On torsion of nonlocal Lam strain gradient FG elastic beams

The nonlocal strain gradient theory of elasticity is the focus of numerous studies in literature. Eringen's nonlocal integral convolution and Lam's strain gradient model are unified by a variational methodology which leads to well-posed structural problems of technical interest. The proposed nonlocal Lam strain gradient approach is presented for functionally graded (FG) beams under torsion. Static and dynamic responses are shown to be significantly affected by size effects that are assessed in terms of nonlocal and gradient length parameters. Analytical elastic rotations and natural frequencies are established by making recourse to a simple solution procedure which is based on equivalence between integral convolutions and differential equations supplemented with variationally consistent (but non-standard) nonlocal boundary conditions. Effects of Eringen's nonlocal parameter and stretch and rotation gradient parameters on the torsional behavior of FG nano-beams are examined and compared with outcomes in literature. The illustrated methodology is able to efficiently model both stiffening and softening torsional responses of modern composite nano-structures by suitably tuning the small-scale parameters