Journal of Elasticity

, Volume 99, Issue 1, pp 1–17 | Cite as

Evolution of Surfaces and the Kinematics of Membranes

  • N. Kadianakis


The polar decomposition theorem for a two dimensional continuum (a membrane) is used to produce a set of equations that describe the evolution of the geometrical quantities of a moving surface, i.e., the metric, the unit normal, the shape operator, the second fundamental form, the mean and the Gauss curvature. A link to the kinematical quantities of the continuum is also given. The version of the polar decomposition theorem for membranes we use was proved by Chi-Sing Man and H. Cohen (J. Elast. 16:97–104, 1986). Both the geometric and the kinematical framework are coordinate-free, in an attempt to contribute to a coordinate-free description for the kinematics of membranes in analogy to the kinematics of three dimensional continuum bodies as it emerges from the classical works of Noll (Arch. Rat. Mech. Anal. 2:197–226, 1958), Truesdell and Noll (The Non-linear Field Theories of Mechanics, 3rd edn., Springer, Berlin, 2004), Truesdell (A First Course in Rational Continuum Mechanics, vol. 1, Academic Press, San Diego, 1977).

Kinematics Surfaces Membranes 

Mathematics Subject Classification (2000)

53A05 53A17 74K15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. 2, 151–171 (1994) MATHCrossRefGoogle Scholar
  2. 2.
    Appleby, P.G., Kadianakis, N.: A frame-independent description of the principles of classical mechanics. Arch. Rat. Mech. Anal. 95, 1–22 (1986) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Capovilla, R., Guven, J., Santiago, J.A.: Deformations of the geometry of lipid vesicles. J. Phys. A Math. Gen. 36, 6281–6295 (2003) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cohen, H., Berkal, A.B.: Wave propagation in elastic membranes. J. Elast. 2, 45–57 (1972) CrossRefGoogle Scholar
  5. 5.
    Fosdick, R., Tang, H.: Surface transport in continuum mechanics. Math. Mech. Solids 14, 587–598 (2009) CrossRefGoogle Scholar
  6. 6.
    Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd edn. CRC Press, Boca Raton (1998) MATHGoogle Scholar
  7. 7.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323 (1975) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jost, J.: Riemannian Geometry and Geometric Analysis, 5th edn. Springer, Berlin (2008) MATHGoogle Scholar
  9. 9.
    Kadianakis, N.: On the geometry of Lagrangian and Eulerian descriptions in continuum mechanics. Z. Angew. Math. Mech. 79, 131–138 (1999) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Man, C.-S., Cohen, H.: A coordinate-free approach to the kinematics of membranes. J. Elast. 16, 97–104 (1986) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983) MATHGoogle Scholar
  12. 12.
    Murdoch, A.I.: A coordinate-free approach to surface kinematics. Glasg. Math. J. 32, 299–307 (1990) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Murdoch, A.I.: Some fundamental aspects of surface modelling. J. Elast. 80, 33–52 (2005) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Arch. Rat. Mech. Anal. 72, 61–78 (1979) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Rat. Mech. Anal. 2, 197–226 (1958) MATHCrossRefGoogle Scholar
  16. 16.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 4. Publish or Perish, Boston (1979) Google Scholar
  17. 17.
    Truesdell, C.: A First Course in Rational Continuum Mechanics, vol. 1. Academic Press, San Diego (1977) MATHGoogle Scholar
  18. 18.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004) Google Scholar
  19. 19.
    Yano, K.: Integral Formulas in Riemannian Geometry. Dekker, New York (1970) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations