Skip to main content
SpringerLink
Log in

Analytical and Numerical Investigations of Two Special Classes of Generalized Continuum Media

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

In this paper, the micromorphic theory and the second gradient theory are proposed-where the micromorphic model can be reduced to the second gradient model with the vanishing relative deformation between macrodeformation gradient and microdeformation. Analytical solutions for the simple shear problem in the case of a general small strain isotropic elasticity micromorphic model and the second gradient model are presented, respectively. Besides, uniaxial tension of a constrained layer with two different boundary conditions is also analytically solved. Finally, the micromorphic theory is implemented numerically within a two-dimensional plane strain finite element framework by developing two isoparametric elements.

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. Nix, W.D., Mechanical properties of thin films. Metallurgical and Materials Transactions A, 1989, 20(11): 2217–2245.

    Article  Google Scholar 

  2. Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W., Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 1994, 42(2): 475–488.

    Article  Google Scholar 

  3. Olurin, O.B., Fleck, N.A. and Ashby, M.F., Indentation resistance of an aluminium foam. Scripta Materialia, 2000, 43(11): 983–989.

    Article  Google Scholar 

  4. Wu, B. and Wei, Y., Simulations of mechanical behavior of polycrystalline copper with nano-twins. Acta Mechanica Solida Sinica, 2008, 21(3): 189–197.

    Article  MathSciNet  Google Scholar 

  5. Ericksen, J.L. and Truesdell, C., Exact theory of stress and strain in rods and shells. Archive for Rational Mechanics and Analysis, 1957, 1(1): 295–323.

    Article  MathSciNet  Google Scholar 

  6. Eringen, A.C., Nonlinear theory of continuous media. New York: McGraw-Hill Book Company, 1962.

    Google Scholar 

  7. Mindlin, R.D., Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, 16(1): 51–78.

    Article  MathSciNet  Google Scholar 

  8. Fleck, N.A. and Hutchinson, J.W., Strain gradient plasticity. Advances in Applied Mechanics, 1997, 33: 295–361.

    Article  Google Scholar 

  9. Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W., Mechanism-based strain gradient plasticity—I. Theory. Journal of the Mechanics and Physics of Solids, 1999, 47(6): 1239–1263.

    Article  MathSciNet  Google Scholar 

  10. Huang, Y., Qu, S., Hwang, K.C., Li, M. and Gao, H., A conventional theory of mechanism-based strain gradient plasticity. International Journal of Plasticity, 2004, 20(4–5): 753–782.

    Article  Google Scholar 

  11. Yi, D., Wang, T.C. and Chen, S., New strain gradient theory and analysis. Acta Mechanica Solida Sinica, 2009, 22(1): 45–52.

    Article  Google Scholar 

  12. Qiu, X., Huang, Y., Wei, Y., Gao, H. and Hwang, K.C., The flow theory of mechanism-based strain gradient plasticity. Mechanics of Materials, 2003, 35(3–6): 245–258.

    Article  Google Scholar 

  13. Mindlin, R.D., On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures, 1968, 4(1): 109–124.

    Article  Google Scholar 

  14. Germain, P., The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM Journal on Applied Mathematics, 1973, 25(3): 556–575.

    Article  Google Scholar 

  15. Suhubi, E.S. and Eringen, A.C., Nonlinear theory of micro-elastic solids. II. International Journal of Engineering Science, 1964, 2: 389–404.

    Article  MathSciNet  Google Scholar 

  16. Eringen, A.C., Microcontinuum Field Theories, I. Foundations and Solids, New York: Springer-Verlag, 1999.

    Book  Google Scholar 

  17. Eringen, A.C., Theory of thermo-microstretch elastic solids. International Journal of Engineering Science, 1990, 28(12): 1291–1301.

    Article  MathSciNet  Google Scholar 

  18. Forest, S. and Sievert, R., Nonlinear microstrain theories. International Journal of Solids and Structures, 2006, 43(24): 7224–7245.

    Article  MathSciNet  Google Scholar 

  19. Liu, X. and Hu, G., Inclusion problem of microstretch continuum. International Journal of Engineering Science, 2004, 42(8–9): 849–860.

    Article  Google Scholar 

  20. Eringen, A.C., Linear theory of micropolar elasticity. Journal of Mathematics and Mechanics, 1966, 15: 909–923.

    MathSciNet  MATH  Google Scholar 

  21. Dillard, T., Forest, S. and Ienny, P., Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. European Journal of Mechanics/A Solids, 2006, 25(3): 526–549.

    Article  Google Scholar 

  22. Neff, P. and Forest, S., A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. Journal of Elasticity, 2007, 87(2): 239–276.

    Article  MathSciNet  Google Scholar 

  23. Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage. Journal of Engineering Mechanics, 2009, 135(3): 117–131.

    Article  Google Scholar 

  24. Zhang, Z.H., Zhuang, Z., Gao, Y., Liu, Z.L. and Nie, J.F., Cyclic plastic behavior analysis based on the micromorphic mixed hardening plasticity model. Computational Materials Science, 2011, 50(3): 1136–1144.

    Article  Google Scholar 

  25. Kirchner, N. and Steinmann, P., Mechanics of extended continua: modeling and simulation of elastic microstretch materials. Computational Mechanics, 2007, 40(4): 651–666.

    Article  Google Scholar 

  26. Zhang, H.W., Wang, H., Chen, B.S., and Xie, Z.Q., Parametric variational principle based elastic-plastic analysis of cosserat continuum. Acta Mechanica Solida Sinica, 2007, 20(1): 65–74.

    Article  Google Scholar 

  27. Muhlhaus, H.B. and Vardoulakis, L., The thickness of shear bands in granular materials. Geotechnique, 1987, 37(3): 271–283.

    Article  Google Scholar 

  28. Tekoğlu, C. and Onck, P.R., Size effects in two-dimensional Voronoi foams: A comparison between generalized continua and discrete models. Journal of the Mechanics and Physics of Solids, 2008, 56(12): 3541–3564.

    Article  Google Scholar 

  29. Li, X., Liu, Q. and Zhang, J., A micro-macro homogenization approach for discrete particle assembly—Cosserat continuum modeling of granular materials. International Journal of Solids and Structures, 2009, 47(2): 291–303.

    Article  Google Scholar 

  30. Toupin, R.A., Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, 1962, 11(1): 385–414.

    Article  MathSciNet  Google Scholar 

  31. Britta Hirschberger, C., Kuhl, E. and Steinmann, P., On deformational and configurational mechanics of micromorphic hyperelasticity-theory and computation. Computer Methods in Applied Mechanics and Engineering, 2007, 196: 4027–4044.

    Article  MathSciNet  Google Scholar 

  32. Zervos, A., Papanicolopulos, S.A. and Vardoulakis, I., Two finite-element discretizations for gradient elasticity. Journal of Engineering Mechanics, 2009, 135(3): 203–213.

    Article  Google Scholar 

  33. Zervos, A., Finite elements for elasticity with microstructure and gradient elasticity. International Journal for Numerical Methods in Engineering, 2008, 73(4): 564–595.

    Article  MathSciNet  Google Scholar 

  34. Vernerey, F., Liu, W.K. and Moran, B., Multi-scale micromorphic theory for hierarchical materials. Journal of the Mechanics and Physics of Solids, 2007, 55(12): 2603–2651.

    Article  MathSciNet  Google Scholar 

  35. Abaqus/Standard 6.9, 2009. Abaqus Users’ Manual. Available from: www.simulia.com .

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing, 100084, China

    Zhaohui Zhang, Zhanli Liu, Yuan Gao & Zhuo Zhuang

  2. Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing, 100084, China

    Junfang Nie

  3. Naval Academy of Armament, Beijing, 100073, China

    Zhaohui Zhang

Authors
  1. Zhaohui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhanli Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Junfang Nie
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhuo Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuo Zhuang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10772096).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Z., Liu, Z., Gao, Y. et al. Analytical and Numerical Investigations of Two Special Classes of Generalized Continuum Media. Acta Mech. Solida Sin. 24, 326–339 (2011). https://doi.org/10.1016/S0894-9166(11)60034-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1016/S0894-9166(11)60034-7

Key words

Search

Navigation