Homogenization and the limit of vanishing hardening in Hencky plasticity with non-convex potentials

  • Martin Jesenko
  • Bernd Schmidt


We prove a homogenization result for Hencky plasticity functionals with non-convex potentials. We also investigate the influence of a small hardening parameter and show that homogenization and taking the vanishing hardening limit commute.

Mathematics Subject Classification

49J45 74C05 74Q05 



The authors wish to thank the unknown referee for the invested effort and the valuable suggestions.


  1. 1.
    Alberti, G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinb. Sect. A 123, 239–274 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alberti, G., Bianchini, S., Crippa, G.: On the \(L^{p}\)-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30(1), 349–367 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alibert, J., Bouchitté, G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4(1), 129–147 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139(3), 201–238 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, Oxford University Press, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Anzellotti, G., Giaquinta, M.: Existence of the displacement field for an elastoplastic body subject to Hencky’s law and von Mises yield condition. Manuscr. Math. 32(1–2), 101–136 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Anzellotti, G., Giaquinta, M.: On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pures Appl. (9) 61(3), 219–244 (1982)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization. The MPS-SIAM Series on Optimization. SIAM, Philadelphia (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Babadjian, J.: Traces of functions of bounded deformation. Indiana Univ. Math. J. 64(4), 1271–1290 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ball, J., Kirchheim, B., Kristensen, J.: Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11(4), 333–359(2000)Google Scholar
  11. 11.
    Barroso, A., Fonseca, I., Toader, R.: A relaxation theorem in the space of functions of bounded deformation. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 29(1), 19–49 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bartels, S., Mielke, A., Roubiček, T.: Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation. SIAM J. Numer. Anal. 50, 951–976 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bhattacharya, K.: Microstructure of Martensite. Why It Forms and How It Gives Rise to the Shape-memory Effect. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  14. 14.
    Borchers, W., Sohr, H.: On the equations \( {\rm rot} v = g \) and \( {\rm div} u = f \) with zero boundary conditions. Hokkaido Math. J. 19(1), 67–87 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bouchitté, G., Fonseca, I., Mascarenhas, L.: A global method for relaxation. Arch. Ration. Mech. Anal. 145(1), 51–98 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Braides, A.: Homogenization of some almost periodic functionals. Rend. Accad. Naz. Sci. XL 103, 313–322 (1985)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bredies, K.: Symmetric tensor fields of bounded deformation. Ann. Mat. Pura Appl. (4) 192(5), 815–851 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  19. 19.
    Demengel, F., Temam, R.: Convex functions of a measure and applications. Indiana Univ. Math. J. 33(5), 673–709 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Demengel, F., Temam, R.: Convex function of a measure: the unbounded case. In: Hiriart-Urruty, J. (ed.) FERMAT Days 85: Mathematics for Optimization. North-Holland, Amsterdam (1986)Google Scholar
  21. 21.
    Demengel, F., Qi, T.: Convex function of a measure obtained by homogenization. SIAM J. Math. Anal. 21(2), 409–435 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    De Philippis, G., Rindler, F.: On the structure of \({\fancyscript {A}}\)-free measures and applications. Ann. Math. (2) 184(3), 1017–1039 (2016)Google Scholar
  23. 23.
    De Philippis, G., Rindler, F.: Characterization of generalized Young measures generated by symmetric gradients. Arch. Ration. Mech. Anal. 224(3), 1087–1125 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    DiPerna, R., Majda, A.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Duvaut, G., Lions, J.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  26. 26.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam, Oxford; American Elsevier Publishing, New York (1976)Google Scholar
  27. 27.
    Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\) Spaces. Springer Monographs in Mathematics. Springer, New York (2007)zbMATHGoogle Scholar
