Acta Mechanica Sinica

, Volume 35, Issue 5, pp 1001–1020 | Cite as

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

  • Sara Grbčić
  • Gordan JelenićEmail author
  • Dragan Ribarić
Research Paper


Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.


Micropolar theory Finite element method Linked interpolation Quadrilateral elements 



The research presented in this paper has been financially supported by the Croatian Science Foundation (Grants HRZZ-IP-11-2013-1631 and HRZZ-IP-2018-01-1732), Young Researchers’ Career Development—Training of Doctoral Students, as well as a French Government Scholarship.


  1. 1.
    Nowacki, W.: Theory of Micropolar Elasticity. Springer, Vienna (1972)zbMATHGoogle Scholar
  2. 2.
    Lakes, R.S.: Size effects and micromechanics of a porous solid. J. Mater. Sci. 18, 2572–2580 (1983). CrossRefGoogle Scholar
  3. 3.
    Lakes, R.S.: Reduced warp in torsion of reticulated foam due to Cosserat elasticity: experiment. Z. Angew. Math. Phys. 67, 46 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diebels, S., Geringer, A.: Micromechanical and macromechanical modelling of foams: identification of Cosserat parameters. ZAMM J. Appl. Math. Mech./Z. Angew. Math. Mech. 94, 414–420 (2014). CrossRefGoogle Scholar
  5. 5.
    Eringen, A.C.: Microcontinuum Field Theories: I. Foundations and Solids. Springer, New York (2012)zbMATHGoogle Scholar
  6. 6.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Herman, Paris (1909), in FrenchGoogle Scholar
  7. 7.
    Malvern, L.E.: Introduction to the Mechanics of a Continious Medium. Prentice-Hall Inc, New Jersey (1969)Google Scholar
  8. 8.
    Neuber, H.: On the General Solution of Linear-Elastic Problems in Isotropic and Anisotropic Cosserat Continua, pp. 153–158. Springer, Berlin (1966). CrossRefzbMATHGoogle Scholar
  9. 9.
    Toubal, L., Karama, M., Lorrain, B.: Stress concentration in a circular hole in composite plate. Compos. Struct. 68, 31–36 (2005). CrossRefGoogle Scholar
  10. 10.
    Dyszlewicz, J.: Micropolar Theory of Elasticity. Springer Science & Business Media (2004). CrossRefGoogle Scholar
  11. 11.
    Eremeyev, V., Lebedev, L., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Berlin (2013). CrossRefzbMATHGoogle Scholar
  12. 12.
    Gauthier, R., Jahsman, W.E.: A quest for micropolar elastic constants. J. Appl. Mech. 42, 369–374 (1975). CrossRefzbMATHGoogle Scholar
  13. 13.
    Yang, J.F.C., Lakes, R.S.: Transient study of couple stress effects in compact bone: torsion. J. Biomech. Eng. 103, 275–279 (1981). CrossRefGoogle Scholar
  14. 14.
    Yang, J.F.C., Lakes, R.S.: Experimental study of micropolar and couple stress elasticity in compact bone in bending. J. Biomech. 15, 91–98 (1982). CrossRefGoogle Scholar
  15. 15.
    Lakes, R.S., Nakamura, S., Behiri, J.C., et al.: Fracture mechanics of bone with short cracks. J. Biomech. 23, 967–975 (1990). CrossRefGoogle Scholar
  16. 16.
    Anderson, W.B., Lakes, R.S.: Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J. Mater. Sci. 29, 6413–6419 (1994). CrossRefGoogle Scholar
  17. 17.
    Rueger, Z., Lakes, R.S.: Cosserat elasticity of negative Poisson’s ratio foam: experiment. Smart Mater. Struct. 25, 054004 (2016). CrossRefGoogle Scholar
  18. 18.
    Lakes, R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22, 55–63 (1986). CrossRefGoogle Scholar
  19. 19.
    Chen, C.P., Lakes, R.S.: Holographic study of conventional and negative Poisson’s ratio metallic foams: elasticity, yield and micro-deformation. J. Mater. Sci. 26, 5397–5402 (1991). CrossRefGoogle Scholar
  20. 20.
    Wang, X.L., Stronge, W.J.: Micropolar theory for two-dimensional stresses in elastic honeycomb. Proc. R. Soc. A Math. Phys. Eng. Sci. 455, 2091–2116 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mora, R.J., Waas, A.M., Arbor, A.: Evaluation of the micropolar elasticity constants for honeycombs. Acta Mech. 192, 1–16 (2007). CrossRefzbMATHGoogle Scholar
  22. 22.
    Duan, S., Weibin, W., Daining, F.: A predictive micropolar continuum model for a novel three-dimensional chiral lattice with size effect and tension-twist coupling behavior. J. Mech. Phys. Solids 121, 23–46 (2018). MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yoder, M., Thompson, L., Summers, J.: Size effects in lattice structures and a comparison to micropolar elasticity. Int. J. Solids Struct. 143, 245–261 (2018). CrossRefGoogle Scholar
  24. 24.
    Zhang, W., Neville, R., Zhang, D., et al.: The two-dimensional elasticity of a chiral hinge lattice metamaterial. Int. J. Solids Struct. 141–142, 254–263 (2018). CrossRefGoogle Scholar
  25. 25.
    Merkel, A., Luding, S.: Enhanced micropolar model for wave propagation in ordered granular materials. Int. J. Solids Struct. 106–107, 91–105 (2017). CrossRefGoogle Scholar
  26. 26.
    Niu, B., Yan, J.: A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material. Acta Mech. Sin. 32, 456–468 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Romeo, M.: Surface waves in hexagonal micropolar dielectrics. Int. J. Solids Struct. 87, 39–47 (2016). CrossRefGoogle Scholar
  28. 28.
    Liebenstein, S., Zaiser, M.: Determining Cosserat constants of 2D cellular solids from beam models. Mater. Theory 2, 2 (2018). CrossRefGoogle Scholar
  29. 29.
    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). CrossRefGoogle Scholar
  30. 30.
    Bažant, Z., Christensen, M.: Analogy between micropolar continuum and grid frameworks under initial stress. Int. J. Solids Struct. 8, 327–346 (1972). CrossRefzbMATHGoogle Scholar
  31. 31.
    Besdo, D.: Towards a Cosserat-theory describing motion of an originally rectangular structure of blocks. Arch. Appl. Mech. 80, 25–45 (2010). CrossRefzbMATHGoogle Scholar
  32. 32.
    Hassanpour, S., Heppler, G.R.: Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Math. Mech. Solids 22, 224–242 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Grbčić, S., Ibrahimbegović, A., Jelenić, G.: Variational formulation of micropolar elasticity using 3D hexahedral finite-element interpolation with incompatible modes. Comput. Struct. 205, 1–14 (2018). CrossRefGoogle Scholar
  34. 34.
    Nakamura, S., Benedict, R., Lakes, R.: Finite element method for orthotropic micropolar elasticity. Int. J. Eng. Sci. 22, 319–330 (1984). CrossRefzbMATHGoogle Scholar
  35. 35.
    Providas, E., Kattis, M.A.: Finite element method in plane Cosserat elasticity. Comput. Struct. 80, 2059–2069 (2002). CrossRefGoogle Scholar
  36. 36.
    Li, L., Xie, S.: Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in MEMS. Int. J. Mech. Sci. 46, 1571–1587 (2004). CrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, H., Wang, H., Liu, G.: Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies. Acta Mech. Sin. 21, 388–394 (2005). CrossRefzbMATHGoogle Scholar
  38. 38.
    Korepanov, V.V., Matveenko, V.P., Shardakov, I.N.: Finite element analysis of two- and three-dimensional static problems in the asymmetric theory of elasticity as a basis for the design of experiments. Acta Mech. 223, 1739–1750 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wheel, M.A.: A control volume-based finite element method for plane micropolar elasticity. Int. J. Numer. Methods Eng. 75, 992–1006 (2008). CrossRefzbMATHGoogle Scholar
  40. 40.
    Beveridge, A.J., Wheel, M.A., Nash, D.H.: A higher order control volume based finite element method to predict the deformation of heterogeneous materials. Comput. Struct. 129, 54–62 (2013). CrossRefGoogle Scholar
  41. 41.
    Hassanpour, S., Heppler, G.R.: Comprehensive and easy-to-use torsion and bending theories for micropolar beams. Int. J. Mech. Sci. 114, 71–87 (2016). CrossRefGoogle Scholar
  42. 42.
