Quadrilateral 2D linked-interpolation finite elements for micropolar continuum


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.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Nowacki, W.: Theory of Micropolar Elasticity. Springer, Vienna (1972)

    MATH  Google Scholar 

  2. 2.

    Lakes, R.S.: Size effects and micromechanics of a porous solid. J. Mater. Sci. 18, 2572–2580 (1983). https://doi.org/10.1007/BF00547573

    Article  Google 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). https://doi.org/10.1007/s00033-016-0632-4

    MathSciNet  Article  MATH  Google 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). https://doi.org/10.1002/zamm.201200271

    Article  Google Scholar 

  5. 5.

    Eringen, A.C.: Microcontinuum Field Theories: I. Foundations and Solids. Springer, New York (2012)

    MATH  Google Scholar 

  6. 6.

    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Herman, Paris (1909), in French

  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). https://doi.org/10.1007/978-3-662-29364-5_16

    Book  MATH  Google Scholar 

  9. 9.

    Toubal, L., Karama, M., Lorrain, B.: Stress concentration in a circular hole in composite plate. Compos. Struct. 68, 31–36 (2005). https://doi.org/10.1016/j.compstruct.2004.02.016

    Article  Google Scholar 

  10. 10.

    Dyszlewicz, J.: Micropolar Theory of Elasticity. Springer Science & Business Media (2004). https://doi.org/10.1007/978-3-540-45286-7

    Book  Google Scholar 

  11. 11.

    Eremeyev, V., Lebedev, L., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-28353-6

    Book  MATH  Google Scholar 

  12. 12.

    Gauthier, R., Jahsman, W.E.: A quest for micropolar elastic constants. J. Appl. Mech. 42, 369–374 (1975). https://doi.org/10.1115/1.3423583

    Article  MATH  Google 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). https://doi.org/10.1115/1.3138292

    Article  Google 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). https://doi.org/10.1016/0021-9290(82)90040-9

    Article  Google 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). https://doi.org/10.1016/0021-9290(90)90311-P

    Article  Google 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). https://doi.org/10.1007/BF00353997

    Article  Google Scholar 

  17. 17.

    Rueger, Z., Lakes, R.S.: Cosserat elasticity of negative Poisson’s ratio foam: experiment. Smart Mater. Struct. 25, 054004 (2016). https://doi.org/10.1088/0964-1726/25/5/054004

    Article  Google Scholar 

  18. 18.

    Lakes, R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22, 55–63 (1986). https://doi.org/10.1016/0020-7683(86)90103-4

    Article  Google 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). https://doi.org/10.1007/BF02403936

    Article  Google 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). https://doi.org/10.1098/rspa.1999.0394

    MathSciNet  Article  MATH  Google 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). https://doi.org/10.1007/s00707-007-0446-8

    Article  MATH  Google 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). https://doi.org/10.1016/j.jmps.2018.07.016

    MathSciNet  Article  Google 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). https://doi.org/10.1016/j.ijsolstr.2018.03.013

    Article  Google 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). https://doi.org/10.1016/j.ijsolstr.2018.02.027

    Article  Google 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). https://doi.org/10.1016/j.ijsolstr.2016.11.029

    Article  Google 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). https://doi.org/10.1007/s10409-015-0492-8

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Romeo, M.: Surface waves in hexagonal micropolar dielectrics. Int. J. Solids Struct. 87, 39–47 (2016). https://doi.org/10.1016/j.ijsolstr.2016.02.235

    Article  Google Scholar 

  28. 28.

    Liebenstein, S., Zaiser, M.: Determining Cosserat constants of 2D cellular solids from beam models. Mater. Theory 2, 2 (2018). https://doi.org/10.1186/s41313-017-0009-x

    Article  Google 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). https://doi.org/10.1016/j.ijsolstr.2018.03.023

    Article  Google 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). https://doi.org/10.1016/0020-7683(72)90093-5

    Article  MATH  Google 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). https://doi.org/10.1007/s00419-009-0366-2

    Article  MATH  Google 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). https://doi.org/10.1177/1081286515581183

    MathSciNet  Article  MATH  Google 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). https://doi.org/10.1016/j.compstruc.2018.04.005

    Article  Google Scholar 

  34. 34.