  28. 28.
    Fonseca, I., Müller, S., Pedregal, P.: Analysis of oscillation and concentration effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011)zbMATHGoogle Scholar
  30. 30.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)zbMATHGoogle Scholar
  31. 31.
    Grubb, G.: Pseudo-differential boundary problems in \( L_{p} \) spaces. Commun. Partial Differ. Equ. 15(3), 289–340 (1990)CrossRefzbMATHGoogle Scholar
  32. 32.
    Hajlasz, P.: On approximate differentiability of functions with bounded deformation. Manuscr. Math. 91(1), 61–72 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Han, W., Reddy, B.: Plasticity. Mathematical Theory and Numerical Analysis. Springer, New York (1999)zbMATHGoogle Scholar
  34. 34.
    Hlávaček, I., Nečas, J.: Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. Elsevier, Amsterdam (1980)zbMATHGoogle Scholar
  35. 35.
    Jesenko, M.: PhD-thesis Universität Augsburg (2016)Google Scholar
  36. 36.
    Jesenko, M., Schmidt, B.: Closure and commutability results for \(\Gamma \)-limits and the geometric linearization and homogenization of multiwell energy functionals. SIAM J. Math. Anal. 46(4), 2525–2553 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Khachaturyan, A.G.: Some questions concerning the theory of phase transformations in solids. Sov. Phys. Solid State 8, 2163–2168 (1967)Google Scholar
  38. 38.
    Khachaturyan, A.G.: Theory of Structural Transformations in Solids. Wiley, New York (1983)Google Scholar
  39. 39.
    Khachaturyan, A.G., Shatalov, G.A.: Theory of macroscopic periodicity for a phase transition in the solid state. Sov. Phys. JETP 29, 557–561 (1969)Google Scholar
  40. 40.
    Kirchheim, B., Kristensen, J.: On rank one convex functions that are homogeneous of degree one. Arch. Ration. Mech. Anal. 221(1), 527–558 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kohn, R., Temam, R.: Dual spaces of stresses and strains, with applications to Hencky plasticity. Appl. Math. Optim. 10(1), 1–35 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kristensen, J.: Finite functionals and Young measures generated by gradients of Sobolev functions. Mat-report 1994-34. Mathematical Institute, Technical University of Denmark (1994)Google Scholar
  43. 43.
    Kristensen, J., Rindler, F.: Relaxation of signed integral functionals in \( BV \). Calc. Var. Partial Differ. Equ. 37(1–2), 29–62 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in \( W^{1,1} \) and \( BV \). Arch. Ration. Mech. Anal. 197(2), 539–598 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Licht, C., Michaille, G.: Global–local subadditive ergodic theorems and application to homogenization in elasticity. Ann. Math. Blaise Pascal 9(1), 21–62 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Marcellini, P.: Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. 4(117), 139–152 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mora, M.: Relaxation of the Hencky model in perfect plasticity. J. Math. Pures Appl. (9) 106(4), 725–743 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Müller, S.: Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Ration. Mech. Anal. 99(3), 189–212 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pedregal, P.: Parametrized Measures and Variational Principles. Birkhäuser Verlag, Basel (1997)CrossRefzbMATHGoogle Scholar
  50. 50.
    Roitburd, A.L.: Orientational and habit relationships between crystalline phases in solid state transformations. Soviet Phys. Crystallography. 12, 499 (1968) (English translation of Kristallografiya 12, 567 ff. (1967))Google Scholar
  51. 51.
    Roitburd, A.L.: Martensitic transformation as a typical phase transformation in solids. Solid state physics 33, 317–390 (1978)CrossRefGoogle Scholar
  52. 52.
    Schmidt, B.: \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn. 20(6), 375–396 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Suquet, P.: Sur les équations de la plasticité: existence et rgularit des solutions. J. Méc. 20(1), 3–39 (1981)zbMATHGoogle Scholar
  54. 54.
    Temam, R.: Mathematical Problems in Plasticity. Gauthier-Villars, Paris (1985)zbMATHGoogle Scholar
  55. 55.
    Wu, Z., Yin, J., Wang, C.: Elliptic & Parabolic Equations. World Scientific, Hackensack (2006)CrossRefzbMATHGoogle Scholar
  56. 56.
    Zhang, K.: An approximation theorem for sequences of linear strains and its applications. ESAIM Control Optim. Calc. Var. 10, 224–242 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany
  2. 2.Institut für MathematikUniversität AugsburgAugsburgGermany

Personalised recommendations