Aifantis versus Lam strain gradient models of Bishop elastic rods

In this paper, the size-dependent static behavior of Bishop rods is investigated by Lam and Aifantis strain gradient formulations of elasticity. Appropriate constitutive boundary conditions are established for both the theories by making recourse to a variational approach. Unlike contributions of literature, no higher-order kinematic and static boundary conditions, which have not a clear physical meaning, are required to close the relevant gradient problems. The proposed methodology leads to mathematically well-posed elastostatic problems and is illustrated by examining size effects in selected thick rods of nanotechnological interest. Exact solutions of Bishop nano-rods are detected for a variety of loading systems and kinematic boundary conditions. Peculiar properties, merits, and implications of both the strain gradient formulations, equipped with the proper boundary conditions, are illustrated and commented. The outcomes can be useful for the design and optimization of rod-like thick components of nanoelectromechanical systems.