    Hassanpour, S., Heppler, G.R.: Theory of micropolar gyroeastic continua. Acta Mech. 227, 1469–1491 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ma, T., Wang, Y., Yuan, L., et al.: Timoshenko beam model for chiral materials. Acta Mech. Sin. 34, 549–560 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Jelenić, G., Papa, E.: Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order. Arch. Appl. Mech. 81, 171–183 (2011). CrossRefzbMATHGoogle Scholar
  45. 45.
    Ribarić, D., Jelenić, G.: Higher-order linked interpolation in quadrilateral thick plate finite elements. Finite Elem. Anal. Des. 51, 67–80 (2012). MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ribarić, D., Jelenić, G.: Higher-order linked interpolation in triangular thick plate finite elements. Eng. Comput. 31, 69–109 (2014). CrossRefzbMATHGoogle Scholar
  47. 47.
    Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966). MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover Publications Inc., New York (1994)zbMATHGoogle Scholar
  49. 49.
    Jeffreys, H.: On isotropic tensors. Math. Proc. Camb. Philos. Soc. 73, 173–176 (1973). MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lakes, R.S.: Physical meaning of elastic constants in cosserat, void, and microstretch elasticity. Mech. Mater. Struct. 11, 1–13 (2016). MathSciNetCrossRefGoogle Scholar
  51. 51.
    Cowin, S.C.: An incorrect inequality in micropolar elasticity theory. ZAMP, Z. Angew. Math. Phys. 21, 494–497 (1970). CrossRefzbMATHGoogle Scholar
  52. 52.
    Tessler, A., Dong, S.: On a hierarchy of conforming Timoshenko beam elements. Comput. Struct. 14, 335–344 (1981). CrossRefGoogle Scholar
  53. 53.
    Auricchio, F., Taylor, R.: A shear deformable plate element with an exact thin limit. Comput. Methods Appl. Mech. Eng. 118, 393–412 (1994). MathSciNetCrossRefGoogle Scholar
  54. 54.
    Ibrahimbegović, A.: Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods. Springer, London (2009)CrossRefGoogle Scholar
  55. 55.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method Volume 1: The Basis. Butterworth-Heinemann, Oxford (2000)zbMATHGoogle Scholar
  56. 56.
    Taylor, R.: FEAP—Finite Element Analysis Program. University of California at Berkeley (2014). Accessed 31 Jan 2019
  57. 57.
    Reddy, J.N.: An Introduction to the Finite Element Method, 2nd edn. McGrawHill Inc, Texas (1994)Google Scholar
  58. 58.
    Wilson, E.L., Ibrahimbegović, A.: Use of incompatible displacement modes for the calculation of element stiffnesses or stresses. Finite Elem. Anal. Des. 7, 229–241 (1990). CrossRefGoogle Scholar
  59. 59.
    Grbčić, S.: Linked interpolation and strain invariance in finite-element modelling of micropolar continuum, [Ph.D. Thesis], University of Rijeka and Université de Technologie de Compiègne Sorbonne Universités (2018)Google Scholar
  60. 60.
    Bauer, S., Schäfer, M., Grammenoudis, P., et al.: Three-dimensional finite elements for large deformation micropolar elasticity. Comput. Methods Appl. Mech. Eng. 199, 2643–2654 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Kirsch, E.: Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Z. V. Dtsch. Ing. 42, 797–807 (1898), in GermanGoogle Scholar
  62. 62.
    Geuzaine, C., Remacle, J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009). CrossRefzbMATHGoogle Scholar
  63. 63.
    Nakamura, S., Lakes, R.S.: Finite element analysis of stress concentration around a blunt crack in a cosserat elastic solid. Comput. Methods Appl. Mech. Eng. 66, 257–266 (1988)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Sara Grbčić
    • 1
    • 2
  • Gordan Jelenić
    • 1
    Email author
  • Dragan Ribarić
    • 1
  1. 1.Faculty of Civil EngineeringUniversity of RijekaRijekaCroatia
  2. 2.Université de Technologie de Compiègne, Sorbonne UniversitésCompiègneFrance

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