# Mechanics of Thin Elastic Shells

• Yury Vetyukov
Chapter
Part of the Foundations of Engineering Mechanics book series (FOUNDATIONS)

## Abstract

We begin with the asymptotic splitting of the equations of the three-dimensional theory of elasticity for a thin plate into a problem over the thickness and the equations of the classical plate model. All groups of the three-dimensional equations (equilibrium, compatibility, etc.) are processed separately. Both the material anisotropy and inhomogeneity in the thickness direction are included in the analysis along with the electromechanical coupling in the form of piezoelectric effects. The classical nonlinear theory of curved shells follows with the direct approach to a material surface with five degrees of freedom of particles: three translations and two rotations. Equations of equilibrium, boundary conditions and general constitutive relations are the consequences of the principle of virtual work. Further we transform the equations to the differential operator of the reference configuration with the Piola tensors, which allows linearizing the equations in the vicinity of a pre-stressed and pre-deformed state. Several example problems for a cylindrical shell are considered. A novel finite element scheme with a smooth approximation of the surface of the shell concludes the chapter. The method is presented in Mathematica environment for linear plate problems, and then a general implementation for large deformations of curved shells is discussed. We demonstrate the convergence properties of the numerical scheme on several examples, some of which have been considered in the literature before. This chapter quotes extensively from Eliseev and Vetyukov (Acta Mech. 209(1–2):43–57, 2010) with permission from Springer Science and Business Media, from Vetyukov et al. (Int. J. Solids Struct., 48(1):12–23, 2011) with permission from Elsevier, from Vetyukov (Z. Angew. Math. Mech., 94(1–2):150–163, 2014) with permission from Wiley-VCH and from Eliseev and Vetyukov (Shell Structures: Theory and Applications, vol. 3, pp. 81–84, CRC Press, London, 2014) with permission from Taylor & Francis Group. Other results of the author, previously presented in Vetyukov and Belyaev (Proceedings of the Tenth International Conference on Computational Structures Technology, p. 19, Civil-Comp Press, Stirlingshire, 2010) and Vetyukov and Krommer (Proceedings of SPIE—The International Society for Optical Engineering, vol. 7647, 2010), were included in the material of the present chapter.

## Keywords

Cylindrical Shell Constitutive Relation Tangent Plane Virtual Work Reference Configuration
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.

## Supplementary material

310026_1_En_4_MOESM1_ESM.zip (4.9 mb)
(ZIP 4.9 MB)

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