Toader, Zanini: An Artificial Viscosity Approach to Quasistatic Crack Growth

Toader, Rodica and Zanini, Chiara:
An Artificial Viscosity Approach to Quasistatic Crack Growth
Bollettino dell'Unione Matematica Italiana Serie 9 2 (2009), fasc. n.1, p. 1-35, (English)
pdf (424 Kb), djvu (322 Kb). | MR 2493642 | Zbl 1180.35521

Sunto

We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\epsilon$-gradient flow of the energy functional, as the "viscosity" parameter $\epsilon$ tends to zero.

Referenze Bibliografiche

[1] J. CASADO-DIAZ - G. DAL MASO, A simplified model for the evolution of a fracture in a membrane, Preprint (2000). | Zbl 0959.49014

[2] A. CHAMBOLLE, A density result in two-dimensional linearized elasticity, and applications. Arch. Rational Mech. Anal., 167 (2003), 211-233. | fulltext (doi) | MR 1978582 | Zbl 1030.74007

[3] G. DAL MASO, G.A. FRANCFORT AND R. TOADER, Quasistatic Crack Growth in Nonlinear Elasticity, Arch. Rational Mech. Anal., 176 (2005), 165-225. | fulltext (doi) | MR 2186036 | Zbl 1064.74150

[4] G. DAL MASO - R. TOADER, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rational Mech. Anal., 162 (2002), 101-135. | fulltext (doi) | MR 1897378 | Zbl 1042.74002

[5] G. DAL MASO - R. TOADER, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799. | fulltext (doi) | MR 1946723 | Zbl 1205.74149

[6] K. DEIMLING, Ordinary Differential Equations in Banach Spaces. Lect. Notes Math. 596, Springer-Verlag, Berlin-New York, 1977. | MR 463601 | Zbl 0361.34050

[7] G.A. FRANCFORT - C.J. LARSEN, Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math., 56 (2003), 1465-1500. | fulltext (doi) | MR 1988896 | Zbl 1068.74056

[8] G. A. FRANCFORT - J.-J. MARIGO, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. | fulltext (doi) | MR 1633984 | Zbl 0966.74060

[9] A. FRIEDMAN - B. HU - J. J. L. VELAZQUEZ, The evolution of stress intensity factors in the propagation of two dimensional cracks, European J. Appl. Math., 11 (2000), 453-471. | fulltext (doi) | MR 1799921 | Zbl 0969.74056

[10] A. A. GRIFFITH, The Phenomena of Rupture and Flow in Solids, Philos. Trans. R. Soc. London Ser. A, 221 (1920), 163-198.

[11] P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. | MR 775683 | Zbl 0695.35060

[12] P. GRISVARD, Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 22. Masson, Paris; Springer-Verlag, Berlin, 1992. | MR 1173209

[13] J. HALE, Ordinary Differential Equations, Pure and Applied Mathematics, XXI, Krieger, Florida, 1980. | MR 587488

[14] D. KNEES - A. MIELKE, Energy release rate for cracks in finite-strain elasticity. Math. Methods Applied Sciences, 31 (2008), 501-528. | fulltext (doi) | MR 2394124 | Zbl 1132.74038

[15] S. G. KRANTZ - H. R. PARKS, The Implicit Function Theorem. History, theory and applications. Birkhäuser, Boston, 2002. | fulltext (doi) | MR 1894435 | Zbl 1012.58003

[16] A. MIELKE, Evolution of Rate-Independent Systems, Handbook of Differential Equations, Evolutionary Equations, v. 2, C.M. Dafermos, E. Feireisl (eds.) 461-559 Elsevier, Amsterdam, 2005. | MR 2549367 | Zbl 1120.47062

[17] C. ZANINI, Variational Techniques for Quasistatic Evolutionary Models, PhD Thesis, 2006, see http://digitallibrary.sissa.it/handle/1963/1874

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche