Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams

Nonlocal strain gradient integral model of elasticity, extension of the fully nonlocal integral law, is widely adopted to assess size effects in nano-beams. The bending moment is sum of convolutions of elastic curvature and of its derivative with a smoothing kernel. For nanomechanical problems on unbounded domains, such as in wave propagation, the nonlocal strain gradient integral relation is equivalent to a differential law with constitutive conditions of vanishing at infinity. For bounded nano-beams, the constitutive boundary conditions (CBC) must be added to close the constitutive model. The formulation of these CBC is an original contribution of the paper. Equivalence between nonlocal strain gradient integral model of elasticity and the differential problem with CBC is proven. It is shown that the CBC do not conflict with equilibrium and provide a viable approach to study size-dependent phenomena in nano-beams of applicative interest. Theoretical outcomes are illustrated by examining the static scheme of a nano-actuator modelled by a nano-cantilever inflected by an end-point load. The relevance of a proper formulation of boundary conditions is elucidated by comparing the numerical results with previous attempts in literature.