Skip to main content
SpringerLink
Log in

Gradient Damage Models Coupled with Plasticity and Nucleation of Cohesive Cracks

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In the framework of rate-independent systems, a family of elastic-plastic-damage models is proposed through a variational formulation. Since the goal is to account for softening behaviors until the total failure, the dissipated energy contains a gradient damage term in order to limit localization effects. The resulting model owns a great flexibility in the possible coupled responses, depending on the constitutive parameters. Moreover, considering the one-dimensional quasi-static problem of a bar under simple traction and constructing solutions with localization of damage, it turns out that in general a cohesive crack appears at the center of the damage zone before the rupture. The associated cohesive law is obtained in a closed form in terms of the parameters of the model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L., Lemenant, A., Royer-Carfagni, G.: A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence. J. Elast. 1–35 (2012). doi:10.1007/s10659-012-9390-5

  2. Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via Γ-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990). doi:10.1002/cpa.3160430805

  3. Amor H, Marigo J.J, Maurini C.: Variational approach to brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids 57(8), 1209–1229 (2009)

    Article  ADS  MATH  Google Scholar 

  4. Barenblatt, G.I.: The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Elsevier, New York, 55–129, 1962. doi:10.1016/S0065-2156(0870121-2)

  5. Benallal A, Marigo J.J.: Bifurcation and stability issues in gradient theories with softening. Model. Simul. Mater. Sci. Eng. 15(1), S283–S295 (2007)

    Article  ADS  Google Scholar 

  6. Bourdin, B., Francfort, G.A., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48(4), 797–826 (2000)

  7. Bourdin B, Francfort G.A, Marigo J.J.: The variational approach to fracture. J. Elast. 91(1–3), 5–148 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Braides, A.: Γ-convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, Vol 22. Oxford University Press, Oxford, (2002)

  9. Comi, C.: Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mech. Cohesive Frict. Mater. 4(1), 17–36 (1999). doi:10.1002/(SICI)1099-1484(199901)4:1<17::AID-CFM55>3.0.CO;2-6

  10. Comi, C., Mariani, S., Negri, M., Perego, U.: A one-dimensional variational formulation for quasi-brittle fracture. J. Mech. Mater. Struct. 1(8), 1323–1343 (2006). doi:10.2140/jomms.2006.1.1323

  11. Dal Maso, G., DeSimone, A., Mora, M.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180(2), 237–291 (2006). doi:10.1007/s00205-005-0407-0

  12. Dal-Maso G, Toader R.: A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Rational Mech. Anal. 162, 101–135 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Del Piero, G.: A variational approach to fracture and other inelastic phenomena. J. Elast 112(1), 3–77 (2013). doi:10.1007/s10659-013-9444-3

  14. Del Piero, G., Lancioni, G., March, R.: Diffuse cohesive energy in plasticity and fracture. Tech. Mech. 32(2), 174–188 (2012)

  15. Del Piero, G., Truskinovsky, L.: Elastic bars with cohesive energy. Contin. Mech. Thermodyn. 21, 141–171 (2009)

  16. Dimitrijevic, B.J., Hackl, K.: A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Methods Biomed. Eng. 27, 1199—1210 (2011)

  17. Dugdale, D.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960). doi:10.1016/0022-5096(60)90013-2

  18. Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, (1992)

  19. Francfort, G.A., Giacomini, A.: Small-strain heterogeneous elastoplasticity revisited. Commun. Pure Appl. Math. LXV, 1185—1241 (2012)

  20. Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998). doi:10.1016/S0022-5096(98)00034-9

  21. Germain, P., Nguyen, Q.S., Suquet, P.: Continuum thermodynamics. J. Appl. Mech. 50, 1010–1020 (1983)

  22. Lemaitre, J., Chaboche, J.: Mécanique des matériaux solides. Bordas (1985)

  23. Lorentz, E., Cuvilliez, S., Kazymyrenko, K.: Convergence of a gradient damage model toward a cohesive zone model. Comptes Rendus Mécanique 339(1), 20–26 (2011). doi:10.1016/j.crme.2010.10.010

  24. Marigo, J.J.: Formulation d’une loi d’endommagement d’un matériau élastique. Comptes Rendus de l’Académie des Sciences de Paris Série II 292(19), 1309–1312 (1981)

  25. Marigo, J.J.: Constitutive relations in plasticity, damage and fracture mechanics based on a work property. Nucl. Eng. Des. 114, 249–272 (1989)

  26. Mielke, A.: Evolution of rate-independent systems. North-Holland, Amsterdam, 461–559 (2005). doi:10.1016/S1874-5717(06)80009-5

  27. Peerlings, R., de Borst, R., Brekelmans, W., Geers, M.: Gradient-enhanced damage modelling of concrete fracture. Mech. Cohes. Frict. Mater. 3, 323–342 (1998)

  28. Pham, K., Amor, H., Marigo, J.J., Maurini, C.: Gradient damage models and their use to approximate brittle fracture. Int. J. Damage Mech. 20(4), 618–652 (2011). doi:10.1177/1056789510386852

  29. Pham, K., Marigo, J.J.: Approche variationnelle de l’endommagement: I. Les concepts fondamentaux. Comptes Rendus Mécanique 338(4), 191–198 (2010)

  30. Pham, K., Marigo, J.J.: Approche variationnelle de l’endommagement: II. Les modèles à à gradient. Comptes Rendus Mécanique 338(4), 199–206 (2010)

  31. Pham, K., Marigo, J.J.: From the onset of damage until the rupture: construction of the responses with damage localization for a general class of gradient damage models. Continuum Mech. Thermodyn. 25(2–4), 147–171 (2013)

  32. Pham K, Marigo J.J.: Stability of homogeneous states with gradient damage models: size effects and shape effects in the three-dimensional setting. J. Elast. 110(1), 63–93 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pham, K., Marigo, J.J., Maurini, C.: The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. J. Mech. Phys. Solids 59(6), 1163–1190 (2011). doi:10.1016/j.jmps.2011.03.010

  34. Sicsic, P., Marigo, J.J.: From gradient damage laws to Griffith’s theory of crack propagation. J. Elast. 113(1), 55–74 (2013)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Università di Roma, Via Eudossiana 18, 00184, Rome, Italy

    Roberto Alessi & Stefano Vidoli

  2. Laboratoire de Mécanique des Solides, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Jean-Jacques Marigo

Authors
  1. Roberto Alessi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Jacques Marigo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Vidoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Jacques Marigo.

Additional information

Communicated by G. Dal Maso

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alessi, R., Marigo, JJ. & Vidoli, S. Gradient Damage Models Coupled with Plasticity and Nucleation of Cohesive Cracks. Arch Rational Mech Anal 214, 575–615 (2014). https://doi.org/10.1007/s00205-014-0763-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0763-8

Search

Navigation