期刊
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
卷 193, 期 3, 页码 539-583出版社
SPRINGER
DOI: 10.1007/s00205-009-0216-y
关键词
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资金
- National Science Foundation [DMS-0505660]
- Department of Energy through Caltech's ASCI ASAP Center for the Simulation of the Dynamic Response of Materials
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 integral(F(t)) psi(nu(x, t))dH(N-2)(x) where F(t) is the front at time t and nu 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.
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