    Nakamura, S., Benedict, R., Lakes, R.: Finite element method for orthotropic micropolar elasticity. Int. J. Eng. Sci. 22, 319–330 (1984). https://doi.org/10.1016/0020-7225(84)90013-2

    Article  MATH  Google Scholar 

  35. 35.

    Providas, E., Kattis, M.A.: Finite element method in plane Cosserat elasticity. Comput. Struct. 80, 2059–2069 (2002). https://doi.org/10.1016/S0045-7949(02)00262-6

    Article  Google 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). https://doi.org/10.1016/j.ijmecsci.2004.10.004

    Article  MATH  Google 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). https://doi.org/10.1007/s10409-005-0041-y

    Article  MATH  Google 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). https://doi.org/10.1007/s00707-012-0640-1

    MathSciNet  Article  MATH  Google 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). https://doi.org/10.1002/nme.2293

    Article  MATH  Google 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). https://doi.org/10.1016/j.compstruc.2013.08.006

    Article  Google 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). https://doi.org/10.1016/j.ijmecsci.2016.05.007

    Article  Google Scholar 

  42. 42.

    Hassanpour, S., Heppler, G.R.: Theory of micropolar gyroeastic continua. Acta Mech. 227, 1469–1491 (2016). https://doi.org/10.1007/s00707-016-1573-x

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Ma, T., Wang, Y., Yuan, L., et al.: Timoshenko beam model for chiral materials. Acta Mech. Sin. 34, 549–560 (2018). https://doi.org/10.1007/s10409-017-0735-y

    MathSciNet  Article  MATH  Google 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). https://doi.org/10.1007/s00419-009-0403-1

    Article  MATH  Google 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). https://doi.org/10.1016/j.finel.2011.10.003

    MathSciNet  Article  Google Scholar 

  46. 46.

    Ribarić, D., Jelenić, G.: Higher-order linked interpolation in triangular thick plate finite elements. Eng. Comput. 31, 69–109 (2014). https://doi.org/10.1108/EC-03-2012-0056

    Article  MATH  Google Scholar 

  47. 47.

    Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966). https://doi.org/10.2307/24901442

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover Publications Inc., New York (1994)

    MATH  Google Scholar 

  49. 49.

    Jeffreys, H.: On isotropic tensors. Math. Proc. Camb. Philos. Soc. 73, 173–176 (1973). https://doi.org/10.1017/S0305004100047587

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Lakes, R.S.: Physical meaning of elastic constants in cosserat, void, and microstretch elasticity. Mech. Mater. Struct. 11, 1–13 (2016). https://doi.org/10.2140/jomms.2016.11.217

    MathSciNet  Article  Google Scholar 

  51. 51.

    Cowin, S.C.: An incorrect inequality in micropolar elasticity theory. ZAMP, Z. Angew. Math. Phys. 21, 494–497 (1970). https://doi.org/10.1007/BF01627956

    Article  MATH  Google Scholar 

  52. 52.

    Tessler, A., Dong, S.: On a hierarchy of conforming Timoshenko beam elements. Comput. Struct. 14, 335–344 (1981). https://doi.org/10.1016/0045-7949(81)90017-1

    Article  Google 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). https://doi.org/10.1016/0045-7825(94)90009-4

    MathSciNet  Article  Google Scholar 

  54. 54.

    Ibrahimbegović, A.: Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods. Springer, London (2009)

    Book  Google Scholar 

  55. 55.

    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method Volume 1: The Basis. Butterworth-Heinemann, Oxford (2000)

    MATH  Google Scholar 

  56. 56.

    Taylor, R.: FEAP—Finite Element Analysis Program. University of California at Berkeley (2014). http://projects.ce.berkeley.edu/feap. 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). https://doi.org/10.1016/0168-874X(90)90034-C

    Article  Google 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)

  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). https://doi.org/10.1016/j.cma.2010.05.002

    MathSciNet  Article  MATH  Google Scholar 

  61. 61.

