Abstract
This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in L∞.
Similar content being viewed by others
References
Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Rational Mech. Anal. 111, 291–322 (1990)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)
Babadjian, J.-F.: Réduction dimensione lle pow des milieux hétéogéncs dioué ou jusurés, PhD thesis, University Paris Nord (2005)
Ball, J.M., Kirchheim, B., Kristensen, J.: Regularity of quasiconvex envelopes. Calc. Var. Partial Diff. Eqn. 11, 333–359 (2000)
Bhattacharya, K., Fonseca, I., Francfort, G.A.: An asymptotic study of the debonding of thin films. Arch. Rational Mech. Anal. 161, 205–229 (2002)
Bouchitté, G., Fonseca, I., Leoni, G., Mascarenhas, L.: A global method for relaxation in W1,p and in SBVp. Arch. Rational Mech. Anal. 165, 187–242 (2002)
Braides, A., Chiad'o Piat, V.: Integral representation results for functionals defined on SBVΩRm. J. Math. Pures Appl. 75, 595–626 (1996)
Braides, A., Defranceschi, A., Vitali, E.: Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135, 297–356 (1996)
Braides, A., Fonseca, I.: Brittle thin films. Appl. Math. Optim. 44, 299–323 (2001)
Braides, A., Fonseca, I., Francfort, G.A.: 3D-2D Asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49, 1367–1404 (2000)
Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin (1989)
Dal Maso, G.: An Introduction to γ-Convergence. Birkhäuser, Boston (1993)
Dal Maso, G., Francfort, G.A., Toader, R.: Quasi-static crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176, 165–225 (2005)
Dal Maso, G., Francfort, G.A., Toader, R.: 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, 247–265 (2005)
Ekeland, I., Temam, R.: Analyse Convexe et Problémes Variationnels. Dunod, Gauthiers-Villars, Paris (1974)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Boca Raton, CRC Press (1992)
Fonseca, I. Francfort, G.A.: Relaxation in BV versus quasiconvexification in W1,p; a model for the interaction between fracture and damage. Calc. Var. Partial Diff. Eqn. 3, 407–446 (1995)
Francfort, G.A., Larsen, C.J.: Existence and convergence for quasistatic evolution in brittle fracture. Comm. Pure Appl. Math. 56, 1465–1500 (2003)
Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)
Giacomini, A.: Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture. Calc. Var. Partial Differential Equations 22, 129–172 (2005)
Giacomini, A., Ponsiglione, M.: A γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications, to appear in Arch. Rational Mech. Anal
Le Dret, H., Raoult, A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74, 549–578 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000) 74K30, 49J45, 74K30, 35R35, 49Q20
Rights and permissions
About this article
Cite this article
Babadjian, JF. Quasistatic evolution of a brittle thin film. Calc. Var. 26, 69–118 (2006). https://doi.org/10.1007/s00526-005-0369-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-005-0369-y