Instability by Extension of an Elastic Nanorod

The static stability of an elastic, incompressible nanorod subjected to an extensional force is analyzed. The force is applied to a rigid rod that is welded to the free end of the nanorod. The material behavior of the nanorod is described using a two-phase local/nonlocal stress-driven model. Mathematically, the problem is formulated as a system of nonlinear differential equations suitable for nonlinear analysis. For the analysis, the Liapunov-Schmidt method is employed. Depending on a geometric parameter (the length of the rigid rod) and nonlocal parameters (the small length-scale parameter and the phase parameter), the buckling load and post-buckling behavior of the nanorod are determined. The results show that the nonlocal effect increases the buckling load, indicating a stiffening effect. An increase in the length of the rigid rod decreases the buckling load. Regarding the post-buckling behavior, it is shown that both supercritical and subcritical bifurcations can occur, depending on the values of the geometric and nonlocal parameters. The occurrence of a subcritical bifurcation, which is highly undesirable in real-world constructions, is a novel effect not observed in the classical Bernoulli-Euler theory.