A well-posed non-local theory in 1D linear elastodynamics

We show how to construct a well-posed theory of purely non-local elasticity by kernel modification. Specifically, we modify the classical Helmholtz kernel so that the constitutive boundary conditions associated with it are replaced by constraints that emerge from the natural boundary conditions of the problem at hand. The procedure is illustrated by two examples, one dealing with a statically indeterminate problem and the other concerning free vibrations of a cantilever beam. The defining feature of the method is that the modified kernel is no longer a difference kernel. This outcome is a consequence of the incorporation of the problem's boundary conditions, which affects the kernel near the boundaries and, consequently, induces a different mechanical response in dependence of the distance from those. In contrast, negligible changes are found in the interior of the material. Still, the modified kernel remains symmetric and positive definite, which property guarantees that the strain energy is quadratic and positive definite, and it complies with the impulsivity requirement, by which it reverts to the classical local theory in the limit of a vanishing non-local length-scale. Kernel modification is conceptually different from the two-phase approach under many respects, most notably because it gets away from the need to introduce extra boundary conditions besides those naturally associated with the physics of the problem.