Skip to main content
SpringerLink
Log in

Quasistatic evolution in non-associative plasticity revisited

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

The mathematical treatment of evolutionary non-associative elasto-plasticity is still in its infancy. In particular, all existence results thus far rely on a spatially mollified stress admissibility constraint. Further, the evolution is formulated in a rescaled time from which it is very difficult to infer any useful information on the “real” time evolution. We propose a causal spatio-temporal mollification of the stress admissibility constraint that, while no more far-fetched than a purely spatial one, produces a more elegant and complete evolution for such models, and this in the “real” time variable.

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

Notes

  1. The Armstrong–Frederick model is the simplest plastic model that phenomenologically captures the so-called Bauschinger effect, a kind of hysteretic behavior often observed in metals under cyclic loadings [11].

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, Oxford (2000)

    MATH  Google Scholar 

  2. Babadjian, J.-F., Francfort, G.A., Mora, M.G.: Quasistatic evolution in non-associative plasticity—the cap model. SIAM J. Math. Anal. 44, 245–292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dal Maso, G., DeSimone, A., Mora, M.G.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180, 237–291 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dal Maso, G., DeSimone, A., Solombrino, F.: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions. Calc. Var. Partial Differ. Equ. 44, 495–541 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dal Maso, G., Scala, R.: Quasistatic evolution in perfect plasticity as limit of dynamic processes. J. Dyn. Differ. Equ. 26, 915–954 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dal Maso, G., Solombrino, F.: Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case. Netw. Heterog. Media 5, 97–132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Efendiev, M., Mielke, A.: On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13, 151–167 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Francfort, G.A., Giacomini, A.: Small strain heterogeneous elasto-plasticity revisited. Commun. Pure Appl. Math. 65–9, 1185–1241 (2012)

    Article  MATH  Google Scholar 

  10. Francfort, G.A., Stefanelli, U.: Quasi-static evolution for the Armstrong–Frederick hardening-plasticity model. Appl. Math. Res. Express AMRX 2013(2), 297–344 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Frederick, C.O., Armstrong, P.J.: A mathematical representation of the multiaxial Bauschinger effect, CEGB Report RD/B/N731, 1966, reproduced with foreword in Mater. High Temp. 24-1, 1–26 (2007)

  12. Laborde, P.: A nonmonotone differential inclusion. Nonlinear Anal. Theory Math. Appl. 11, 757–767 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lubliner, J.: Plasticity Theory. Macmillan Publishing Company, New York (1990)

    MATH  Google Scholar 

  14. Mainik, A., Mielke, A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ. 22, 73–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mielke, A.: Evolution of rate-independent systems. In: Dafermos, C.M., Feireisl, E. (eds.) Evolutionary Equations. Handbook of Differential Equations, vol. II, pp. 461–559. Elsevier/North-Holland, Amsterdam (2005)

    Chapter  Google Scholar 

  16. Mielke, A., Rossi, R., Savaré, G.: Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. 25, 585–615 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mielke, A., Rossi, R., Savaré, G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. 18, 36–80 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mielke, A., Rossi, R., Savaré, G.: Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. 18, 2107–2165 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  20. Solombrino, F.: Quasistatic evolution problems for nonhomogeneous elastic plastic materials. J. Convex Anal. 16, 89–119 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Suquet, P.-M.: Sur les équations de la plasticité: existence et régularité des solutions. J. Méc. 20, 3–39 (1981)

    MATH  Google Scholar 

  22. Temam, R.: Mathematical problems in plasticity. Gauthier-Villars, Paris (1985). Translation of Problèmes mathématiques en plasticité, Gauthier-Villars, Paris (1983)

Download references

Acknowledgements

The research has been partially supported by the National Science Fundation Grant DMS-1615839, by INdAM–GNAMPA under Project 2016 “Multiscale analysis of complex systems with variational methods”, and by the Università degli Studi di Pavia through the 2017 Blue Sky Research Project “Plasticity at different scales: micro to macro”.

Author information

Authors and Affiliations

  1. LAGA, Université Paris-Nord, Avenue J.-B. Clément, 93430, Villetaneuse, France

    G. A. Francfort

  2. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    G. A. Francfort

  3. Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100, Pavia, Italy

    M. G. Mora

Authors
  1. G. A. Francfort
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. G. Mora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. G. Mora.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Francfort, G.A., Mora, M.G. Quasistatic evolution in non-associative plasticity revisited. Calc. Var. 57, 11 (2018). https://doi.org/10.1007/s00526-017-1284-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1284-8

Mathematics Subject Classification

Search

Navigation