Archive for Rational Mechanics and Analysis

, Volume 193, Issue 3, pp 539–583

# Fracture Paths from Front Kinetics: Relaxation and Rate Independence

Article

## 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.

## Keywords

Crack Front Fracture Path Rate Problem Dissipation Potential Front Speed

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