Fracture Paths from Front Kinetics: Relaxation and Rate Independence |
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Authors: | C J Larsen M Ortiz C L Richardson |
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Institution: | (1) Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA;(2) Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA |
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Abstract: | 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
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. |
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