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Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture

  • Stefano Almi
  • Matteo NegriEmail author
Article
  • 30 Downloads

Abstract

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After recasting the staggered scheme by means of gradient flows, we characterize the time-continuous limits of the discrete solutions in terms of balanced viscosity evolutions, parametrized by their arc-length with respect to the \(L^2\)-norm (for the phase field) and the \(H^1\)-norm (for the displacement field). By a careful study of the energy balance we deduce that time-continuous evolutions may still exhibit an alternate behavior in discontinuity times.

Notes

References

  1. 1.
    Almi, S., Belz, S.: Consistent finite-dimensional approximation of phase-field models of fracture. Ann. Mat. Pura Appl. (4)198(4), 1191–1225, 2019MathSciNetCrossRefGoogle Scholar
  2. 2.
    Almi, S., Belz, S., Negri, M.: Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics. ESAIM Math. Model. Numer. Anal. 53(2), 659–699, 2019MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambati, M., Gerasimov, T., De Lorenzis, L.: A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 55(2), 383–405, 2015MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel 2005zbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence. Commun. Pure Appl. Math. 43(8), 999–1036, 1990MathSciNetCrossRefGoogle Scholar
  6. 6.
    Amor, H., Marigo, J.J., Maurini, C.: Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids57, 1209–1229, 2009ADSCrossRefGoogle Scholar
  7. 7.
    Babadjian, J.F., Millot, V.: Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements. Ann. Inst. H. Poincaré Anal. Non Linéaire31(4), 779–822, 2014ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Balder, E.J.: An extension of Prohorov’s theorem for transition probabilities with applications to infinite-dimensional lower closure problems. Rend. Circ. Mat. Palermo (2)34(3), 427–447, 1985 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourdin, B.: Numerical implementation of the variational formulation for quasi-static brittle fracture. Interfaces Free Bound. 9(3), 411–430, 2007MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bourdin, B., Francfort, G.A., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids48(4), 797–826, 2000ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)North-Holland Publishing Co., Amsterdam 1973zbMATHGoogle Scholar
  12. 12.
    Burke, S., Ortner, C., Süli, E.: An adaptive finite element approximation of a variational model of brittle fracture. SIAM J. Numer. Anal. 48(3), 980–1012, 2010MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chambolle, A.: An approximation result for special functions with bounded deformation. J. Math. Pures Appl. (9)83(7), 929–954, 2004MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chambolle, A., Conti, S., Francfort, G.A.: Approximation of a brittle fracture energy with a constraint of non-interpenetration. Arch. Ration. Mech. Anal. 228(3), 867–889, 2018MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chambolle, A., Crismale, V.: A density result in \({GSBD}^p\) with applications to the approximation of brittle fracture energies. Arch. Ration. Mech. Anal. 232(3), 1329–1378, 2019MathSciNetCrossRefGoogle Scholar
  16. 16.
    Comi, C., Perego, U.: Fracture energy based bi-dissipative damage model for concrete. Int. J. Solids Struct. 38(36), 6427–6454, 2001 CrossRefGoogle Scholar
  17. 17.
    Dal Maso, G.: Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS)15(5), 1943–1997, 2013MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176(2), 165–225, 2005MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\)Spaces. Springer Monographs in MathematicsSpringer, New York 2007Google Scholar
  20. 20.
    Giacomini, A.: Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22(2), 129–172, 2005MathSciNetCrossRefGoogle Scholar
  21. 21.
    Herzog, R., Meyer, C., Wachsmuth, G.: Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. J. Math. Anal. Appl. 382(2), 802–813, 2011MathSciNetCrossRefGoogle Scholar
  22. 22.
    Iurlano, F.: A density result for GSBD and its application to the approximation of brittle fracture energies. Calc. Var. Partial Differ. Equ. 51(1–2), 315–342, 2014MathSciNetCrossRefGoogle Scholar
  23. 23.
    Karma, A., Kessler, D.A., Levine, H.: Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87, 045–501, 2001Google Scholar
  24. 24.
    Knees, D., Negri, M.: Convergence of alternate minimization schemes for phase field fracture and damage. Math. Models Methods Appl. Sci. 27(9), 1743–1794, 2017MathSciNetCrossRefGoogle Scholar
  25. 25.
    Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23(4), 565–616, 2013MathSciNetCrossRefGoogle Scholar
  26. 26.
    Knees, D., Rossi, R., Zanini, C.: A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains. Nonlinear Anal. Real World Appl. 24, 126–162, 2015MathSciNetCrossRefGoogle Scholar
  27. 27.
    Knees, D., Rossi, R., Zanini, C.: Balanced viscosity solutions to a rate-independent system for damage. Eur. J. Appl. Math. 30(1), 117–175, 2019MathSciNetCrossRefGoogle Scholar
  28. 28.
    Łojasiewicz, S.: Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier (Grenoble)43(5), 1575–1595, 1993MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mielke, A.: Evolution of rate-independent systems. In: Dafermos, C., Feireisl, E. (eds.) Evolutionary Equations. Handbook of Differential Equations, vol. II, pp. 461–559. Elsevier, Amsterdam 2005CrossRefGoogle Scholar
  30. 30.
    Mielke, A., Rossi, R., Savaré, G.: Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. (JEMS)18(9), 2107–2165, 2016MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York 2015CrossRefGoogle Scholar
  32. 32.
    Negri, M.: A unilateral \(L^2\)-gradient flow and its quasi-static limit in phase-field fracture by alternate minimization. Adv. Calc. Var. 12(1), 1–29, 2019MathSciNetCrossRefGoogle Scholar
  33. 33.
    Negri, M., Kimura, M.: Weak solutions for gradient flows under monotonicity contraints. arxiv:1908.10111
  34. 34.
    Thomas, M.: Quasistatic damage evolution with spatial BV-regularization. Discrete Contin. Dyn. Syst. Ser. S6(1), 235–255, 2013MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wu, J.Y.: A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J. Mech. Phys. Solids103, 72–99, 2017ADSMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fakultät für Mathematik - TUMGarching bei MünchenGermany
  2. 2.Department of MathematicsUniversity of PaviaPaviaItaly

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