Journal of Nonlinear Science

, Volume 29, Issue 3, pp 1041–1094 | Cite as

Fatigue Effects in Elastic Materials with Variational Damage Models: A Vanishing Viscosity Approach

  • Roberto Alessi
  • Vito CrismaleEmail author
  • Gianluca Orlando


We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depending on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time step \(\tau \) of the time discretisation and later the viscosity parameter \(\varepsilon \) go to zero. As \(\tau \rightarrow 0\), we find \(\varepsilon \)-approximate viscous evolutions; then, as \(\varepsilon \rightarrow 0\), we find a rescaled approximate evolution satisfying an energy-dissipation balance.


Fatigue Gradient damage models Variational methods Vanishing viscosity approach 

Mathematics Subject Classification

74C05 74A45 74R20 35Q74 49J45 



The authors wish to thank Adriana Garroni for several interesting discussions and fruitful advices. Roberto Alessi has been supported by the MATHTECH-CNR-INdAM project and the MIUR-DAAD Joint Mobility Program: “Variational approach to fatigue phenomena with phase-field models: modeling, numerics and experiments”. Vito Crismale has been supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and acknowledges the financial support from the Laboratory Ypatia and the CMAP. He is currently funded by the Marie Skłodowska-Curie Standard European Fellowship No. 793018. Gianluca Orlando has been supported by the Alexander von Humboldt Foundation. Vito Crismale and Gianluca Orlando acknowledge the kind hospitality of the Department of Mathematics of Sapienza University of Rome, where part of this research was developed.


  1. Alessi, R., Marigo, J.-J., Vidoli, S.: Gradient damage models coupled with plasticity and nucleation of cohesive cracks. Arch. Ration. Mech. Anal. 214, 575–615 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alessi, R., Marigo, J.-J., Vidoli, S.: Gradient damage models coupled with plasticity: variational formulation and main properties. Mech. Mater. 80(Part B), 351–367 (2015)CrossRefGoogle Scholar
  3. Alessi, R., Ambati, M., Gerasimov, T., Vidoli, S., De Lorenzis, L.: Comparison of Phase-Field Models of Fracture Coupled with Plasticity, pp. 1–21. Springer, Cham (2018a)Google Scholar
  4. Alessi, R., Marigo, J.-J., Maurini, C., Vidoli, S.: Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: one-dimensional examples. Int. J. Mech. Sci. 149, 559–576 (2018b)CrossRefGoogle Scholar
  5. Alessi, R., Vidoli, S., De Lorenzis, L.: A phenomenological approach to fatigue with a variational phase-field model: the one-dimensional case. Eng. Fract. Mech. 190, 53–73 (2018c)CrossRefGoogle Scholar
  6. Aubin, J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)MathSciNetzbMATHGoogle Scholar
  7. Bouchitté, G., Mielke, A., Roubíček, T.: A complete-damage problem at small strains. Z. Angew. Math. Phys. 60, 205–236 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Brezis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland, American Elsevier, Amsterdam-London, New York (1973)zbMATHGoogle Scholar
  9. Crismale, V.: Globally stable quasistatic evolution for a coupled elastoplastic-damage model. ESAIM Control Optim. Calc. Var. 22, 883–912 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Crismale, V.: Globally stable quasistatic evolution for strain gradient plasticity coupled with damage. Ann. Mat. Pura Appl. (4) 196, 641–685 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Crismale, V., Lazzaroni, G.: Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model. Calc. Var. Partial Differ. Equ. 55, Art. 17, 54 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Crismale, V., Orlando, G.: A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in \(W^{1, n}\). NoDEA Nonlinear Differ. Equ. Appl. 25, Art. 16, 20 (2018)CrossRefzbMATHGoogle Scholar
  13. Crismale, V., Lazzaroni, G., Orlando, G.: Cohesive fracture with irreversibility: quasistatic evolution for a model subject to fatigue. Math. Models Methods Appl. Sci. 28, 1371–1412 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  15. Dal Maso, G., DeSimone, A., Solombrino, F.: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differ. Equ. 40, 125–181 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dal Maso, G.: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution. Calc. Var. Partial Differ. Equ. 44, 495–541 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Dal Maso, G., Orlando, G., Toader, R.: Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case. Calc. Var. Partial Differ. Equ. 55, Art. 45, 39 (2016)MathSciNetzbMATHGoogle Scholar
  18. Duchoň, M., Maličký, P.: A Helly theorem for functions with values in metric spaces. Tatra Mt. Math. Publ. 44, 159–168 (2009)MathSciNetzbMATHGoogle Scholar
  19. Dunford, N., Schwartz, J.T.: Linear operators. Part I, Wiley Classics Library. John, New York (1988). General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience PublicationGoogle Scholar
  20. Francfort, G.A., Garroni, A.: A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182, 125–152 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Frémond, M.: Non-smooth Thermomechanics. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  22. Gröger, K.: A \(W^{1, p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283, 679–687 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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, 802–813 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23, 565–616 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Knees, D., Rossi, R., Zanini, C.: Balanced viscosity solutions to a rate-independent system for damage. Eur. J. Appl. Math. 29, 1–59 (2018)CrossRefzbMATHGoogle Scholar
  27. Mielke, A., Roubíček, T.: Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16, 177–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  29. Mielke, A., Zelik, S.: On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13, 67–135 (2014)MathSciNetzbMATHGoogle Scholar
  30. Mielke, A., Rossi, R., Savaré, G.: Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. (JEMS) 18, 2107–2165 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Minotti, L., Savaré, G.: Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems. Arch. Ration. Mech. Anal. 227, 477–543 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Negri, M.: A unilateral \(L^{2}\)-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement. Adv. Calc. Var. (2016).
  33. Nochetto, R.H., Savaré, G., Verdi, C.: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math. 53, 525–589 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Pham, K., Marigo, J.-J.: From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models. Contin. Mech. Thermodyn. 25, 147–171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Schijve, J.: Fatigue of Structures and Materials. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  36. Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Skibicki, D.: Phenomena and Computational Models of Non-proportional Loadings. Springer, Cham (2014)Google Scholar
  38. Stephens, R., Fatemi, A., Stephens, R., Fuchs, H.: Metal Fatigue in Engineering. A Wiley-Interscience publication. Wiley, New York (2000)Google Scholar
  39. Suresh, S.: Fatigue of Materials. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  40. Thomas, M.: Quasistatic damage evolution with spatial BV-regularization. Discrete Contin. Dyn. Syst. Ser. S 6, 235–255 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain—existence and regularity results. ZAMM Z. Angew. Math. Mech. 90, 88–112 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Civil and Industrial EngineeringUniversità di PisaPisaItaly
  2. 2.CMAP, École Polytechnique, UMR CNRS 7641Palaiseau CedexFrance
  3. 3.Zentrum MathematikTechnische Universität MünchenGarching bei MünchenGermany

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