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From finite to linear elastic fracture mechanics by scaling

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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.

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References

  1. 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)

    Google Scholar 

  2. Ball, J.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337–403 (1977)

    Article  MATH  Google Scholar 

  3. Ball, J.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002)

  4. Bažant, Z.: Scaling of structural strength. Butterworth-Heinemann, Oxford (2005)

    MATH  Google Scholar 

  5. Bourdin, B., Francfort, G., Marigo, J.J.: The variational approach to fracture. J. Elasticity 91, 5–148 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ciarlet, P.: Mathematical Elasticity. Three-Dimensional Elasticity. North-Holland Publishing Co., Amsterdam (1988)

    Google Scholar 

  7. Dal Maso, G.: An introduction to \(\Gamma \)-convergence. Birkhäuser, Boston (1993)

  8. Dal Maso, G., Francfort, G., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176(2), 165–225 (2005)

    Google Scholar 

  9. 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)

    Google Scholar 

  10. Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Anal. 10(2–3), 165–183 (2002)

  11. DeSimone, A.: Hysteresis and imperfection sensitivity in small ferromagnetic particles. Meccanica 30(5), 591–603 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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)

    MATH  MathSciNet  Google Scholar 

  14. Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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)

    Article  MATH  MathSciNet  Google Scholar 

  16. Griffith, A.: The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. 18, 163–198 (1920)

    Google Scholar 

  17. Gurtin, M.: An introduction to continuum mechanics, vol. 158. Academic Press Inc., New York (1981)

    MATH  Google Scholar 

  18. 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)

    Google Scholar 

  19. Inglis, C.: Stress in a plate due to the presence of sharp corners and cracks. Trans. R. Inst. Naval Architects 60, 219–241 (1913)

    Google Scholar 

  20. Knees, D., Mielke, A.: Energy release rate for cracks in finite-strain elasticity. Math. Methods Appl. Sci. 31(5), 501–528 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Knees, D., Zanini, C., Mielke, A.: Crack growth in polyconvex materials. Phys. D 239(15), 1470–1484 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. 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)

    Google Scholar 

  25. Mielke, A., Theil, F.: On rate-independent hysteresis models. Nonlinear Differ. Eqs. Appl. (NoDEA) 11, 151–189 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Negri, M.: A comparative analysis on variational models for quasi-static brittle crack propagation. Adv. Calc. Var. (2010)

  27. Negri, M., Ortner, C.: Quasi-static propagation of brittle fracture by Griffith’s criterion. Math. Models Methods Appl. Sci. 18(11), 1895–1925 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schmidt, B.: Linear \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn. 20(6), 375–396 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Spadaro, E.: Non-uniqueness of minimizers for strictly polyconvex functionals. Arch. Rational Mech. Anal. 193, 659–678 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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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”.

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Correspondence to M. Negri.

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Communicated by J. Ball.

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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

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