Analytical solutions of the simple shear problem for micromorphic models and other generalized continua


To draw conclusions as regards the stability and modelling limits of the investigated continuum, we consider a family of infinitesimal isotropic generalized continuum models (Mindlin–Eringen micromorphic, relaxed micromorphic continuum, Cosserat, micropolar, microstretch, microstrain, microvoid, indeterminate couple stress, second gradient elasticity, etc.) and solve analytically the simple shear problem of an infinite stripe. A qualitative measure characterizing the different generalized continuum moduli is given by the shear stiffness \(\mu ^{*}\). This stiffness is in general length-scale dependent. Interesting limit cases are highlighted, which allow to interpret some of the appearing material parameter of the investigated continua.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    The following energy expression has been used in [41]: \(W \left( \varvec{\nabla u}, \varvec{P}, \varvec{\nabla P}\right) = \mu \left\Vert \hbox {sym} \, \varvec{\nabla u} \right\Vert ^2 + \lambda /2 \, \hbox {tr}^2 \left( \varvec{\nabla u} \right) + \alpha \, \mu \left\Vert \varvec{\nabla u} - \varvec{P} \right\Vert ^2 + \alpha \, \lambda /2 \, \hbox {tr}^2 \left( \varvec{\nabla u} - \varvec{P} \right) +\mu \, L_{c}^2/2 \, \left\Vert \varvec{\nabla P} \right\Vert ^2 \). This formulations is not reconcilable with the relaxed micromorphic model even if we neglect the curvature part.


  1. 1.

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

    Article  Google Scholar 

  2. 2.

    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)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cosserat, E., Cosserat, F.: Théorie des Corps déformables. Hermann, Paris (1909)

    Google Scholar 

  4. 4.

    d’Agostino, M.V., Barbagallo, G., Ghiba, I.D., Eidel, B., Neff, P., Madeo, A.: Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. J. Elas 139, 299 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Diebels, S., Steeb, H.: Stress and couple stress in foams. Comput. Mater. Sci. 28(3–4), 714–722 (2003)

    Article  Google Scholar 

  6. 6.

    Dunn, M., Wheel, M.: Size effect anomalies in the behaviour of loaded 3d mechanical metamaterials. Phil. Mag. 100(2), 139–156 (2020)

    Article  Google Scholar 

  7. 7.

    Eringen, A. C.: Mechanics of micromorphic continua. In: Mechanics of Generalized Continua, pp. 18–35. Springer, Berlin (1968)

  8. 8.

    Forest, S.: Generalized continua from the theory to engineering applications. In: Altenbach, H., Eremeyev, V. (eds.) Micromorphic Media, vol. 541, pp. 249–300. Springer, Berlin (2013)

    Google Scholar 

  9. 9.

    Forest, S.: Questioning size effects as predicted by strain gradient plasticity. J. Mech. Behavior Mater. 22(3–4), 101–110 (2013)

    Article  Google Scholar 

  10. 10.

    Forest, S.: Micromorphic approach to materials with internal length. In: Encyclopedia of Continuum Mechanics, pp. 1–11. Springer, Berlin, Heidelberg (2018)

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct. 48(18), 2496–2510 (2011)

    Article  Google Scholar 

  13. 13.

    Hütter, G.: Application of a microstrain continuum to size effects in bending and torsion of foams. Int. J. Eng. Sci. 101, 81–91 (2016)

    Article  Google Scholar 

  14. 14.

    Hütter, G.: On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua. J. Mech. Phys. Solids 127, 62–79 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Hütter, G., Mühlich, U., Kuna, M.: Micromorphic homogenization of a porous medium: elastic behavior and quasi-brittle damage. Continuum Mech. Thermodyn. 27(6), 1059–1072 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Iltchev, A., Marcadon, V., Kruch, S., Forest, S.: Computational homogenisation of periodic cellular materials: application to structural modelling. Int. J. Mech. Sci. 93, 240–255 (2015)

    Article  Google Scholar 

  17. 17.

    Jeong, J., Neff, P.: Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15(1), 78–95 (2010)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kruch, S., Forest, S.: Computation of coarse grain structures using a homogeneous equivalent medium. Le Journal de Physique IV 8(PR8), Pr8–197 (1998)

    Google Scholar 

  19. 19.

    Liebenstein, S., Sandfeld, S., Zaiser, M.: Size and disorder effects in elasticity of cellular structures: from discrete models to continuum representations. Int. J. Solids Struct. 146, 97–116 (2018)

    Article  Google Scholar 

  20. 20.

