Skip to main content
SpringerLink
Log in

A micromechanical approach for the Cosserat modeling of composites

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

The present paper deals with the homogenization problem of periodic composite materials, considering a Cosserat continuum at the macro-level and a Cauchy continuum at the micro-level. Consistently with the strain-driven approach, the two levels are linked by a kinematic map based on a third order polynomial expansion. Because of the assumed regular texture of the composite material, a Unit Cell (UC) is selected; then, the problem of determining the displacement perturbation fields, arising when second or third order polynomial boundary conditions are imposed on the UC, is investigated. A new micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is proposed. The identification of the linear elastic 2D Cosserat constitutive parameters is performed, by using the Hill–Mandel technique, based on the macrohomogeneity condition. The influence of the selection of the UC is analyzed and some critical issues are outlined. Numerical examples for a specific composite with cubic symmetry are shown.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Addessi D, De Bellis ML, Sacco E (2013) Micromechanical analysis of heterogeneous materials subjected to overall cosserat strains. Mech Res Commun 54:27–34

    Article  Google Scholar 

  2. Addessi D, Sacco E (2012) A multi-scale enriched model for the analysis of masonry panel. Int J Solids Struct 49:865–880

    Article  Google Scholar 

  3. Anthoine A (1995) Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int J Solids Struct 32(2):137–163

    Article  MATH  Google Scholar 

  4. Bacigalupo A (2014) Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits. Meccanica 49(6):1407–1425

    Article  MathSciNet  MATH  Google Scholar 

  5. Bacigalupo A, Gambarotta L (2010) Second-order computational homogenization of heterogeneous materials with periodic microstructure. ZAMM Z Angew Math Mech 90(10–11):796–811

    Article  MathSciNet  MATH  Google Scholar 

  6. Bacigalupo A, Gambarotta L (2012) Computational two-scale homogenization of periodic masonry: characteristic lengths and dispersive waves. Comput Method Appl Mech 213–216:16–28

    Article  MathSciNet  Google Scholar 

  7. Bacigalupo A, Gambarotta L (2013) Multi-scale strain-localization analysis of a layered strip with debonding interfaces. Int J Solids Struct 50:2061–2077

    Article  Google Scholar 

  8. Bakhvalov N, Panasenko G (1989) Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials. Springer, New York

    MATH  Google Scholar 

  9. Bigoni D, Drugan WJ (2007) Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J Appl Mech 74:741–753

    Article  MathSciNet  Google Scholar 

  10. Boutin C (1996) Microstructural effects in elastic composites. Int J Solids Struct 33(7):1023–1051

    Article  MATH  Google Scholar 

  11. Bouyge F, Jasiuk I, Boccara S, Ostoja-Starzewski M (2002) A micromechanically based couple-stress model of an elastic orthotropic two-phase composite. Eur J Mech A Solid 21:465–481

    Article  MATH  Google Scholar 

  12. Bouyge F, Jasiuk I, Ostoja-Starzewski M (2001) A micromechanically based couple-stress model of an elastic two-phase composite. Int J Solids Struct 38:1721–1735

    Article  MATH  Google Scholar 

  13. De Bellis M, Addessi D (2011) A Cosserat based multi-scale model for masonry structures. Int J Multiscale Comput 9(5):543–563

    Article  Google Scholar 

  14. Feyel F (2003) A multilevel finite element method (fe2) to describe the response of highly non-linear structures using generalized continua. Comput Method Appl Mech 192:3233–3244

    Article  MATH  Google Scholar 

  15. Forest S (1998) Mechanics of generalized continua: construction by homogeneization. J. Phys. IV France 8

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaczmarczyk L, Pearce C, Bicanic N (2008) Scale transition and enforcement of rve boundary conditions in second-order computational homogenization. Int J Numer Methods Eng 74:506–522

    Article  MATH  Google Scholar 

  19. Kouznetsova V, Geers M, Brekelmans W (2004a) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Method Appl Mech 193:5525–5550

    Article  MATH  Google Scholar 

  20. Kouznetsova VG (2002) Computational homogenization for the multi-scale analysis of multi-phase materials. Ph.D. thesis, Technische Universiteit Eindhoven

  21. Kouznetsova VG, Geers MGD, Brekelmans WAM (2004b) Size of a representative volume element in a second-order computational homogenization framework. Int J Multiscale Comput Eng 2:575–598

    Article  Google Scholar 

  22. Li J (2011) A micromechanics-based strain gradient damage model for fracture prediction of brittle materials part I: homogenization methodology and constitutive relations. Int J Solids Struct 48(24):3336–3345

    Article  Google Scholar 

  23. Miehe C (2002) Strain-driven homogenization of inelastic microstructures andcomposites based on an incremental variational formulation. Int J Numer Methods Eng 55:1285–1322

    Article  MathSciNet  MATH  Google Scholar 

  24. Sánchez-Palencia E (1980) Non homogeneous media and vibration theory, vol 127 of Lecture Notes in Physics. Springer, Berlin

  25. Sanchez-Palencia E (1987) Boundary layers and edge effects in composites. In: Sánchez-Palencia E, Zaoui A (eds.) Homogenization techniques for composite media. Lecture notes in physics 272. Springer, Berlin, p 121–192

  26. Smyshlyaev VP, Cherednichenko K (2000) On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. J Mech Phys Solids 48:1325–1357

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Suquet P (1987) Elements of homogenization for inelastic solid mechanics. In: Sánchez-Palencia E, Zaoui A (eds.) Homogenization Techniques for Composite Media. Lecture Notes in Physics 272. Springer, Berlin, p 193–278

  28. Taylor R (2011) FEAP-A finite element analysis program, Version 8.3. Department of Civil and Environmental Engineering, University of California at Berkeley, California

  29. Tran TH, Monchiet V, Bonnet G (2012) A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media. Int J Solids Struct 49:783–792

    Article  Google Scholar 

  30. Trovalusci P, Masiani R (2003) Non-linear micropolar and classical continua for anisotropic discontinuous materials. Int J Solids Struct 40(5):1281–1297

    Article  MATH  Google Scholar 

  31. van der Sluis O, Schreurs PJG, Brekelmans WAM, Meijer HEH (2000) Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling. Mech Mater 32:449–462

    Article  Google Scholar 

  32. van der Sluis O, Vosbeek PHJ, Schreurs PJG, Meijer HEH (1999) Homogenization of heterogeneous polymers. Int J Solids Struct 36:3193–3214

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Strutturale e Geotecnica Sapienza, Universitá di Roma, via Eudossiana 18, 00184, Rome, Italy

    Daniela Addessi & Maria Laura De Bellis

  2. Dipartimento di Ingegneria Civile e Meccanica, Universitá di Cassino e del Lazio Meridionale, Via G.Di Biasio 43, 03043, Cassino, Italy

    Elio Sacco

Authors
  1. Daniela Addessi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Laura De Bellis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elio Sacco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniela Addessi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Addessi, D., De Bellis, M.L. & Sacco, E. A micromechanical approach for the Cosserat modeling of composites. Meccanica 51, 569–592 (2016). https://doi.org/10.1007/s11012-015-0224-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-015-0224-y

Keywords

Search

Navigation