Wave dispersion and bifurcation analyses of eikonal gradient-enhanced isotropic damage models

Often used to describe material degradation, local continuum damage models inevitably lead to an ill-posed problem. In quasi-static analysis, the singularity of the acoustic tensor implies the loss of ellipticity of the boundary value problem. In dynamics, the hyperbolic equilibrium equations become elliptic, which defines the loss of hyperbolicity, and waves cannot propagate in a local medium experiencing strain-softening. Classic gradient-enhanced damage models are known to preserve the well-posedness of the initial value problem, and waves can propagate in such a medium in a dispersive way. This paper addresses the question of wave dispersion and localization analysis in a gradient-enhanced medium with evolving non-local interactions. Particular attention is given to the gradient-enhanced Eikonal continuum, where damage-dependent interactions are considered. The regularization properties of the model are first studied through a one-dimensional problem of dispersive wave propagation. The analysis is completed by a three-dimensional bifurcation study based on a geometric representation, which generalizes the wave dispersion results. It is shown analytically that the gradient-enhanced Eikonal model preserves the localization-limiting character of non-local approaches and exhibits an intrinsic re-localizing nature for high initial damage levels. Like classic non-local approaches, loading waves propagate and can be transformed into a unique stationary harmonic localization wave, associated with a critical wavelength. • Localization analysis of a gradient-enhanced isotropic eikonal non-Local damage model. • Damage dependent non-local interactions. • One-dimensional wave dispersion analysis and three-dimensional bifurcation analysis. • The eikonal model preserves the localization-limiting character of standard gradient-enhanced non-local approaches. • The eikonal model exhibits an intrinsic re-localizing nature for high initial damage levels.