While damage mechanical models have been developed for decades now, the transition from diffuse degradation to a single fracture zone still remains a difficulty. Indeed, beyond a certain loading level, damage models generally lead to localized solutions of the associated continuum mechanics problem. This localized area is used to describe a crack but, without regularization, leads to an ill-posed problem that results into well-known issues such as spurious mesh dependency. Different solutions have been proposed in the literature, relying on a spatial non-local treatment of the regularization. The driving forces are averaged to introduce an internal length that allows mesh independency. The averaging can be conducted directly or through an auxiliary gradient problem. The internal length may be constant, in which case all interactions are not cut by a fully damaged zone, or it can evolve with stress to account for boundary conditions and cut the interactions across a crack. The aim of this paper is to present a non-local gradient regularization technique called Eikonal non-local gradient. It is a gradient regularization model where the classical internal length is replaced by a varying metric originating from the acoustic wave speed. The potentially anisotropic metric evolves with damage, as in phase-field models, thus cutting interactions when the damage reaches one. The first part of the paper presents this model and the associated properties on a 1D problem in tension. A comparison to other classical methods is also given. This example shows the good behavior of this regularization technique. It is shown that the evolving characteristic length leads to a rather sharp and brittle post-peak behavior. A plasticity model is then coupled in order to decrease the brittleness of the mechanical response. This choice is illustrated to better understand the interactions between damage, regularization and plasticity.