A well-posed theory of linear non-local elasticity

We address ill-posedness of Eringen's non-local theory of elasticity, as a result of the implicit presence of boundary conditions, termed constitutive (CBCs), embedded in the choice of the attenuation function (kernel). Such CBCs supplement the natural boundary conditions of the problem, hence the problem becomes overdetermined and, almost inevitably, ill-posed. Although this feature is true in general, it is especially manifest when the kernel is the Green function of a differential operator. To guarantee well-posedness for any loading, we propose a method by which the kernel is modified only in terms of the CBCs, which are selected to coincide with the natural boundary conditions of the problem. Taking the Helmholtz kernel as an example, we show that, after modification, the self-adjoint character of the integral operator is preserved, which guarantees that the attached elastic energy is a (positive definite) quadratic functional. By eigenfunction expansion of the kernel, we prove, through some examples, that the results obtained from the differential formulation correspond to those given by the integral problem, a result largely disputed in the literature. Along the process, we explain the inevitable appearance of scenarios, sometimes named paradoxes, which lead to solutions that match those of local elasticity. This outcome simply emerges whenever any particular problem produces a local curvature field which matches one of the kernel's eigenfunctions and it is in no way connected with the problem of the CBCs. Given that the modified kernel remains close to the original kernel away from the boundaries, results correspond to Eringen's in the limit of an infinite domain. Comparison is also made with respect to the Two Phase Non-local Model (TPNM), that also warrants well-posedness for any load and yet, in contrast to this approach, requires extra nonphysical boundary conditions.