Skip to main content
SpringerLink
Log in

Micromorphic homogenization of a porous medium: elastic behavior and quasi-brittle damage

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

Today it is well known that the classical Cauchy continuum theory is insufficient to describe the deformation behavior of solids if gradients occur over distances which are comparable to the microstructure of the material. This becomes crucial e.g., for small specimens or during localization of deformation induced by material degradation (damage). Higher-order continuum approaches like micromorphic theories are established to address such problems. However, such theories require the formulation of respective constitutive laws, which account for the microstructural interactions. Especially in damage mechanics such laws are mostly formulated in a purely heuristic way, which leads to physical and numerical problems. In the present contribution, the fully micromorphic constitutive law for a porous medium is obtained in closed form by homogenization based on the minimal boundary conditions concept. It is shown that this model describes size effects of porous media like foams adequately. The model is extended toward quasi-brittle damage overcoming the physical and numerical limitations of purely heuristic approaches.

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. Eringen A.C., Suhubi E.S.: Nonlinear theory of simple micro-elastic solids—I. Int. J. Eng. Sci. 2(2), 189–203 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  2. Lakes R.S.: Size effects and micromechanics of a porous solid. J. Mater. Sci. 18(9), 2572–2580 (1983)

    Article  ADS  Google Scholar 

  3. Lakes R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22(1), 55–63 (1986)

    Article  Google Scholar 

  4. Tekoğlu C., Gibson L., Pardoen T., Onck P.: Size effects in foams: experiments and modeling. Prog. Mater. Sci. 56(2), 109–138 (2011)

    Article  Google Scholar 

  5. Branke, D.: Homogenisierungsmethode für den Übergang vom CAUCHY- zum COSSERAT-Kontinuum. Dissertation; TU Dresden (2012)

  6. Tekoğlu C., Onck P.R.: Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models. J. Mech. Phys. Solids 56(12), 3541–3564 (2008)

    Article  MATH  ADS  Google Scholar 

  7. de Borst R., Sluys L., Muhlhaus H.B., Pamin J.: Fundamental issues in finite element analyses of localization of deformation. Eng. Comput. 10(2), 99–121 (1993)

    Article  Google Scholar 

  8. Forest S., Sab K.: Cosserat overall modeling of heterogeneous materials. Mech. Res. Commun. 25(4), 449–454 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jänicke R., Diebels S., Sehlhorst H.G., Düster A.: Two-scale modelling of micromorphic continua. Continuum Mech. Thermodyn. 21(4), 297–315 (2009)

    Article  MATH  ADS  Google Scholar 

  10. Jänicke R., Steeb H.: Wave propagation in periodic microstructures by homogenisation of extended continua. Comput. Mater. Sci. 52(1), 209–211 (2012)

    Article  Google Scholar 

  11. Zybell L., Mühlich U., Kuna M.: Constitutive equations for porous plane-strain gradient elasticity obtained by homogenization. Arch. Appl. Mech. 79(4), 359–375 (2009)

    Article  MATH  ADS  Google Scholar 

  12. Gologanu, M., Leblond, J., Perrin, G., Devaux, J.: Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P., (ed.) Continuum micromechanics. No. 377 in Cism Courses And Lectures: International Centre For Mechanical Sciences Series, pp. 61–130. Springer, New York (1997)

  13. Bigoni D., Drugan W.J.: Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech. 74(4), 741–753 (2006)

    Article  MathSciNet  Google Scholar 

  14. Yuan X., Tomita Y., Andou T.: A micromechanical approach of nonlocal modeling for media with periodic microstructures. Mech. Res. Commun. 35(1–2), 126–133 (2008)

    Article  MATH  Google Scholar 

  15. Li J.: Establishment of strain gradient constitutive relations by homogenization. Comptes Rendus Mécanique 339(4), 235–244 (2011)

    Article  MATH  ADS  Google Scholar 

  16. Forest S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)

    Article  Google Scholar 

  17. Forest, S., Ammar, K., Appolaire, B.: Micromorphic vs. phase-field approaches for gradient viscoplasticity and phase transformations. In: Markert B., (ed.) Advances in Extended and Multifield Theories for Continua; Lecture Notes in Applied and Computational Mechanics, vol. 59, pp. 69–88. Springer, Berlin (2011)

  18. Hütter G., Linse T., Mühlich U., Kuna M.: Simulation of ductile crack initiation and propagation by means of a non-local GTN-model under small-scale yielding. Int. J. Solids Struct. 50, 662–671 (2013)

