Snap-through and Bifurcation of Nano-arches on Elastic Foundation By the Strain Gradient and Nonlocal Theories

This paper presents the snap-through and bifurcation elastic stability analysis of nano-arch type structures with the Winkler foundation under transverse loadings by the strain gradient and stress gradient (nonlocal) theories. The equations of equilibrium are derived by using the variational method and virtual displacement theorem of minimum total potential energy. In the elastic stability analysis, von Karman's nonlinear strain component is included, with the deformation represented by a series solution. It is concluded that in general, the strain gradient theory pushes the system away from instability as compared to the classical theory. However, the nonlocal theory does the reverse and causes the system to experience instability earlier than that of the classical theory. Moreover, theories with different small-size considerations change the mechanism of instability in different ways. For example, in similar conditions, the strain gradient theory causes the system to reach a snap-through point, while the nonlocal theory causes the system to stop at a bifurcation critical point.