    Kirsch, E.: Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Z. V. Dtsch. Ing. 42, 797–807 (1898), in German

  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). https://doi.org/10.1002/nme.2579

    Article  MATH  Google 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)

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Gordan Jelenić.

Appendix A: Shape functions

Appendix A: Shape functions

Appendix A.1: Quadrilateral finite elements

The shape functions in Eq. (34) are given for \(\xi _1=\xi _4=-1,~\xi _2=\xi _3=+1,~\eta _1=\eta _2=-1,~\eta _3=\eta _4=+1\) as follows, a being the node number according to Fig. 1. For Q4, they are defined as \(N_a(\xi ,\eta )=\frac{1}{4}(1+\xi _a \xi )(1+\eta _a \eta )\). For Q9, with \(\xi _8=-1\), \(\xi _5=\xi _7=\xi _9=0\), \(\xi _6=+1,~\eta _5=-1,~\eta _6=\eta _8=\eta _9=0\) and \(\eta _7=+1\), they are given as

$$\begin{aligned}&\text {vertex nodes:}&N_a&=\frac{1}{4}\xi \eta (\xi +\xi _a)(\eta +\eta _a),\\&\text {edge nodes 5 and 7: }&N_a&=\frac{1}{2}\eta (1-\xi ^2)(\eta +\eta _a),\\&\text {edge nodes 6 and 8: }&N_a&=\frac{1}{2}\xi (\xi +\xi _a)(1-\eta ^2),\\&\text {central node:}&N_9&=(1-\xi ^2)(1-\eta ^2),\\ \end{aligned}$$

while for Q16, with \(\xi _{11}=\xi _{12}=-1\), \(\xi _5=\xi _{10}=\xi _{13}=\xi _{16}=-\frac{1}{3}\), \(\xi _6=\xi _{9}=\xi _{14}=\xi _{15}=+\frac{1}{3}\), \(\xi _{7}=\xi _{8}=+1\), \(\eta _{5}=\eta _{6}=-1\), \(\eta _{7}=\eta _{12}=\eta _{13}=\eta _{14}=-\frac{1}{3}\), \(\eta _{8}=\eta _{11}=\eta _{15}=\eta _{16}=+\frac{1}{3}\) and \(\eta _{9}=\eta _{10}=+1\), they are given as

$$\begin{aligned}&\text {vertex nodes:}\\&\quad N_a =\frac{81}{256}(1+\xi _a\xi )(1+\eta _a\eta )\left( \frac{1}{9}-\xi ^2\right) \left( \frac{1}{9}-\eta ^2\right) ,\\&\text {edge nodes with } \xi _a=\pm 1 \text { and } \eta _a=\pm \frac{1}{3}:\\&\quad N_a = \frac{243}{256}(1-\xi ^2)\left( \eta ^2-\frac{1}{9}\right) \left( \frac{1}{3}+3\xi _a\xi \right) (1+\eta _a\eta ),\\&\text {edge nodes with } \xi _a=\pm \frac{1}{3} \text { and } \eta _a=\pm 1:\\&\quad N_a = \frac{243}{256}(1-\eta ^2)\left( \xi ^2-\frac{1}{9}\right) \left( \frac{1}{3}+3\eta _a\eta \right) (1+\xi _a\xi ),\\&\text {internal nodes:}\\&\quad N_a =\frac{729}{256}(1-\xi ^2)(1-\eta ^2)\left( \frac{1}{3}+3\eta _a\eta \right) \left( \frac{1}{3}+3\xi _a\xi \right) . \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Grbčić, S., Jelenić, G. & Ribarić, D. Quadrilateral 2D linked-interpolation finite elements for micropolar continuum. Acta Mech. Sin. 35, 1001–1020 (2019). https://doi.org/10.1007/s10409-019-00870-1

Download citation


  • Micropolar theory
  • Finite element method
  • Linked interpolation
  • Quadrilateral elements