Nonlocal Theory of Plates and Shells Based on Legendre’s Polynomial Expansion

Abstract

New nonlocal models of plates and shells based on Legendre’s polynomial series expansion have been developed here. The 3-D dynamic equations of the nonlocal elasticity have been presented in an orthogonal system of coordinates. For the development of 2-D models of plates and shells the curvilinear system of coordinates related to the middle surface of the shell has been used along with special hypothesizes based on assumptions that take into account the fact that the plate and shells are thin. Higher order theory is based on the expansion of the 3-D equations of the nonlocal theory of elasticity into Fourier series in terms of Legendre polynomials. The stress and strain tensors, as well as vectors of displacements have been expanded into Fourier series in terms of Legendre polynomials with respect to thickness. Thereby, all equations of the nonlocal theory of elasticity have been transformed to the corresponding equations for the Legendre polynomials coefficients. Then, in the same way as in the classical theory of elasticity, a system of differential equations in terms of displacements with initial and boundary conditions for the Legendre polynomials coefficients has been obtained. All equations for higher order theory of nonlocal plates in Cartesian and polar coordinates as well as for cylindrical and spherical shells in coordinates related to the shells geometry have been developed and presented here in detail. The obtained equations can be used for calculating stress-strain and for modelling thin walled structures in macro, micro and nano scale when taking into account size dependent and nonlocal effects.