Theory of Planar Surface Continua

  • Marcus AßmusEmail author
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.


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© 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

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