    Madeo, A., Ghiba, I.D., Neff, P., Münch, I.: A new view on boundary conditions in the Grioli-Koiter-Mindlin-Toupin indeterminate couple stress model. Euro. J. Mech.-A/Solids 59, 294–322 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Mazière, M., Forest, S.: Strain gradient plasticity modeling and finite element simulation of lüders band formation and propagation. Continuum Mech. Thermodyn. 27(1–2), 83–104 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Münch, I., Neff, P., Madeo, A., Ghiba, I.D.: The modified indeterminate couple stress model: why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. Z. Angew. Math Me. 97(12), 1524–1554 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Neff, P.: On material constants for micromorphic continua. In: Trends in Applications of Mathematics to Mechanics, STAMM Proceedings, Seeheim, pp. 337–348. Shaker–Verlag (2004)

  25. 25.

    Neff, P.: The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric. Z. Angew. Math. Me. 86(11), 892–912 (2006)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Neff, P., Eidel, B., d’Agostino, M.V., Madeo, A.: Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization. J. Elast. 139, 269–298 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Neff, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Thermodyn. 26(5), 639–681 (2014)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Neff, P., Jeong, J.: A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. Z. Angew. Math. Me. 89(2), 107–122 (2009)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Neff, P., Jeong, J., Fischle, A.: Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mech. 211(3–4), 237–249 (2010)

    Article  Google Scholar 

  30. 30.

    Neff, P., Münch, I.: Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure. Continuum Mech. Thermodyn. 21(3), 195–221 (2009)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Neff, P., Münch, I., Ghiba, I.D., Madeo, A.: On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of AR Hadjesfandiari and GF Dargush. Int. J. Solids Struct. 81, 233–243 (2016)

    Article  Google Scholar 

  32. 32.

    Nourmohammadi, N., O’Dowd, N.P., Weaver, P.M.: Effective bending modulus of thin ply fibre composites with uniform fibre spacing. Int. J. Solids Struct 196, 26 (2020)

    Article  Google Scholar 

  33. 33.

    Pham, R.D., Hütter, G.: Influence of topology and porosity on size effects in cellular materials with hexagonal structure under shear, tension and bending. arXiv preprint arXiv:2009.10404 (2020)

  34. 34.

    Rizzi, G., Dal Corso, F., Veber, D., Bigoni, D.: Identification of second-gradient elastic materials from planar hexagonal lattices. part ii: Mechanical characteristics and model validation. Int. J. Solids Struct. 176, 19–35 (2019)

    Article  Google Scholar 

  35. 35.

    Rizzi, G., Hütter, G., Madeo, A., Neff, P.: Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations). arXiv preprint (2020)

  36. 36.

    Rizzi, G., Hütter, G., Madeo, A., Neff, P.: Analytical solutions of the simple shear problem for certain types of micromorphic continuum models—including full derivations. arXiv preprint arXiv:2006.02391 (2020)

  37. 37.

    Rueger, Z., Lakes, R.S.: Experimental study of elastic constants of a dense foam with weak Cosserat coupling. J. Elast. 137(1), 101–115 (2019)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Shaat, M.: A reduced micromorphic model for multiscale materials and its applications in wave propagation. Compos. Struct. 201, 446–454 (2018)

    Article  Google Scholar 

  39. 39.

    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  Google Scholar 

  40. 40.

    Yoder, M., Thompson, L., Summers, J.: Size effects in lattice-structured cellular materials: material distribution. J. Mater. Sci. 54(18), 11858–11877 (2019)

    Article  Google Scholar 

  41. 41.

    Zhang, Z., Liu, Z., Gao, Y., Nie, J., Zhuang, Z.: Analytical and numerical investigations of two special classes of generalized continuum media. Acta Mech. Solida Sin. 24(4), 326–339 (2011)

    Article  Google Scholar 

Download references


AM acknowledge funding from the French Research Agency ANR, “METASMART” (ANR-17CE08-0006). AM and GR acknowledges support from IDEXLYON in the framework of the “Programme Investissement d’Avenir” ANR-16-IDEX-0005.

Author information



Corresponding author

Correspondence to Gianluca Rizzi.

Ethics declarations

Conflict of interest

The authors declared that they have no conflicts of interest to this work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rizzi, G., Hütter, G., Madeo, A. et al. Analytical solutions of the simple shear problem for micromorphic models and other generalized continua. Arch Appl Mech (2021).

Download citation


  • Generalized continua
  • Simple shear
  • Shear stiffness
  • Characteristic length
  • Size-effect
  • Micromorphic continuum
  • Cosserat continuum
  • Gradient elasticity