Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams

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Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams (2019, background): “Nonlocal (strain-driven) stress fields in bounded structural domains are indeed not included in the affine manifold of equilibrated stresses fields, as recognized in literature [2, 13, 15, 16, 19, 33–35, 37–39] on the basis of the original contribution in [28].”
Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams (2017, background): “For structural schemes of applicative interest no solution exists [3].”
Nonlocal elasticity in nanobeams: the stress-driven integral model (2017, methodology): “Failure of the strain-driven nonlocal elastic law was finally clarified by proving non-existence of solutions for all elastic equilibrium problems of engineering interest ( Romano et al., 2017 ).”
Longitudinal vibrations of nano-rods by stress-driven integral elasticity (2019, background): “This occurrence is due to conflicting constitutive boundary conditions and equilibrium requirements [28], [29].”
Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory (2017, unknown): “However, using the mixed constitutive model with ξ 1 > 0, the ill-posedness of the purely nonlocal problem is eliminated, and therefore true solutions can be achieved using this formulation (Romano et al., 2017).”