Higher modes of buckling in shear deformable nanobeams

The size-dependent buckling instability of shear deformable nanobeams rested on a two-parameter elastic foundation is studied through the stress-driven nonlocal theory of elasticity and the kinematic assumptions of the Timoshenko beam theory. The small-scale size effects are taken into account by nonlocal constitutive relationships, which define the strains at each point as integral convolutions in terms of the stresses in all the points and a kernel. In this manner, the nonlocal elasticity formulation is well-posed and does not include inconsistencies usually arising using other nonlocal models. The size-dependent governing differential equations in terms of the transverse displacement and the cross-sectional rotation are decoupled, and closed form solutions are presented for the displacement functions. Proper boundary conditions are imposed and the buckling problem is reduced to finding roots of a determinant of a matrix, whose elements are given explicitly for different classical edge conditions. The closed form treatment of the problem avoids the numerical instabilities usually occurring within numerical techniques, and allows to find also higher buckling loads and shape modes. Several nanobeams rested on the Winkler or Pasternak elastic foundations and characterized by different boundary conditions, shear deformability, and nonlocality are considered and the critical loads and shape modes are presented, including those for the higher modes of buckling. Excellent agreements are found with the available approximate numerical results in the literature and novel insightful findings are presented and discussed, which are in accordance with experimental observations.