    Article  Google Scholar 

  19. Forest S., Sievert R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43(24), 7224–7245 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Germain P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25(3), 556–575 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  21. Forest S.: Homogenization methods and the mechanics of generalized continua—part 2. Theor. Appl. Mech. 28-29, 113–143 (2002)

    Article  MathSciNet  Google Scholar 

  22. Forest S., Trinh D.K.: Generalized continua and non-homogeneous boundary conditions in homogenisation methods. Z. Angew. Math. Mech. 91(2), 90–109 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jänicke R., Steeb H.: Minimal loading conditions for higher order numerical homogenisation schemes. Arch. Appl. Mech. 82(8), 1075–1088 (2012)

    Article  MATH  Google Scholar 

  24. Forest S. : Mechanics of Cosserat media—an introduction. Ecole des Mines de Paris, Paris (2005)

  25. Jänicke, R.: Micromorphic media: interpretation by homogenisation. Dissertation; Universität des Saarlandes; Saarbrücken (2010)

  26. Mühlich U., Zybell L., Kuna M.: Estimation of material properties for linear elastic strain gradient effective media. Eur. J. Mech. A-Solid 31(1), 117–130 (2012)

    Article  MATH  Google Scholar 

  27. Aifantis E.C.: The physics of plastic deformation. Int. J. Plast. 3(3), 211–247 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  28. Forest S.: Questioning size effects as predicted by strain gradient plasticity. J. Mech. Behav. Mater. 22, 101–110 (2013)

    Article  Google Scholar 

  29. Aifantis K.E., Willis J.R.: The role of interfaces in enhancing the yield strength of composites and polycrystals. J. Mech. Phys. Solids 53(5), 1047–1070 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Mazière, M., Forest, S.: Strain gradient plasticity modeling and finite element simulation of Lüders band formation and propagation. Continuum Mech. Thermodyn. (2013). doi:10.1007/s00161-013-0331-8

  31. Mühlich U., Zybell L., Hütter G., Kuna M.: A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids. Arch. Appl. Mech. 83, 955–967 (2013)

    Article  MATH  Google Scholar 

  32. Waseem A., Beveridge A., Wheel M., Nash D.: The influence of void size on the micropolar constitutive properties of model heterogeneous materials. Eur. J. Mech. A-Solid 40, 148–157 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  33. Jänicke R., Sehlhorst H.G., Duster A., Diebels S.: Micromorphic two-scale modelling of periodic grid structures. Int. J. Multiscale Comput. Eng. 11(2), 161–176 (2013)

    Article  Google Scholar 

  34. Lemaitre J., Chaboche J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  35. Geers M.G.D., Borst R.de, Brekelmans W.A.M., Peerlings R.H.J.: Strain-based transient-gradient damage model for failure analyses. Comput. Methods Appl. Mech. Eng. 160(1–2), 133–153 (1998)

    Article  MATH  ADS  Google Scholar 

  36. Geers M.G.D., Peerlings R.H.J., Brekelmans W.A.M., Borst R.de : Phenomenological nonlocal approaches based on implicit gradient-enhanced damage. Acta Mech. 144(1), 1–15 (2000)

    Article  MATH  Google Scholar 

  37. Dimitrijevic B.J., Hackl K.: A method for gradient enhancement of continuum damage models. Tech. Mech. 28(1), 43–52 (2008)

    Google Scholar 

  38. Jirásek M.: Nonlocal models for damage and fracture: comparison of approaches. Int. J. Solids Struct. 35(31–32), 4133–4145 (1998)

    Article  MATH  Google Scholar 

  39. Simone A., Wells G.N., Sluys L.J.: From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput. Methods Appl. Mech. Eng. 192(41–42), 4581–4607 (2003)

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. TU Bergakademie Freiberg, Freiberg, Germany

    Geralf Hütter, Uwe Mühlich & Meinhard Kuna

Authors
  1. Geralf Hütter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Mühlich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meinhard Kuna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Geralf Hütter.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hütter, G., Mühlich, U. & Kuna, M. Micromorphic homogenization of a porous medium: elastic behavior and quasi-brittle damage. Continuum Mech. Thermodyn. 27, 1059–1072 (2015). https://doi.org/10.1007/s00161-014-0402-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-014-0402-5

Keywords

Search

Navigation