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Variational fracture: twenty years after

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Abstract

In this work, we propose to succinctly review, in non technical terms, the current mathematical state of brittle fracture first put forth by A. A. Griffith in his seminal work Griffith (Philos Trans R Soc Lond CCXXI-A:163–198, 1920), then re-interpreted under the label “variational fracture” in Francfort and Marigo (J. Mech Phys Solids 46(8):1319–1342, 1998). and subsequent works. We will only address the sharp theory and will limit ourselves to a theoretical exposition, leaving phase-field approximations and other possible implementations to other articles within this volume.

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Notes

  1. Our admittedly personal account of that story will keep the number of references to a minimum. Narrative continuity is our goal, not taxonomy. Nor for that matter exhaustion: many topics worthy of investigation will not even get a mention, be it non-interpenetration, dynamics, energy dissipating evolutions, ....

  2. We find telltale signs in the work of J. Rice (see e.g. (Rice 1979, Sect. 3) but have been at pains to isolate a definitive statement of its origin in the literature.

  3. Irreversibility may be debatable when dealing with elastomers or gels.

  4. The symbol \(_\lfloor \) means “restricted to”.

  5. Although quite natural, this assumption prohibits jumps on \(\partial _d\Omega \) so that a tearing experiment with \(u=d\) on one side of a pre-crack and \(u=-d\) on the other side is not allowed.

  6. It may be so that the physics underlying proper phase-field models is actually a better fit for brittle fracture than the sharp theory. As already stated, ours is only a Griffith universe beyond which we dare not venture for fear of getting sucked into a modeling wormhole.

  7. This will not produce an exact solution, because that solution will exhibit a discontinuity of the normal stress at the points \((\ell (t),y), \; y\ne 0\).

  8. To our knowledge one has to appeal to results in Grisvard (1985) which deal exclusively with polygonal domains.

  9. Actually the result is more general and does not allow cracks whose length is asymptotically like that of a line segment as that length tends to 0. Also the assumption that the pre-crack is straight can be greatly relaxed (see Babadjian et al 2015)

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  1. Université Paris-Nord, 93430, Villetaneuse, France

    G. A. Francfort

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Correspondence to G. A. Francfort.

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The author wishes to acknowledge A. Dumas (fils) who thoroughly investigated a case of rupture in Dumas (1854), this some seventy years before A.A. Griffith.

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Francfort, G.A. Variational fracture: twenty years after. Int J Fract 237, 3–13 (2022). https://doi.org/10.1007/s10704-020-00508-5

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