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Published September 2009 | public
Journal Article

Fracture Paths from Front Kinetics: Relaxation and Rate Independence


Crack fronts play a fundamental role in engineering models for fracture: they are the location of both crack growth and the energy dissipation due to growth. However, there has not been a rigorous mathematical definition of crack front, nor rigorous mathematical analysis predicting fracture paths using these fronts as the location of growth and dissipation. Here, we give a natural weak definition of crack front and front speed, and consider models of crack growth in which the energy dissipation is a function of the front speed, that is, the dissipation rate at time t is of the form $$\int_{F(t)}\psi(v(x, t)) {\rm d}{\mathcal {H}^{N - 2}}(x)$$ where F(t) is the front at time t and v is the front speed. We show how this dissipation can be used within existing models of quasi-static fracture, as well as in the new dissipation functionals of Mielke–Ortiz. An example of a constrained problem for which there is existence is shown, but in general, if there are no constraints or other energy penalties, this dissipation must be relaxed. We prove a general relaxation formula that gives the surprising result that the effective dissipation is always rate-independent.

Additional Information

© 2009 Springer. Received: 8 November 2007. Accepted: 21 January 2008. Published online: 14 February 2009. This research began while CL was a visiting associate in mechanical engineering at Caltech. CL and CR were supported by the National Science Foundation under Grant No. DMS-0505660. MO gratefully acknowledges the support of the Department of Energy through Caltech's ASCI ASAP Center for the Simulation of the Dynamic Response of Materials, and the support received from NSF through an ITR grant on multiscale modeling and simulation and Caltech's Center for Integrative Multiscale Modeling and Simulation. Finally, CR wishes to thank G. Dal Maso for pointing out some helpful references in [2].

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