Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution

Abstract

According to paradoxical behaviors of common differential nonlocal elasticity, employing the two-phase local/nonlocal elasticity, to consider the size effects of nanostructures, has recently attracted the attentions of nano-mechanics researchers. Now, due to more complexity of the two-phase elasticity problems than the differential nonlocal ones, it is essential to achieve efficient methods for studying the mechanical characteristics of two-phase nanostructures. Therefore, in this work, the exact solution corresponding to the vibrations of two-phase Timoshenko nanobeams is provided for the first time. Furthermore, the shear-locking problem is investigated in the case of two-phase finite-element method (FEM), and since the FE model of local/nonlocal nanobeam is more complex than the classic one, due to coupling of all elements together, one of the main aims of the present work is to create an efficient locking-free local/nonlocal FEM with a simple and efficient beam element. To extract the exact natural frequencies, the basic form of two-phase elasticity is replaced with the equal differential equation and the obtained higher-order governing equations are solved by satisfying additional constitutive boundary conditions. To construct the two-phase FE model, an efficient and simple shear-locking-free Timoshenko beam element is introduced, and next, basic form of two-phase elasticity including local and integral form of nonlocal elasticity is utilized. No need for satisfying higher-order boundary conditions, shear-locking-free, simple shape functions and well convergence are advantages of the present two-phase finite element model. Several convergence and comparison studies are conducted, and the reliability and locking-free properties of the present two-phase finite element model are confirmed. Also, the influences of two-phase elasticity on the natural frequencies of Timoshenko nanobeams with different thickness ratios are studied.

Appendix

$$\begin{gathered} \alpha = EI;\,\beta = k_{s} GA; \hfill \\ K_{e} = \left[ {\begin{array}{*{20}c} {\frac{12\alpha }{{l_{e}^{3} }}} & {\frac{6\alpha }{{l_{e}^{2} }}} & 0 & { - \frac{12\alpha }{{l_{e}^{3} }}} & {\frac{6\alpha }{{l_{e}^{2} }}} & 0 \\ {\frac{6\alpha }{{l_{e}^{2} }}} & {\frac{4\alpha }{{l_{e} }}} & {\frac{\alpha }{{l_{e} }}} & { - \frac{6\alpha }{{l_{e}^{2} }}} & {\frac{2\alpha }{{l_{e} }}} & { - \frac{\alpha }{{l_{e} }}} \\ 0 & {\frac{\alpha }{{l_{e} }}} & {\frac{\alpha }{{l_{e} }} + \frac{{\beta l_{e} }}{3}} & 0 & { - \frac{\alpha }{{l_{e} }}} & { - \frac{\alpha }{{l_{e} }} + \frac{{\beta l_{e} }}{6}} \\ { - \frac{12\alpha }{{l_{e}^{3} }}} & { - \frac{6\alpha }{{l_{e}^{2} }}} & 0 & {\frac{12\alpha }{{l_{e}^{3} }}} & { - \frac{6\alpha }{{l_{e}^{2} }}} & 0 \\ {\frac{6\alpha }{{l_{e}^{2} }}} & {\frac{2\alpha }{{l_{e} }}} & { - \frac{\alpha }{{l_{e} }}} & { - \frac{6\alpha }{{l_{e}^{2} }}} & {\frac{4\alpha }{{l_{e} }}} & {\frac{\alpha }{{l_{e} }}} \\ 0 & { - \frac{\alpha }{{l_{e} }}} & { - \frac{\alpha }{{l_{e} }} + \frac{{\beta l_{e} }}{6}} & 0 & {\frac{\alpha }{{l_{e} }}} & {\frac{\alpha }{{l_{e} }} + \frac{{\beta l_{e} }}{3}} \\ \end{array} } \right]; \hfill \\ \end{gathered}$$

$$\begin{gathered} m_{1} = \rho A;\,m_{2} = \rho I; \hfill \\ M = \left[ {\begin{array}{*{20}c} {\frac{{13l_{e} m_{1} }}{35} + \frac{{6m_{2} }}{{5l_{e} }}} & {\frac{1}{210}\left( {11l_{e}^{2} m_{1} + 21m_{2} } \right)} & { - \frac{{m_{2} }}{2}} & {\frac{{9l_{e} m_{1} }}{70} - \frac{{6m_{2} }}{5l}} & {\frac{1}{420}\left( { - 13l_{e}^{2} m_{1} + 42m_{2} } \right)} & { - \frac{{m_{2} }}{2}} \\ {\frac{1}{210}\left( {11l_{e}^{2} m_{1} + 21m_{2} } \right)} & {\frac{1}{105}\left( {l_{e}^{3} m_{1} + 14l_{e} m_{2} } \right)} & {\frac{{l_{e} m_{2} }}{12}} & {\frac{1}{420}\left( {13l_{e}^{2} m_{1} - 42m_{2} } \right)} & { - \frac{1}{420}l_{e} \left( {3l_{e}^{2} m_{1} + 14m_{2} } \right)} & { - \frac{{l_{e} m_{2} }}{12}} \\ { - \frac{{m_{2} }}{2}} & {\frac{{l_{e} m_{2} }}{12}} & {\frac{{l_{e} m_{2} }}{3}} & {\frac{{m_{2} }}{2}} & { - \frac{{l_{e} m_{2} }}{12}} & {\frac{{l_{e} m_{2} }}{6}} \\ {\frac{{9l_{e} m_{1} }}{70} - \frac{{6m_{2} }}{5l}} & {\frac{1}{420}\left( {13l_{e}^{2} m_{1} - 42m_{2} } \right)} & {\frac{{m_{2} }}{2}} & {\frac{{13l_{e} m_{1} }}{35} + \frac{{6m_{2} }}{5l}} & {\frac{1}{210}\left( { - 11l_{e}^{2} m_{1} - 21m_{2} } \right)} & {\frac{{m_{2} }}{2}} \\ {\frac{1}{420}\left( { - 13l_{e}^{2} m_{1} + 42m_{2} } \right)} & { - \frac{1}{420}l_{e} \left( {3l_{e}^{2} m_{1} + 14m_{2} } \right)} & { - \frac{{l_{e} m_{2} }}{12}} & {\frac{1}{210}\left( { - 11l_{e}^{2} m_{1} - 21m_{2} } \right)} & {\frac{1}{105}\left( {l_{e}^{3} m_{1} + 14l_{e} m_{2} } \right)} & {\frac{{l_{e} m_{2} }}{12}} \\ { - \frac{{m_{2} }}{2}} & { - \frac{{l_{e} m_{2} }}{12}} & {\frac{{l_{e} m_{2} }}{6}} & {\frac{{m_{2} }}{2}} & {\frac{{l_{e} m_{2} }}{12}} & {\frac{{l_{e} m_{2} }}{3}} \\ \end{array} } \right]; \hfill \\ \end{gathered}$$