On the finite element formulation for second-order strain gradient nonlocal beam theories

Finite element (FE) formulation of nanobeams using second-order positive/negative strain gradient theories is presented. The variational statement for beam, written using Hamilton's principle involving virtual work of stress/moment resultants, with C-1 continuity requirement leads to unsymmetric nonlocal terms whereas the symmetrization of the nonlocal terms leads to C-2 continuity requirement. The C-2 continuous FE with proper boundary conditions is accurate for the modeling of nanobeams using second-order strain gradient theories. The negative strain gradient model predicts results matching with the experimental investigations on microbeams. The strain gradient theory based plate FE is also formulated by extending C-2 continuous beam FE formulation.