  Recently, it was claimed that the two-phase local/nonlocal constitutive
models give well-posed nonlocal field problems and eliminates the ill-posedness
of the fully nonlocal constitutive models. In this study, it is demonstrated
that, both, the fully nonlocal and the two-phase local/nonlocal constitutive
models secrete ill-posed nonlocal boundary value problems. Moreover, it is
revealed that all Eringen integral and differential nonlocal constitutive
models secrete unsolvable nonlocal boundary value problems. In this study, it
is demonstrated that solutions of nonlocal elasticity problems are exist, and
Eringen constitutive model cannot determine these solutions. To overcome the
limitations of Eringen constitutive models, novel integral and differential
iterative nonlocal residual constitutive models are proposed. Using these two
constitutive models, the sum of the nonlocal residual field at a point is
iteratively formed. Then, this nonlocal residual is imposed to the local
boundary value problem. Thus, the nonlocal elasticity is obtained in the form
of a local boundary value problem with an imposed nonlocal residual field.
Using any of these constitutive models, a solution is guaranteed for a nonlocal
field problem. To show the effectiveness of the proposed constitutive models,
the nonlocal field problems of beams with different natural boundary conditions
are considered. The results of the proposed integral and differential
constitutive models are identical and feasible.
