Higher order theory of micropolar plates and shells

Abstract

AbstractNew higher order models of micropolar plates and shells have been developed here. The 3‐D dynamic equations of the micropolar elasticity have been presented in an orthogonal system of coordinates using a generalized variational principle. For the creation of 2‐D models of plates and shells the curvilinear system of coordinates related to the middle surface of the shell have been used along with a special hypothesis based on assumptions that take into account the fact that the rod is thin. Higher order theory is based on a generalized variational principle and the expansion of the 3‐D equations of the micropolar theory of elasticity into Fourier series in terms of Legendre polynomials. The stress and strain tensors, as well as vectors of displacements and rotation have been expanded into Fourier series in terms of Legendre polynomials with respect to thickness. Thereby, all equations of the micropolar theory of elasticity (including generalized Hooke's law) 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 and rotation with initial and boundary conditions for the Legendre polynomials coefficients have been obtained. All equations for higher order theory of micropolar 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 the stress‐strain and for modelling thin walled structures in macro, micro and nano scale when taking into account micropolar couple stress and rotation effects.