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Continuum Mechanics and Thermodynamics

, Volume 29, Issue 3, pp 853–894 | Cite as

Stochastic homogenization of rate-independent systems and applications

  • Martin HeidaEmail author
Original Article

Abstract

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We prove some convergence results with respect to stochastic two-scale convergence, which are related to classical \(\Gamma \)-convergence results. The main result is a general \(\liminf \)-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rate-independent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandtl–Reuss plasticity, Tresca friction on a macroscopic surface and Tresca friction on microscopic fissures.

Keywords

Stochastic homogenization Rate-independent Hysteresis Two-scale convergence Plasticity Tresca friction Coulomb friction 

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Notes

Acknowledgements

This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 “Scaling Cascades in Complex Systems,” Project C05 Effective models for interfaces with many scales. The Author also thanks the reviewers for the very helpful suggestions.

References

  1. 1.
    Alber, H.-D.: Evolving microstructure and homogenization. Contin. Mech. Thermodyn. 12(4), 235–286 (2000)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alber, H.-D., Nesenenko, S.: Justification of homogenization in viscoplasticity: from convergence on two scales to an asymptotic solution in \({L}^2({\Omega })\). J. Multiscale Model. 1, 223–244 (2009)CrossRefGoogle Scholar
  3. 3.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ben-Zion, Y.: Collective behavior of earthquakes and faults: continuum-discrete transitions, progressive evolutionary changes, and different dynamic regimes. Rev. Geophys. 46(4), 1–70 (2008)CrossRefGoogle Scholar
  5. 5.
    Berberian, K.: Measure and Integration. Macmillan Company, London (1970)zbMATHGoogle Scholar
  6. 6.
    Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)zbMATHGoogle Scholar
  7. 7.
    Cioranescu, D., Damlamian, A., Orlik, J.: Homogenization via unfolding in periodic elasticity with contact on closed and open cracks. Asymptot. Anal. 82(3–4), 201–232 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, New York (1988)zbMATHGoogle Scholar
  9. 9.
    Goldfinger, Chris, Ikeda, Yasutaka, Yeats, Robert S., Ren, Junjie: Superquakes and supercycles. Seismol. Res. Lett. 84(1), 24–32 (2013)CrossRefGoogle Scholar
  10. 10.
    Hanke, H.: Rigorous derivation of two-scale and effective damage models based on microstructure evolution. Ph.D. thesis at Mathematisch-Naturwissenschaftliche Fakultät, Humbold University Berlin (2014)Google Scholar
  11. 11.
    Heida, M.: An extension of the stochastic two-scale convergence method and application. Asymptot. Anal. 72(1), 1–30 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Heida, M.: Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities. Appl. Anal. 91(7), 1243–1264 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Heida, M., Schweizer, B.: Stochastic homogenization of plasticity equations. ESAIM-COCV (Preprint) (2017)Google Scholar
  14. 14.
    Heida, Martin, Schweizer, Ben: Non-periodic homogenization of infinitesimal strain plasticity equations. ZAMM J. Appl. Math. Mech. 96(1), 5–23 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hummel, Hans-Karl: Homogenization for heat transfer in polycrystals with interfacial resistances. Appl. Anal. 75(3–4), 403–424 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hummel, H.K.: Homogenization of Periodic and Random Multidimensional Microstructures. Ph.D. thesis, Technische Universität Bergakademie Freiberg (1999)Google Scholar
  17. 17.
    Kelley, J.L.: General Topology. D. Van Nostrand Company, New York (1955)zbMATHGoogle Scholar
  18. 18.
    Krengel, Ulrich: Ergodic Theorems, vol. 6. Walter de Gruyter, Berlin (1985)CrossRefGoogle Scholar
  19. 19.
    Matheron, G.: Random sets and integral geometry. Wiley, USA (1975)Google Scholar
  20. 20.
    Mecke, J.: Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen. Probab. Theory Related Fields 9(1), 36–58 (1967)zbMATHGoogle Scholar
  21. 21.
    Mielke, A.: Evolution of rate-independent systems. Evolut. Equ. 2, 461–559 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mielke, A.: Deriving effective models for multiscale systems via evolutionary \(\Gamma \)-convergence In: Control of Self-Organizing Nonlinear Systems, pp 235–251. Springer International Publishing, Switzerland (2015)Google Scholar
  23. 23.
    Mielke, A., Roubicek, T.: Rate-Independent Systems. Springer, Berlin (2015)CrossRefGoogle Scholar
  24. 24.
    Nesenenko, S.: Homogenization in viscoplasticity. SIAM J. Math. Anal. 39(1), 236–262 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Orlik, J., Shiryaev, V.: Integral Methods in Science and Engineering: Theoretical and Computational Advances, chapter Evolutional Contact with Coulomb Friction on a Periodic Microstructure, pp. 455–470. Springer, Cham (2015)Google Scholar
  26. 26.
    Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pp. 835–873. North-Holland, Amsterdam (1981)Google Scholar
  27. 27.
    Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Schweizer, B., Veneroni, M.: Periodic homogenization of the Prandtl-Reuss model with hardening. J. Multiscale Model. 2, 69–106 (2010)CrossRefGoogle Scholar
  29. 29.
    Tempel’man, A.A.: Ergodic theorems for general dynamical systems. Trudy Moskovskogo Matematicheskogo Obshchestva 26, 95–132 (1972)MathSciNetGoogle Scholar
  30. 30.
    Valadier, M., Castaing, C.: Convex Analysis and Measurable Multi-functions. Springer, Berlin (1977)zbMATHGoogle Scholar
  31. 31.
    Visintin, A.: Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity. Contin. Mech. Thermodyn. 18(3–4), 223–252 (2006)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Visintin, A.: Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity. Proc. R. Soc. Edinb. Sect. A 138(6), 1363–1401 (2008)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Visintin, Augusto: Differential Models of Hysteresis, vol. 111. Springer, Berlin (1994)zbMATHGoogle Scholar
  34. 34.
    Zaehle, M.: Random processes of hausdorff rectifiable closed sets. Math. Nachr. 108, 49–72 (1982)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhikov, V.V.: On an extension of the method of two-scale convergence and its applications. Sb. Math. 191(7), 973–1014 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhikov, V.V., Kozlov, S.M., Olejnik, O.A.: Homogenization of differential operators and integral functionals. Transl. from the Russian by G. A. Yosifian. Springer, Berlin. xi, 570 p., (1994)Google Scholar
  37. 37.
    Zhikov, V.V., Pyatniskii, A.L.: Homogenization of random singular structures and random measures. Izv. Math. 70(1), 19–67 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.WIASBerlinGermany

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