Abstract
This paper deals with quasi-static crack growth in thin films. We show that, when the thickness of the film tends to zero, any three-dimensional quasi-static crack evolution converges to a two-dimensional one, in a sense related to the Г - convergence of the associated total energy. We extend the prior analysis of [2] by adding conservative body and surface forces which allow us to remove the boundedness assumption on the deformation field.
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References
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.
J.-F. Babadjian, Quasistatic evolution of a brittle thin film, Calc. Var. and PDEs 26(1), 2006, 69–118.
J.-F. Babadjian: Lower semicontinuity of quasiconvex bulk energies in SBV and integral representation in dimension reduction, SIAM J. Math. Anal. 39(6), 2008, 1921–1950.
B. Bourdin, G.A. Francfort and J.-J. Marigo: The Variational Approach to Fracture, Springer, Amsterdam, 2008.
A. Braides and I. Fonseca: Brittle thin films, Appl. Math. Optim. 44, 2001, 299–323.
A. Braides, I. Fonseca and G.A. Francfort: 3D-2D Asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49, 2000, 1367–1404.
G. Dal Maso: An Introduction to Γ- Convergence, Birkhäuser, Boston, 1993.
G. Dal Maso, G.A. Francfort and R. Toader: Quasi-static crack growth in nonlinear elasticity, Arch. Rational Mech. Anal. 176, 2005, 165–225.
G. Dal Maso, G.A. Francfort and R. Toader: Quasi-static evolution in brittle fracture: The case of bounded solutions, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi. Quaderni di Matematica 14, 2005, 247–265.
G.A. Francfort and J.-J. Marigo: Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46, 1998, 1319–1342.
G.A. Francfort and G.J. Larsen: Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56, 2003, 1465–1500.
G. Friesecke, R. James and S. Müller: A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence, Arch. Rational Mech. Anal. 180(2), 2006, 183–236.
D. Fox, A. Raoult and J.C. Simo: A justification of nonlinear properly invariant plate theories, Arch. Rational. Mech. Anal. 25, 1992, 157–199.
A. Giacomini and M. Ponsiglione: A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications, Arch. Rational Mech. Anal. 180, 2006, 399–447.
H. Le Dret and A. Raoult: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 74, 1995, 549–578.
A. Mielke, T. Roubicek and U. Stefanelli: Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. and PDEs 31, 2008, 387–416.
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Babadjian, JF. (2010). Stability of Quasi-Static Crack Evolution through Dimensional Reduction. In: Hackl, K. (eds) IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials. IUTAM Bookseries, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9195-6_1
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DOI: https://doi.org/10.1007/978-90-481-9195-6_1
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