Fracture Paths from Front Kinetics: Relaxation and Rate Independence

• Published in: Archive for rational mechanics and analysis Vol. 193; no. 3; pp. 539 - 583
• Main Authors: Larsen, C. J., Ortiz, M., Richardson, C. L.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2009; Springer
• Subjects: Classical Mechanics; Complex Systems; Exact sciences and technology; Fluid- and Aerodynamics; Fracture mechanics (crack, fatigue, damage...); Fundamental areas of phenomenology (including applications); Mathematical and Computational Physics; Physics; Physics and Astronomy; Solid mechanics; Structural and continuum mechanics; Theoretical; Fracture Path; Crack Front; Dissipation Potential; Front Speed; Rate Problem; Static model; Energy dissipation; Energy function; Rupture; Quasi static theory; Kinetics; Trajectory; Modeling; Crack; Crack propagation
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-009-0216-y

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.