One-dimensional Eikonal Non-Local (ENL) 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 by considering that interaction distances between material points are computed from solving an eikonal equation with an 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 an isotropic damage mechanics framework, the damage variable tends to unity on a single integration point, and no damage diffusion occurs. In a one-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). Conversely, Jirásek 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.

Path-following methods for unstable structural responses induced by strain softening: a critical review

“…Moreover, no regularization methods will be used to induce sharp snap-back responses in the numerical examples. The presented path-following techniques can, however, be used even in that case without significant changes (see, e.g., [13,15,23,24,46]).…”

“…These methods were applied to non-linear FE simulations where material non-linearities were represented using Continuum Damage Mechanics models [15,23,24], phase-field formulations [16], and Thick Level-Set damage models [25]. Path-following methods were also employed in strong discontinuity simulations, where cracks were modeled using zero-thickness interface FEs [15,26,27], in the eXtended Finite Element Method (X-FEM) [28] and in the Embedded Finite Element Method (E-FEM) [22,[29][30][31].…”

One-dimensional study of boundary effects and damage diffusion for regularized damage models

“…Equation ( 10) can be integrated analytically, 5 and the effective distance between points x and y reads [37,47]:…”

“…According to such an approach, the main modification with respect to the INL formulation is the use of the lx y instead of l x y in (5). Recent works have shown that the ENLI approach is suitable for modeling strain localization processes [37,38,40,47].…”

“…Indeed, damage tends to increase the effective distances. When it tends to the unity, the effective distance between two points separated by the highly damaged zone tends to infinity [37,38,47], so that the non-local interaction between the considered points vanishes. To the authors' knowledge, no numerical implementation of such an approach exists in the literature.…”