Variational nonlocal gradient elasticity for nano-beams

In the paper ( Zaera, Serrano, & Fernandez-Saez, 2019 ) it is claimed that the nonlocal strain gradient theory (NSGT) leads to ill-posed structural problems. This conclusion was motivated by observing that Constitutive Boundary Conditions (CBC) conflict with non-standard Kinematic and Static Higher-Order Boundary Conditions (KHOBC) - (SHOBC). In the present study, it is shown that no ill-posedness holds if NSGT is established by an adequate variational formulation, with appropriate test fields. KHOBC and SHOBC have nothing to do with the proper mathematical formulation and thus they have not to be prescribed, while standard kinematic and static boundary conditions and CBC have to be imposed to close the relevant nonlocal gradient problem. This conclusion follows from a well-posed abstract variational scheme conceived for nonlocal gradient inflected beams. The treatment provides as special cases most of the size-dependent models adopted in Engineering Science to assess size-effects in nanostructures, such as NSGT, strain-driven and stress-driven local-nonlocal elasticity approaches. Additionally, a well-posed Nonlocal Stress Gradient (NStressG) model is presented, coupling the stress-driven nonlocal strategy (Romano and Barretta, 2017) with the stress gradient elasticity. The presented methodology is elucidated and validated by investigating the structural behavior of a variety of inflected nano-beams of nanotechnological interest, such as sensors and actuators. NStressG is able to predict both softening and hardening nonlocal responses and, unlike the special stress gradient elasticity model, leads to well-posed structural problems in Nanomechanics.