Part of the Advanced Structured Materials book series (STRUCTMAT, volume 22)

# Shells and Plates with Surface Effects

Chapter

## Abstract

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.

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