Modified couple stress theory for three-dimensional elasticity in curvilinear coordinate system: application to micro torus panels

This research considers size effects in the linear three-dimensional elasticity analysis of micro-tori. The fundamental relations (displacement form) are derived for isotropic toroidal shells in the framework of the modified couple stress theory in the curvilinear coordinate system to predict the mechanical responses. A numerical solution for the displacement field is obtained using the GDQ method. The numerical results are in a close agreement with those found by the finite element and the Galerkin method. Parametric studies are conducted to explore the effect of size-dependency, micro-tori geometry, meridional and circumferential angle, toroidal shell thickness, and different boundary conditions on the distribution of the displacement fields. Numerical results for displacement also show that natural frequencies of micro-toroidal shells, predicted by modified couple stress theory, are less than those predicted by the classical theory, due to the significant effect of length scale parameter (related to material microstructures) on the mechanical responses. The use of general curvilinear coordinates in toroidal structures enables us to also study the mechanical behavior of irregular geometries, cap-shaped panel, saddle-shaped panel, and sectorial-shaped panel.