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Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue

    https://doi.org/10.1142/S0218202518500379Cited by:16 (Source: Crossref)

    In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

    Communicated by G. Dal Maso

    AMSC: 74C05, 74R99, 35Q74, 74G65, 35A35

    References

    • 1. R. Abdelmoula, J.-J. Marigo and T. Weller, Construction d’une loi de fatigue à partir d’un modèle de forces cohésives: cas d’une fissure en mode III, C. R. Mecanique 337 (2009) 53–59. Crossref, Web of ScienceGoogle Scholar
    • 2. R. Alessi, J.-J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal. 214 (2014) 575–615. Crossref, Web of ScienceGoogle Scholar
    • 3. R. Alessi, J.-J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity: Variational formulation and main properties, Mech. Materials 80 (2015) 351–367. Crossref, Web of ScienceGoogle Scholar
    • 4. S. Almi, Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening, ESAIM Control Optim. Calc. Var. 23 (2017) 791–826. Crossref, Web of ScienceGoogle Scholar
    • 5. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs (Oxford Univ. Press, 2000). CrossrefGoogle Scholar
    • 6. M. Artina, F. Cagnetti, M. Fornasier and F. Solombrino, Linearly constrained evolutions of critical points and an application to cohesive fractures, Math. Models Methods Appl. Sci. 27 (2017) 231–290. Link, Web of ScienceGoogle Scholar
    • 7. J.-F. Babadjian and A. Giacomini, Existence of strong solutions for quasi-static evolution in brittle fracture, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014) 925–974. Google Scholar
    • 8. M. Barchiesi, G. Lazzaroni and C. I. Zeppieri, A bridging mechanism in the homogenisation of brittle composites with soft inclusions, SIAM J. Math. Anal. 48 (2016) 1178–1209. Crossref, Web of ScienceGoogle Scholar
    • 9. G. I. Barenblatt, The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech. 7 (1962) 55–129. CrossrefGoogle Scholar
    • 10. G. Bouchitté, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew. Math. 458 (1995) 1–18. Web of ScienceGoogle Scholar
    • 11. B. Bourdin, G. A. Francfort and J.-J. Marigo, The Variational Approach to Fracture (Springer, 2008). Reprinted from J. Elasticity 91 (2008) 5–148. Google Scholar
    • 12. H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (North-Holland, 1973). Google Scholar
    • 13. F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci. 18 (2008) 1027–1071. Link, Web of ScienceGoogle Scholar
    • 14. F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measures approach, ESAIM: Control Optim. Calc. Var. 17 (2011) 1–27. Crossref, Web of ScienceGoogle Scholar
    • 15. A. Chambolle, A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal. 167 (2003) 211–233. Crossref, Web of ScienceGoogle Scholar
    • 16. S. Conti, M. Focardi and F. Iurlano, Phase field approximation of cohesive fracture models, Ann. Inst. H. Poincaré Anal. Non Linéaire 44 (2015) 1033–1067. Google Scholar
    • 17. S. D. Cramer and B. S. Covino (eds.), Corrosion: Fundamentals, Testing, and Protection, ASM Handbook, Vol. 13A (ASM International, 2003). CrossrefGoogle Scholar
    • 18. V. Crismale, Globally stable quasistatic evolution for a coupled elastoplastic-damage model, ESAIM Control Optim. Calc. Var. 22 (2016) 883–912. Crossref, Web of ScienceGoogle Scholar
    • 19. V. Crismale, Globally stable quasistatic evolution for strain gradient plasticity coupled with damage, Ann. Mat. Pura Appl. 196 (2017) 641–685. Crossref, Web of ScienceGoogle Scholar
    • 20. V. Crismale and G. Lazzaroni, Viscoupros apximation of quasistatic evolutions for a coupled elastoplastic-damage model, Calc. Var. Partial Differential Equations 55(17) (2016). Google Scholar
    • 21. V. Crismale and G. Lazzaroni, Quasistatic crack growth based on viscous approximation: A model with branching and kinking, NoDEA Nonlinear Differential Equations Appl. 24(7) (2017). Google Scholar
    • 22. V. Crismale and G. Orlando, A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in W1,n, NoDEA Nonlinear Differential Equations Appl. 25(16) (2018). Google Scholar
    • 23. G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005) 165–225. Crossref, Web of ScienceGoogle Scholar
    • 24. G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) 257–290. Crossref, Web of ScienceGoogle Scholar
    • 25. G. Dal Maso, G. Orlando and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case, Calc. Var. Partial Differential Equations 55(45) (2016). Google Scholar
    • 26. G. Dal Maso, G. Orlando and R. Toader, Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation, Adv. Cal. Var. 10 (2016) 183–207. Crossref, Web of ScienceGoogle Scholar
    • 27. G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002) 101–135. Crossref, Web of ScienceGoogle Scholar
    • 28. G. Dal Maso and C. Zanini, Quasi-static crack growth for a cohesive zone model with prescribed crack path, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253–279. Crossref, Web of ScienceGoogle Scholar
    • 29. A. Ferriero, Quasi-static evolution for fatigue debonding, ESAIM Control Optim. Calc. Var. 14 (2008) 233–253. Crossref, Web of ScienceGoogle Scholar
    • 30. G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003) 1465–1500. Crossref, Web of ScienceGoogle Scholar
    • 31. G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342. Crossref, Web of ScienceGoogle Scholar
    • 32. M. Friedrich and F. Solombrino, Quasistatic crack growth in 2D-linearized elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018) 27–64. Crossref, Web of ScienceGoogle Scholar
    • 33. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London Ser. A 221 (1920) 163–198. CrossrefGoogle Scholar
    • 34. D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci. 18 (2008) 1529–1569. Link, Web of ScienceGoogle Scholar
    • 35. D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Physica D 239 (2010) 1470–1484. Crossref, Web of ScienceGoogle Scholar
    • 36. M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem, Math. Mech. Solids 11 (2006) 423–447. Crossref, Web of ScienceGoogle Scholar
    • 37. G. Lazzaroni, Quasistatic crack growth in finite elasticity with Lipschitz data, Ann. Mat. Pura Appl. (4) 190 (2011) 165–194. Crossref, Web of ScienceGoogle Scholar
    • 38. G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci. 21 (2011) 2019–2047. Link, Web of ScienceGoogle Scholar
    • 39. G. Lazzaroni and R. Toader, Some remarks on the viscous approximation of crack growth, Discrete Contin. Dyn. Syst. Ser. S 6 (2013) 131–146. Crossref, Web of ScienceGoogle Scholar
    • 40. A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, Vol. 193 (Springer, 2015). CrossrefGoogle Scholar
    • 41. M. Negri and R. Scala, A quasi-static evolution generated by local energy minimizers for an elastic material with a cohesive interface, Nonlinear Anal. Real World Appl. 38 (2017) 271–305. Crossref, Web of ScienceGoogle Scholar
    • 42. M. Negri and E. Vitali, Approximation and characterization of quasi-static H1-evolutions for a cohesive interface with different loading-unloading regimes, to appear on Interfaces Free Bound. Google Scholar
    • 43. T. Roubíček, L. Scardia and C. Zanini, Quasistatic delamination problem, Contin. Mech. Thermodyn. 21 (2009) 223–235. Crossref, Web of ScienceGoogle Scholar
    • 44. M. Valadier, Young measures, in Methods of Nonconvex Analysis, Lecture Notes in Mathematics, Vol. 1446 (Springer, 1990), pp. 152–188. CrossrefGoogle Scholar
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