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 $$\int_{F(t)}\psi(v(x, t)) {\rm d}{\mathcal {H}^{N - 2}}(x)$$ 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.