful method for existence proofs is to build a sequence of approximate solutions, based on time discretization, and take the limit as the time steps go to zero. We show that there are mainly two complications on the cracks of these approximate solutions that we need to rule out in order to show existence. The first one is due to the potential oscillation of the crack path. The second is due to the potential splitting of a crack into two or more nearby cracks, with the same total jump in displacement. We begin by first constructing an example illustrating how oscillations described above can affect the minimality of the limit. Then we prove that the splitting described above can be ruled out for any sequence of unilateral minimizers. With this result, we show how exactly oscillation affect the minimality on the limit of the sequence. We then move to the evolution problem and show the convergence of energy for almost every t. Based on this result we develop a method that allows us to analyze the problem using only a finite set of times. An application of this method is a proof of absolute continuity. Future work will be aimed at using the tools we developed to rule out oscillation and finally to prove existence results under more general assumptions.

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