On the well-posedness of Eringen’s non-local elasticity for harmonic plane wave problems

In this paper, we establish a criterion for well-posedness of Eringen’s non-local elasticity theory for problems of harmonic plane waves in domains with non-empty boundaries, and introduce a novel method for solving well-posed problems. The criterion for well-posedness says that for problems of harmonic plane waves, Eringen’s non-local elasticity theory is well-posed when the constitutive boundary conditions contain all equilibrium boundary conditions, otherwise it is ill-posed in the sense of no solutions. With this well-posedness criterion, it is easy to check whether a non-local harmonic plane wave problem is well-posed or not. If it is a well-posed problem, its solution will be found by employing the novel method. It has been shown that Eringen’s method, which has been used widely to solve problems of non-local harmonic plane waves, does not give their correct solutions. Therefore, it must be replaced by the novel method. As an application of the criterion for well-posedness and the novel method, two well-posed problems of harmonic plane waves are considered including Rayleigh waves and SH waves propagating in traction-free non-local isotropic elastic half-spaces. Exact solutions of these problems have been obtained including explicit expressions of displacements, local and non-local stresses and dispersion equations.