These notes begin with a review of the mainstream theory of brittle fracture, as it has emerged from the works of Griffith and Irwin. We propose a re-formulation of that theory within the confines of the calculus of variations, focussing on crack path prediction. We then illustrate the various possible minimality criteria in a simple 1d-case as well as in a tearing experiment and discuss in some details the only complete mathematical formulation so far, that is that where global minimality for the total energy holds at each time. Next we focus on the numerical treatment of crack evolution and detail crack regularization which turns out to be a good approximation from the standpoint of crack propagation. This leads to a discussion of the computation of minimizing states for a non-convex functional. We illustrate the computational issues with a detailed investigation of the tearing experiment.
KeywordsCrack Length Crack Path Bulk Energy Dissipation Potential Elastic Energy Density
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- G. Alberti. Variational models for phase transitions, an approach via Γ-convergence. In G. Buttazzo, editor, Calculus of Variations and Partial Differential Equations, pages 95–114. Springer-Verlag, 2000.Google Scholar
- L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, 2000.Google Scholar
- B. Bourdin. Une méthode variationnelle en mécanique de la rupture. Théorie et applications numériques. Thèse de doctorat, Université Paris-Nord, 1998.Google Scholar
- A. Braides. Γ-convergence for Beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2002.Google Scholar
- S. Burke, C. Ortner, and E. Süli. An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numer. Anal., 48(3): 980–1012, 2010.Google Scholar
- G. Dal Maso. An introduction to Γ-convergence. Birkhäuser, Boston, 1993.Google Scholar
- A.A. Griffith. The phenomena of rupture and flow in solids. Philos. T. Roy. Soc. A, CCXXIA: 163–198, 1920.Google Scholar
- M.E. Gurtin. Configurational forces as basic concepts of continuum physics, volume 137 of Applied Mathematical Sciences. Springer-Verlag, New York, 2000.Google Scholar
- H. Hahn. Über Annäherung an Lebesgue’sche integrale durch Riemann’sche summen. Sitzungsber. Math. Phys. Kl. K. Akad. Wiss. Wien, 123:713–743, 1914.Google Scholar
- F. Murat. The Neumann sieve. In Nonlinear variational problems (Isola d’Elba, 1983), volume 127 of Res. Notes in Math., pages 24–32. Pitman, Boston, MA, 1985.Google Scholar