On the well-posedness of stress-driven nonlocal elasticity theories

Although nonlocal elasticity theories have been widely used to study various nonlocal problems, including problems of nanostructures and problems of harmonic plane waves with a huge number of published papers, they have not been proven to be well-posed at all for the general case. Therefore, it is extremely necessary and important to prove their well-posedness. In this paper, it has been proven that the stress-driven fully nonlocal elasticity theory is ill-posed for any harmonic plane wave problem in the sense of no solution, while the stress-driven two-phase nonlocal elasticity theory is well-posed for all problems of harmonic plane waves. It implies from this fact that we cannot use the stress-driven fully nonlocal elasticity theory to study problems of harmonic plane waves, while the stress-driven two-phase nonlocal elasticity theory is good tool for this task. It is remarkable that the ill-posedness of the stress-driven fully nonlocal elasticity theory for all problems of harmonic plane waves is contrary to the well-posedness of nanobeam problems of this theory. To illustrate the well-posedness of the stress-driven two-phase nonlocal elasticity theory, the reflection of (shear horizontal) waves from the traction-free surface of stress-driven two-phase nonlocal isotropic elastic half-spaces is investigated.