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Crack growth by vanishing viscosity in planar elasticity

1 Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2 Università degli Studi di Firenze, Dipartimento di Matematica e Informatica “Ulisse Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy
3 Université de Lorraine, Institut Élie Cartan de Lorraine, BP 70239, 54506 Vandoeuvre-lès-Nancy, France

This contribution is part of the Special Issue: Variational Models in Elasticity
Guest Editors: Lucia De Luca; Marcello Ponsiglione
Link: https://www.aimspress.com/newsinfo/1369.html

Special Issues: Variational Models in Elasticity

We show the existence of quasistatic evolutions in a fracture model for brittle materials by a vanishing viscosity approach, in the setting of planar linearized elasticity. Differently from previous works, the crack is not prescribed a priori and is selected in a class of (unions of) regular curves. To prove the result, it is crucial to analyze the properties of the energy release rate showing that it is independent of the crack extension.
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Keywords free-discontinuity problems; brittle fracture; crack propagation; vanishing viscosity; local minimizers; energy derivative; Griffith’s criterion; energy release rate; stress intensity factor

Citation: Stefano Almi, Giuliano Lazzaroni, Ilaria Lucardesi. Crack growth by vanishing viscosity in planar elasticity. Mathematics in Engineering, 2020, 2(1): 141-173. doi: 10.3934/mine.2020008


