Abstract
In the setting of finite elasticity we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions we prove that for large domains the evolution with finite elasticity converges to the evolution with linearized elasticity. In the proof the crucial step is the (locally uniform) convergence of the non-linear to the linear energy release rate, which follows from the combination of several ingredients: the \(\Gamma \)-convergence of re-scaled energies, the strong convergence of minimizers, the Euler–Lagrange equation for non-linear elasticity and the volume integral representation of the energy release.
Similar content being viewed by others
References
Agostiniani, V., Maso, G.D., Simone, A.D.: Linear elasticity obtained from finite elasticity by \(\Gamma \)-convergence under weak coerciveness conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 715–735 (2012)
Ball, J.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337–403 (1977)
Ball, J.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002)
Bažant, Z.: Scaling of structural strength. Butterworth-Heinemann, Oxford (2005)
Bourdin, B., Francfort, G., Marigo, J.J.: The variational approach to fracture. J. Elasticity 91, 5–148 (2008)
Ciarlet, P.: Mathematical Elasticity. Three-Dimensional Elasticity. North-Holland Publishing Co., Amsterdam (1988)
Dal Maso, G.: An introduction to \(\Gamma \)-convergence. Birkhäuser, Boston (1993)
Dal Maso, G., Francfort, G., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176(2), 165–225 (2005)
Dal Maso, G., Lazzaroni, G.: Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(1), 257–290 (2010)
Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Anal. 10(2–3), 165–183 (2002)
DeSimone, A.: Hysteresis and imperfection sensitivity in small ferromagnetic particles. Meccanica 30(5), 591–603 (1995)
DeSimone, A., Kohn, R., Müller, S., Otto, F., Schäfer, R.: Two-dimensional modelling of soft ferromagnetic films. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2016), 2983–2991 (2001)
Efendiev, M., Mielke, A.: On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13(1), 151–167 (2006)
Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)
Friesecke, G., James, R., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55(11), 1461–1506 (2002)
Griffith, A.: The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. 18, 163–198 (1920)
Gurtin, M.: An introduction to continuum mechanics, vol. 158. Academic Press Inc., New York (1981)
Henao, D., Mora-Corral, C.: Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Rational Mech. Anal. 197(2), 619–655 (2010)
Inglis, C.: Stress in a plate due to the presence of sharp corners and cracks. Trans. R. Inst. Naval Architects 60, 219–241 (1913)
Knees, D., Mielke, A.: Energy release rate for cracks in finite-strain elasticity. Math. Methods Appl. Sci. 31(5), 501–528 (2008)
Knees, D., Mielke, A., Zanini, C.: On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18(9), 1529–1569 (2008)
Knees, D., Zanini, C., Mielke, A.: Crack growth in polyconvex materials. Phys. D 239(15), 1470–1484 (2010)
Knowles, J., Sternberg, E.: An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack. J. Elasticity 3, 67–107 (1973)
Mielke, A., Theil, F.: A mathematical model for rate-independent phase transformations with hysteresis. In: Alber, H.D., Balean, R., Farwig, R. (eds.) Proceedings of the Workshop on “Models of Continuum Mechanics in Analysis and Engineering”, pp. 117–129. Shaker-Verlag, Aachen (1999)
Mielke, A., Theil, F.: On rate-independent hysteresis models. Nonlinear Differ. Eqs. Appl. (NoDEA) 11, 151–189 (2004)
Negri, M.: A comparative analysis on variational models for quasi-static brittle crack propagation. Adv. Calc. Var. (2010)
Negri, M., Ortner, C.: Quasi-static propagation of brittle fracture by Griffith’s criterion. Math. Models Methods Appl. Sci. 18(11), 1895–1925 (2008)
Schmidt, B.: Linear \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn. 20(6), 375–396 (2008)
Spadaro, E.: Non-uniqueness of minimizers for strictly polyconvex functionals. Arch. Rational Mech. Anal. 193, 659–678 (2009)
Acknowledgments
This material is based on work supported by INdAM and by ERC under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Ball.
Rights and permissions
About this article
Cite this article
Negri, M., Zanini, C. From finite to linear elastic fracture mechanics by scaling. Calc. Var. 50, 525–548 (2014). https://doi.org/10.1007/s00526-013-0645-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-013-0645-1