Reissner’s mixed variational theorem-based nonlocal Timoshenko beam theory for a single-walled carbon nanotube embedded in an elastic medium and with various boundary conditions

The nonlocal Timoshenko beam theories (TBTs), based on the Reissner mixed variational theorem (RMVT) and principle of virtual displacement (PVD), are derived for the analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium and with various boundary conditions, in which the Eringen nonlocal elasticity theory is used. The strong formulations of RMVT- and PVD-based nonlocal TBTs and their associated possible boundary conditions are presented. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Pasternak foundation model. The generalized displacement and force resultant components induced in the loaded SWCNT are obtained using the meshless collocation method, in which the shape functions of the primary variables are constructed using the differential reproducing kernel (DRK) method. The results show that RMVT-based nonlocal TBT is superior to its PVD-based counterpart in that the convergent rate of the RMVT-based nonlocal TBT is faster than that of the PVD-based one, that the predictions of generalized force resultants obtained using the RMVT-based nonlocal TBT are more accurate than those obtained using the PVD-based nonlocal one, and that the highest order of the base functions used in the RMVT-based nonlocal TBT is lower than that used in the PVD-based one.