Archive for Rational Mechanics and Analysis

, Volume 193, Issue 3, pp 539–583 | Cite as

Fracture Paths from Front Kinetics: Relaxation and Rate Independence

  • C. J. Larsen
  • M. OrtizEmail author
  • C. L. Richardson


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.


Crack Front Fracture Path Rate Problem Dissipation Potential Front Speed 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio L.: A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B (7) 3(4), 857–881 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosio L., Fusco N., Pallara D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000)zbMATHGoogle Scholar
  3. 3.
    Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–179 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dal Maso G., Francfort G.A., Toader R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176(2), 165–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Evans L.C., Gariepy R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  6. 6.
    Francfort G.A., Larsen C.J.: Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56(10), 1465–1500 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Francfort G.A., Marigo J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mielke, A.: Evolution in rate-independent systems (ch. 6). Handbook of Differential Equations, Evolutionary Equations, vol. 2 (Eds. Dafermos C.M. and Feireisl E.). Elsevier B.V., Amsterdam, 461–559, 2005Google Scholar
  9. 9.
    Mielke, A., Ortiz, M.: A class of minimum principles for characterizing the trajectories of dissipative systems. ESAIM Control Optim. Calc. Var. (2007, in press)Google Scholar
  10. 10.
    Paris P., Erdogan F.: A critical analysis of crack propagation laws. Trans. ASME 85, 528–534 (1963)CrossRefGoogle Scholar
  11. 11.
    Rosakis A.J., Duffy J., Freund L.B.: The determination of dynamic fracture toughness of aisi 4340 steel by the shadow spot method. J. Mech. Phys. Solids 32(4), 443–460 (1984)ADSCrossRefGoogle Scholar
  12. 12.
    Yoffe E.H.: The moving griffith crack. Phil. Mag. 42, 739–750 (1951)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA
  2. 2.Graduate Aeronautical LaboratoriesCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations