Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams

In the strain-driven model of nonlocal elasticity proposed by Eringen, the elastic strain is defined by a Fredholm integral equation in which the stress is the output of a convolution between the local response to an elastic strain and a smoothing kernel dependent on a nonlocal parameter. In the wake of this proposal, size effects in nano-beams were investigated in literature by adopting a differential formulation considered to be equivalent to the integral one. Recent improvements have however revealed that equivalence requires also the fulfilment of constitutive boundary conditions. Moreover, this strain-driven nonlocal elastic problem has been shown to be ill-posed, being conflicting with equilibrium requirements. A stress-driven integral constitutive law provides the natural way to get well-posed nonlocal elastic problems for application to nano-structures. The new integral constitutive law is formulated with explicit reference to plane and straight nano-beams according to the standard Bernoulli-Euler structural model. The solution procedure based on the stress-driven nonlocal law is described and adopted for the solution of a simple statically indeterminate scheme, thus showing effectiveness of the new model for the structural design of nano-devices.