Higher order couple stress theory of plates and shells

Abstract

New higher order models of the couple stress plates and shells have been developed here. The 3-D equations of the linear couple stress elasticity have been presented in an orthogonal system of coordinates. For the creation 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 a special hypothesis based on assumptions that consider the fact that the considered plates and shells are thin. Higher order theory is based on the expansion of the 3-D equations of the linear couple stress 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 linear couple stress 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 with boundary conditions for the Legendre polynomials coefficients has been obtained. All equations for higher order theory of the couple stress 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 considering micropolar couple stress and rotation effects. K E Y W O R D Scouple stress, higher order theory, Legendre polynomial, plates, shell 1834

8‐node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

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Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates

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