Shells and Plates with Surface Effects

  • Holm AltenbachEmail author
  • Victor A. Eremeyev
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 22)


The through-the-thickness integration procedure applied to a three-dimensional (3D) slender body leads to exact two-dimensional (2D) equations of plates and shells, see [36]. The procedure can be considered as a specific homogenization technique which results in a 2D generalized media—the non-linear theory of shells of Cosserat type. Within this theory the shell is described as a deformable surface each point of which has 3 translational and 3 rotational degrees of freedom similar to the 3D Cosserat continuum [15]. Below we discuss the through-the-thickness integration procedure applied to the non-classical problem of the theory of surface elasticity [21]. The theory can be applied to modeling of surface effects which are important in mechanics of nanostructured materials [11, 55]. Applying the through-the-thickness integration procedure we reduce 3D equations to 2D ones. The effective (apparent) stiffness properties of the shell are changed in comparison with the classical models of shells. Some examples of a plate bending are discussed taking into account surface effects.


Surface Stress Strain Energy Density Surface Elasticity Shear Correction Factor Deformable Surface 
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.



The second author was supported by the DFG grant No. AL 341/33-1 and by the RFBR with the grant No. 12-01-00038.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für MaschinenbauOtto-von-Guericke-Universtät MagdeburgMagdeburgGermany
  2. 2.Faculty of Mechanical EngineeringOtto-von-Guericke UniversityMagdeburgGermany
  3. 3.South Scientific Center of RASci and South Federal UniversityRostov on DonRussia

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