  • 1. Almi S (2017) 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: 791-826.    
  • 2. Almi S (2018) Quasi-static hydraulic crack growth driven by Darcy's law. Adv Calc Var 11: 161-191.    
  • 3. Almi S, Lucardesi I (2018) Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks. Nonlinear Differ Equ Appl 25: 43.    
  • 4. Argatov II, Nazarov SA (2002) Energy release in the kinking of a crack in a plane anisotropic body. Prikl Mat Mekh 66: 502-514.
  • 5. Babadjian JF, Chambolle A, Lemenant A (2015) Energy release rate for non-smooth cracks in planar elasticity. J Éc polytech Math 2: 117-152.    
  • 6. Brokate M, Khludnev A (2004) On crack propagation shapes in elastic bodies. Z Angew Math Phys 55: 318-329.    
  • 7. Chambolle A (2003) A density result in two-dimensional linearized elasticity, and applications. Arch Ration Mech Anal 167: 211-233.    
  • 8. Chambolle A, Francfort GA, Marigo JJ (2010) Revisiting energy release rates in brittle fracture. J Nonlinear Sci 20: 395-424.    
  • 9. Chambolle A, Giacomini A, Ponsiglione M (2008) Crack initiation in brittle materials. Arch Ration Mech Anal 188: 309-349.    
  • 10. Chambolle A, Lemenant A (2013) The stress intensity factor for non-smooth fractures in antiplane elasticity. Calc Var Partial Dif 47: 589-610.    
  • 11. Costabel M, Dauge M (2002) Crack singularities for general elliptic systems. Math Nachr 235: 29-49.    
  • 12. Crismale V, Lazzaroni G (2017) Quasistatic crack growth based on viscous approximation: A model with branching and kinking. Nonlinear Differ Equ Appl 24: 7.    
  • 13. Dal Maso G, DeSimone A, Solombrino F (2011) Quasistatic evolution for Cam-Clay plasticity: A weak formulation via viscoplastic regularization and time rescaling. Calc Var Partial Dif 40: 125-181.    
  • 14. Dal Maso G, DeSimone A, Solombrino F (2012) Quasistatic evolution for Cam-Clay plasticity: Properties of the viscosity solution. Calc Var Partial Dif 44: 495-541.    
  • 15. Dal Maso G, Francfort GA, Toader R (2005) Quasistatic crack growth in nonlinear elasticity. Arch Ration Mech Anal 176: 165-225.    
  • 16. Dal Maso G, Lazzaroni G (2010) Quasistatic crack growth in finite elasticity with noninterpenetration. Ann Inst H Poincaré Anal Non Linéaire 27: 257-290.    
  • 17. Dal Maso G, Morandotti M (2017) A model for the quasistatic growth of cracks with fractional dimension. Nonlinear Anal 154: 43-58.    
  • 18. Dal Maso G, Orlando G, Toader R (2015) Laplace equation in a domain with a rectilinear crack: Higher order derivatives of the energy with respect to the crack length. Nonlinear Differ Equ Appl 22: 449-476.    
  • 19. Dal Maso G, Toader R (2002) A model for the quasi-static growth of brittle fractures: Existence and approximation results. Arch Ration Mech Anal 162: 101-135.    
  • 20. Dauge M (1988) Smoothness and asymptotics of solutions, In: Elliptic Boundary Value Problems on Corner Domains, Berlin: Springer-Verlag.
  • 21. Destuynder P, Djaoua M (1981) Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile. Math Method Appl Sci 3: 70-87.    
  • 22. Efendiev MA, Mielke A (2006) On the rate-independent limit of systems with dry friction and small viscosity. J Convex Anal 13: 151-167.
  • 23. Fonseca I, Fusco N, Leoni G, et al. (2007) Equilibrium configurations of epitaxially strained crystalline films: Existence and regularity results. Arch Ration Mech Anal 186: 477-537.    
  • 24. Francfort GA, Larsen CJ (2003) Existence and convergence for quasi-static evolution in brittle fracture. Commun Pur Appl Math 56: 1465-1500.    
  • 25. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46: 1319-1342.    
  • 26. Friedrich M, Solombrino F (2018) Quasistatic crack growth in 2d-linearized elasticity. Ann Inst H Poincaré Anal Non Linéaire 35: 27-64.    
  • 27. Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc London A 221: 163-198.    
  • 28. Grisvard P (1985) Monographs and Studies in Mathematics, In: Elliptic Problems in Consmooth Domains, Boston: Pitman.
  • 29. Khludnev AM, Shcherbakov VV (2018) A note on crack propagation paths inside elastic bodies. Appl Math Lett 79: 80-84.    
  • 30. Khludnev AM, Sokolowski J (2000) Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. Eur J Mech A Solids 19: 105-119.    
  • 31. Knees D (2011) A short survey on energy release rates, In: Dal Maso G, Larsen CJ, Ortner C, Mini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Oberwolfach Reports, 8: 1216-1219.
  • 32. Knees D, Mielke A (2008) Energy release rate for cracks in finite-strain elasticity. Math Method Appl Sci 31: 501-528.    
  • 33. Knees D, Mielke A, Zanini C (2008) On the inviscid limit of a model for crack propagation. Math Mod Meth Appl Sci 18: 1529-1569.    
  • 34. Knees D, Rossi R, Zanini C (2013) A vanishing viscosity approach to a rate-independent damage model. Math Mod Meth Appl Sci 23: 565-616.    
  • 35. Knees D, Rossi R, Zanini C (2015) A quasilinear differential inclusion for viscous and rateindependent damage systems in non-smooth domains. Nonlinear Anal Real 24: 126-162.    
  • 36. Knees D, Zanini C, Mielke A (2010) Crack growth in polyconvex materials. Physica D 239: 1470-1484.    
  • 37. Krantz SG, Parks HR (2002) The Implicit Function Theorem: History, theory, and applications, Boston: Birkhäuser Boston Inc.
  • 38. Lazzaroni G, Toader R (2011) A model for crack propagation based on viscous approximation. Math Mod Meth Appl Sci 21: 2019-2047.    
  • 39. Lazzaroni G, Toader R (2011) Energy release rate and stress intensity factor in antiplane elasticity. J Math Pure Appl 95: 565-584.    
  • 40. Lazzaroni G, Toader R (2013) Some remarks on the viscous approximation of crack growth. Discrete Contin Dyn Syst Ser S 6: 131-146.
  • 41. Mielke A, Rossi R, Savaré G (2009) Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin Dyn Syst 25: 585-615.    
  • 42. Mielke A, Rossi R, Savaré G (2012) BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim Calc Var 18: 36-80.    
  • 43. Mielke A, Rossi R, Savaré G (2016) Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J Eur Math Soc 18: 2107-2165.    
  • 44. Mielke A, Roubíček T (2015) Rate-Independent Systems, New York: Springer.
  • 45. Nazarov SA, Specovius-Neugebauer M, Steigemann M (2014) Crack propagation in anisotropic composite structures. Asymptot Anal 86: 123-153.    
  • 46. Negri M (2011) Energy release rate along a kinked path. Math Method Appl Sci 34: 384-396.    
  • 47. Negri M (2014) Quasi-static rate-independent evolutions: Characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim Calc Var 20: 983-1008.    
  • 48. Negri M, Ortner C (2008) Quasi-static crack propagation by Griffith's criterion. Math Mod Meth Appl Sci 18: 1895-1925.    
  • 49. Negri M, Toader R (2015) Scaling in fracture mechanics by Bažant law: From finite to linearized elasticity. Math Mod Meth Appl Sci 25: 1389-1420.    


This article has been cited by

  • 1. L. De Luca, M. Ponsiglione, Variational models in elasticity, Mathematics in Engineering, 2021, 3, 2, 1, 10.3934/mine.2021015
  • 2. Stefano Almi, Irreversibility and alternate minimization in phase field fracture: a viscosity approach, Zeitschrift für angewandte Mathematik und Physik, 2020, 71, 4, 10.1007/s00033-020-01357-x

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