Theory of Planar Surface Continua

  • Marcus Aßmus
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


Planar continua analogous to the direct approach will be introduced at this point, i.e. we will operate on a deformable surface instead of a voluminous body a priori. The description is following the Zhilinean path (Zhilin, Applied mechanics - foundations of shells theory (in russian). Publisher of the Polytechnic University, St. Petersburg, 2006, [11]) though using a more common and unique notation. However, the main hypotheses of the classical mechanics of continuous media hold true. The material surface \(\mathfrak S \) is a coherent and compact set of material space points \(\mathfrak M\). The boundary of this point set is indicated by the domain boundary \(\partial \mathfrak S\). First, let us start with a homogeneous body, i.e. all material points have the same characteristics. Another limitation is in the isotropy assumption, i.e. all directions are equal. Each material point has three translational and two rotational degrees of freedom as kinematic variables, with the rotations introduced as independent degrees of freedom. The continuum hypothesis applies, maintaining the continuity of material points during deformation. The relationship to the three dimensional body \(\mathfrak B\) in whose volume V the surface is embedded can be represented by the following expression, where the assumption \(h=\mathrm {const.}\) holds for the structural thickness.


  1. 1.
    Aßmus M (2018) Global structural analysis at photovoltaic modules: theory, numerics, application (in German). Dissertation, Otto von Guericke University MagdeburgGoogle Scholar
  2. 2.
    Aßmus M, Eisenträger J, Altenbach H (2017) On isotropic linear elastic material laws for directed planes. In: Proceedings of the 11th international conference on shell structures: theory and applications (SSTA 2017), Gdańsk, Poland, pp 57–60.
  3. 3.
    Aßmus M, Eisenträger J, Altenbach H (2017) Projector representation of isotropic linear elastic material laws for directed surfaces. Zeitschrift für Angewandte Mathematik und Mechanik 97(-):1–10.
  4. 4.
    Bertram A, Glüge R (2015) Solid mechanics: theory, modeling, and problems. Springer, Cham.
  5. 5.
    Libai A, Simmonds JG (1983) Nonlinear elastic shell theory. Adv Appl Mech 23:271–371. Scholar
  6. 6.
    Noll W (1958) A mathematical theory of the mechanical behavior of continuous media. Arch Rat Mech Anal 2(1):197–226. Scholar
  7. 7.
    Pal’mov VA (1964) Fundamental equations of the theory of asymmetric elasticity. J Appl Math Mech 28(3):496–505. Scholar
  8. 8.
    Tonti E (1972) On the mathematical structure of a large class of physical theories. Accademia Nazionale Dei Lincei 52(1):48–56MathSciNetGoogle Scholar
  9. 9.
    Tonti E (2013) The mathematical structure of classical and relativistic physics - a general classification diagram. Birkhäuser, Basel.
  10. 10.
    Vlachoutsis S (1992) Shear correction factors for plates and shells. Int J Numer Methods Eng 33(7):1537–1552. Scholar
  11. 11.
    Zhilin PA (2006) Applied mechanics - foundations of shells theory (in russian). Publisher of the Polytechnic University, St. Petersburg.

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of MechanicsOtto von Guericke UniversityMagdeburg, Saxony-AnhaltGermany

Personalised recommendations