Existence and convergence for quasi-static evolution in brittle fracture.

*(English)*Zbl 1068.74056Summary: This paper investigates the mathematical well-posedness of the variational model of quasi-static growth for a brittle crack proposed by G. A. Francfort and J.-J. Marigo [J. Mech. Phys. Solids 46, No. 8, 1319–1342 (1998; Zbl 0966.74060)]. The starting point is a time-discretized version of that evolution which results in a sequence of minimization problems of Mumford-Shah-type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of time-continuous, quasi-static growth is to pass to the limit as the time discretization step tends to \(0\). This is performed with the help of a jump transfer theorem that permits, under weak convergence assumptions on a sequence \(\{u_n\}\) of SBV functions to its BV limit \(u\), the transference of the part of the jump set of any test field that lies in the jump set of \(u\) onto that of the converging sequence \(\{u_n\}\). In particular, it is shown that the notion of minimizer of a Mumford-Shah-type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time-continuous, quasi-static evolution.

##### MSC:

74R10 | Brittle fracture |

74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |

74G65 | Energy minimization in equilibrium problems in solid mechanics |

PDF
BibTeX
Cite

\textit{G. A. Francfort} and \textit{C. J. Larsen}, Commun. Pure Appl. Math. 56, No. 10, 1465--1500 (2003; Zbl 1068.74056)

Full Text:
DOI

##### References:

[1] | Ambrosio, Boll Un Mat Ital B (7) 3 pp 857– (1989) |

[2] | ; ; Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Clarendon, Oxford University Press, New York, 2000. · Zbl 0957.49001 |

[3] | Bourdin, Numer Math 85 pp 609– (2000) |

[4] | Bourdin, J Mech Phys Solids 48 pp 797– (2000) |

[5] | Bower, J Mech Phys Solids 38 pp 443– (1990) |

[6] | Approximation of free-discontinuity problems. Lecture Notes in Mathematics, 1694. Springer, Berlin, 1998. · Zbl 0909.49001 |

[7] | Bucur, Ann Scuola Norm Sup Pisa Cl Sci (4) 29 pp 807– (2000) |

[8] | Carriero, Nonlinear Anal 15 pp 661– (1990) |

[9] | Chambolle, Arch Rat Mech Anal |

[10] | Chambolle, Comm Partial Differential Equations 22 pp 811– (1997) |

[11] | Dal Maso, Acta Math 168 pp 89– (1992) |

[12] | Dal Maso, Arch Ration Mech Anal 162 pp 101– (2002) |

[13] | De Giorgi, Arch Rational Mech Anal 108 pp 195– (1989) |

[14] | ; Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC, Boca Raton, Fla., 1992. · Zbl 0804.28001 |

[15] | Francfort, J Mech Phys Solids 46 pp 1319– (1998) |

[16] | ; Preprint SISSA, 2003. |

[17] | Griffith, Phil Trans Roy Soc London 221-A pp 163– (1920) |

[18] | Larsen, SIAM J Math Anal 29 pp 823– (1998) |

[19] | In preparation. |

[20] | Maddalena, Arch Ration Mech Anal 159 pp 273– (2001) |

[21] | ; The dependence of the fiber pull-out on the surface energy. In preparation. |

[22] | Mumford, Comm Pure Appl Math 42 pp 577– (1989) |

[23] | The Neumann sieve. Nonlinear variational problems (Isola d’Elba, 1983), 24-32. Research Notes in Mathematics, 127. Pitman, Boston, 1985. |

[24] | Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.