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Plastic Flow as an Energy Minimization Problem. Numerical Experiments


In this approach, the plastic part of the deformation field, traditionally described by regular mappings, is interpreted as localized yielding along flow surfaces, with a kinematics analogous to that of crack formation. The resulting deformation is structured, being composed of a bulk and a surface part, respectively due to the elastic distortion of massive material portions and to localized yielding. There is an energetic competition between these two contributions in the energy functional, whose minimization is sought under irreversibility conditions for the inelastic phenomena. Numerical experiments are performed with a regularized variational approach. Paradigmatic examples show that plastic strain concentrates in coarse bands, but the bands may coalesce to form a plastic region, depending upon the shape and size of the body, the presence of pre-existing defects (voids, holes, notches) and the values of the governing parameters.

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  1. 1.

    Recall that the model of [9] is symmetric in tension-compression, so that it does not rule out overlapping of crack lips.


  1. 1.

    Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. XLIII, 999–1036 (1990)

    Article  MathSciNet  Google Scholar 

  2. 2.

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

    MATH  Google Scholar 

  3. 3.

    Ambrosio, L., Lemenant, A., Royer-Carfagni, G.: A variational model for plastic slip and its regularization via Γ-convergence. J. Elast. 110, 201–235 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Bangerth, W., Hartmann, R., Kanschat, G.: deal.II differential equations analysis library, technical reference.

  5. 5.

    Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)

    Article  MathSciNet  Google Scholar 

  6. 6.

    Bažant, Z., Planas, S.T.: Fracture and size-effect in concrete and other Quasi-Brittle materials. CRC Press, New York (1998)

    Google Scholar 

  7. 7.

    Benzarti, K., Freddi, F., Frémond, M.: A damage model to predict the durability of bonded assemblies. Part I: Debonding behavior of FRP strengthened concrete structures. Constr. Build. Mater. 25(2), 547–555 (2011)

    Article  Google Scholar 

  8. 8.

    Bigoni, D., Dal Corso, F.: The unrestrainable growth of a shear band in a prestressed material. Proc. R. Soc. A, Math. Phys. Eng. Sci. 464, 2365–2390 (2008)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  9. 9.

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

    ADS  Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Del Piero, G.: A variational approach to fracture and other inelastic phenomena. J. Elast. 112, 3–77 (2013)

    Article  MATH  Google Scholar 

  11. 11.

    Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960)

    ADS  Article  Google Scholar 

  12. 12.

    Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)

    Article  Google Scholar 

  13. 13.

    Focardi, M.: On the variational approximation of free-discontinuity problems in the vectorial case. Math. Models Methods Appl. Sci. 11, 663–684 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Francfort, G.A., Marigo, J.J.: Revisiting Brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Franco, A., Royer-Carfagni, G.: Energetic balance in the debonding of a reinforcing stringer: effect of the substrate elasticity. Int. J. Solids Struct. 50, 1954–1965 (2013)

    Article  Google Scholar 

  16. 16.

    Freddi, F., Frémond, M.: Damage in domains and interfaces: a coupled predictive theory. J. Mech. Mater. Struct. 1(7), 1205–1233 (2006)

    Article  Google Scholar 

  17. 17.

    Freddi, F., Royer-Carfagni, G.: Regularized variational theories of fracture: a unified approach. J. Mech. Phys. Solids 58, 1154–1174 (2010)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Freddi, F., Royer-Carfagni, G.: Variational fracture mechanics to model compressive splitting of masonry-like materials. Ann. Solid Struct. Mech. 2, 57–67 (2011)

    Article  Google Scholar 

  19. 19.

    Frémond, M.: Non-smooth Thermomechanics. Springer, Heidelberg (2001)

    Google Scholar 

  20. 20.

    Froli, M., Royer-Carfagni, G.: On discontinuous deformation of tensile steel bars: experimental results. J. Eng. Mech. 125, 1243–1250 (1999)

    Article  Google Scholar 

  21. 21.

    Froli, M., Royer-Carfagni, G.: A mechanical model for the elastic-plastic behavior of metallic bars. Int. J. Solids Struct. 37, 3901–3918 (2000)

    Article  MATH  Google Scholar 

  22. 22.

    Körber, F., Siebel, E.: Zur theorie der bildsamen formänderung. Naturwissenschaften 16, 408–412 (1928)

    ADS  Article  Google Scholar 

  23. 23.

    Lancioni, G., Royer-Carfagni, G.: The variational approach to fracture mechanics. A practical applicaton to the French Panthéon in Paris. J. Elast. 95, 1–30 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Nadai, A.: Theory of Flow and Fracture of Solids. McGraw-Hill, New York (1950)

    Google Scholar 

  25. 25.

    Nakanishi, F.: On yield point of mild steel. Rep. Aeronaut. Res. Inst. 6, 83–140 (1931)

    Google Scholar 

  26. 26.

    Ord, A., Vardoulakis, I., Kajewski, R.: Shear band formation in Gosford Sandstone. Int. J. Rock Mech. Min. Sci. 28, 397–409 (1991)

    Article  Google Scholar 

  27. 27.

    Palmer, A.C., Rice, J.R.: The growth of slip surfaces in the progressive failure of over-consolidated clay. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 332, 527–548 (1973)

    ADS  Article  MATH  Google Scholar 

  28. 28.

    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, 1163–1190 (2011)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  29. 29.

    Rice, J.: The localization of plastic deformation. In: Koiter, W.T. (ed.) Proceedings of the 14th International Congress on Theoretical and Applied Mechanics, Vol. I, pp. 207–220. North-Holland, Amsterdam (1976)

    Google Scholar 

  30. 30.

    Rittel, D., Wang, Z.G., Merzer, M.: Adiabatic shear failure and dynamic stored energy of cold work. Phys. Rev. Lett. 96, 075502 (2006)

    ADS  Article  Google Scholar 

  31. 31.

    Wolf, H., König, D., Triantafyllidis, T.: Experimental investigation of shear band patterns in granular material. J. Struct. Geol. 25, 1229–1240 (2003)

    ADS  Article  Google Scholar 

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The authors gratefully acknowledge the Research Fund for Coal and Steel for support under grant RFSR-CT-2012-00026 (“S+G” research project).

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Correspondence to Gianni Royer-Carfagni.

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Freddi, F., Royer-Carfagni, G. Plastic Flow as an Energy Minimization Problem. Numerical Experiments. J Elast 116, 53–74 (2014).

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  • Variational approach
  • Plasticity
  • Yielding
  • Flow lines
  • Fracture

Mathematics Subject Classification

  • 28A75
  • 35R35
  • 74G65
  • 74R05
  • 74R10
  • 74R20