One-dimensional Eikonal non-local damage models: influence of the integration rule for computing interaction distances and indirect loading control on damage localization

Abstract The Eikonal Non-Local approach models damage dependent non-local interactions through considering that interaction distances between material points are computed from solving an Eikonal equation with a isotropic/anisotropic damage dependent metric Riemann function. In the finite element context, such a formulation exhibits good regularization properties and naturally allows one to model strain localization when non-local interactions vanish. In a isotropic damage mechanics framework, the damage variable tends to unity on a single integration point, and no damage diffusion occurs. In a uni-dimensional context, evaluating the interaction distance between two points comes into computing an integral where the integrated function depends on the damage field between them. Desmorat et al. [9] performed the integration using a trapezoidal rule (under the assumption that the damage field is constant between adjacent integration points). In contrast, Jirasek and Desmorat [19] assumed that the damage field is linear between nearby integration points. Mainly motivated by improved integration accuracy, this method leads to a more gradual damaging process and makes it easier to follow the structural response when damage localizes. In this contribution, we show that despite these advantages, such a choice induces parasite damage diffusion issues, making the use of a standard trapezoidal rule more relevant. Moreover, once this choice is made, the structural response during damage localization can be conveniently described using a path-following method based on controlling the non-local strain variation of the localizing element instead of the local strain as in the works cited above.