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On the basis of Reissner's mixed variational theorem (RMVT), rather than the principle of virtual displacement (PVD), the authors presented a nonlocal Timoshenko beam theory (TBT) for the geometrically nonlinear static analysis of multiwalled carbon nanotubes (MWCNT) embedded in an elastic medium. The embedded MWCNT was subjected to mechanical loads on its outer-most surface, with combinations of free, simply supported, and clamped edge conditions. The van der Waals interaction between any pair of walls constituting the MWCNT was considered, and the interaction between the MWCNTand its surrounding medium was simulated using the Pasternak-type foundation model. In the formulation, the governing equations of a typical wall and the associated boundary conditions were derived, in which von Kármán geometrical nonlinearity was considered. Eringen's nonlocal elasticity theory was used to account for the small-length scale effect. The deformations induced in the embedded MWCNT were obtained using the differential quadrature method and a direct iteration approach. In the numerical examples, solutions of the RMVT-based nonlocal TBT converged rapidly, and the convergent solutions of its linear counterpart closely agreed with the analytical and numerical solutions of the PVD-based nonlocal beam theories available in the literature. [ABSTRACT FROM AUTHOR]

A differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving partial differential equations governing a certain physical problem. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the DRK interpolation function, without directly differentiating the DRK interpolation function. In addition, the shape function of the DRK interpolation function at each sampling node is separated into a primitive function processing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied. A point collocation method based on the present DRK interpolation is developed for the analysis of one-dimensional bar problems, two-dimensional potential problems, and plane problems of elastic solids. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach, with excellent accuracy and fast convergence rate. [ABSTRACT FROM AUTHOR]