Mechanics of Biological Membranes from Lattice Homogenization

  • Mohamed AssidiEmail author
  • Francisco Dos Reis
  • Jean François Ganghoffer
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)


The goal of this chapter is to set up a novel methodology for the calculation of the effective mechanical properties of biological membranes viewed as repetitive networks of elastic filaments, basing on the discrete asymptotic homogenization method. We will show that for some lattice configurations, internal structure mechanisms at the unit cell scale lead to additional flexional effects at the continuum scale, accounted for by an internal length associated to a micropolar behavior. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes for a given network topology, as closed form expressions of the geometrical and mechanical micro-parameters. A new approach, based on general beam equations, is proposed to tackle the non-linear constitutive behavior of the network, accounting for large strains and large rotations. Thereby, a perturbed equilibrium problem is set up at the unit cell level, solved by the Newton-Raphson method. This localization problem interacts with the homogenization procedure allowing the construction of the Cauchy and couple stress tensors, both steps leading to an update the network geometry and constitutive behaviour. A classification of lattices with respect to the choice of the equivalent continuum model is proposed: the Cauchy continuum and a micropolar continuum are adopted as two possible effective medium, for a given beam model. The relative ratio of the characteristic length of the micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. Calculation of the equivalent mechanical properties of the peptodoglycan membrane illustrates the proposed methodology.


Couple Stress Persistence Length Beam Length Micropolar Continuum Couple Stress Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boey, S.K., Boal, D.H., Discher, D.E: Simulations of the erythrocyte cytoskeleton at large deformation. Biophysical Journal 75, 1573–1583 (1998)CrossRefGoogle Scholar
  2. 2.
    Boal, D., Seifert, U., Shillcock, J.C: Negative Poisson ratio for two-dimensional networks under tension. Physical Rewiew E 48, 4274–4283 (1993)CrossRefGoogle Scholar
  3. 3.
    Lim, C.T., Zhou, E.H., Quek, S.T: Mechanical models for living cells-a review. Journal of Biomechanics. 39, 195–216 (2006)CrossRefGoogle Scholar
  4. 4.
    Warren, W.E., Byskov, E.: Three-fold symmetry restrictions on two-dimensional micropolar material. European Journal of Mechanics A/Solid 21, 779–792 (2002)CrossRefGoogle Scholar
  5. 5.
    Mourad M., Caillerie D., Raoult A (2003), in Computational Fluid and Solid Mechanics, ed. by Klaus-Jürgen Bathe. Proceeding, vol 2 (Elsevier, Boston, Etats-Unis, 2003), p. 1779–1781Google Scholar
  6. 6.
    Dos Reis, F., Ganghoffer, J.-.F.: Discrete homogenization of architectured materials: implementation of the method in a simulation tool for the systematic prediction of their effective elastic properties. Technische Mechanik 30, 85–109 (2010)Google Scholar
  7. 7.
    Pradel F., Sab K (1998) Homogenization of discrete media. Compte rendu de l’Académie des sciences II(b):699–704Google Scholar
  8. 8.
    Tollenaere, H., Caillerie, D.: Continuous modeling of lattice structures by homogenization. Advances in Engineering Software 29, 699–705 (1998)CrossRefGoogle Scholar
  9. 9.
    Mourad A (2003) Description topologique de l’architecture fibreuse et modélisation mécanique du myocarde. Institut National Polytechnique de GrenobleGoogle Scholar
  10. 10.
    Alkhader, M., Vural, M.: Mechanical response of cellular solids: Role of cellular topology and microstructural irregularity. International Journal of Engineering Science 46, 1035–1051 (2008)CrossRefGoogle Scholar
  11. 11.
    Dao, M., Li, J., Suresh, S.: Molecularly based analysis of deformation of spectrin network and human erythrocyte. Material Sciences and Engineering C 26, 1232–1244 (2006)CrossRefGoogle Scholar
  12. 12.
    Askar, A., Cakmak, A.S: A structural model of a micropolar continuum. International Journal of Engineering Sciences. 6, 583–589 (1968)CrossRefGoogle Scholar
  13. 13.
    Dos Reis F (2010) Homogénéisation automatique de milieux discret périodique. Applications aux mousses polymère et aux milieux auxétiques. Institut National Polytechnique de LorraineGoogle Scholar
  14. 14.
    Trovalusci, P., Masiani, R.: Material symmetries of micropolar continua equivalent to lattices. International Journal of Solids and Structures 36, 2091–2108 (1999)CrossRefGoogle Scholar
  15. 15.
    Mindlin, R.D: Stress functions for Cosserat continuum. International Journal of Solids and Structures 1, 265–271 (1965)CrossRefGoogle Scholar
  16. 16.
    Koch, A.L., Woeste, S.: Elasticity of the sacculus of Escherichia coli. Journal of Bacteriology 174, 327–341 (1992)Google Scholar
  17. 17.
    Boal, D.H.: Mechanics of the Cell. CAMBRIDGE University Press.,   (2002)Google Scholar
  18. 18.
    Stokke, T., Brant, D.A: The reliability of wormlike polysaccharide chain dimension estimated from electron micrographs. Biopolymers 30, 1161–1181 (1990)CrossRefGoogle Scholar
  19. 19.
    Lakes, R.: On the torsional properties of single osteons. Journal of Biomechanics 28, 1409–1410 (1995)CrossRefGoogle Scholar
  20. 20.
    Feyel, F., Chaboche, J.-.L.: \({\hbox{FE}}^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering 183, 309–330 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mohamed Assidi
    • 1
    Email author
  • Francisco Dos Reis
    • 2
  • Jean François Ganghoffer
    • 2
  1. 1.ENSEM-INPLVandoeuvre-les-NancyFrance
  2. 2.LEMTA - ENSEMVandoeuvre CEDEXFrance

Personalised recommendations