Torsion of strain gradient bars

The governing differential equation and both classical and non-classical boundary conditions of strain gradient bars are derived using variational approach. A closed-form analytical solution is obtained for static torsion and the characteristic equation, which gives the natural frequencies, is derived and analytically solved for the free torsional vibrations of the strain gradient microbars. A fixed-fixed microbar is considered as a specific case to investigate the torsional size-dependent static and free-vibration behavior of strain gradient microbars. The results of the current model are compared to those of the modified couple stress and classical theories.