Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams

A debated issue, in applications of ERINGEN's nonlocal model of elasticity to nanobeams, is the paradox concerning the solution of simple beam problems, such as the cantilever under end-point loading. In the adopted nonlocal model, the bending field is expressed as convolution of elastic curvature with a smoothing kernel. The inversion of the nonlocal elastic law leads to solution of a FREDHOLM integral equation of the first kind. It is here shown that this problem admits a unique solution or no solution at all, depending on whether the bending field fulfils constitutive boundary conditions or not. Paradoxical results found in solving nonlocal elastostatic problems of simple beams are shown to stem from incompatibility between the constitutive boundary conditions and equilibrium conditions imposed on the bending field. The conclusion is that existence of a solution of nonlocal beam elastostatic problems is an exception, the rule being non-existence for problems of applicative interest. Numerical evaluations reported in the literature hide or shadow this conclusion since nodal forces expressing the elastic response are not checked against equilibrium under the prescribed data. The cantilever problem is investigated as case study and analytically solved to exemplify the matter.