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.

中文翻译：

---

一维 Eikonal 非局部损伤模型：计算相互作用距离的积分规则和间接加载控制对损伤定位的影响

摘要 Eikonal Non-Local 方法通过考虑材料点之间的相互作用距离是通过求解具有各向同性/各向异性损伤相关度量黎曼函数的 Eikonal 方程来模拟损伤相关的非局部相互作用。在有限元环境中，这样的公式表现出良好的正则化特性，并且当非局部相互作用消失时，自然允许人们对应变局部化进行建模。在各向同性损伤力学框架中，损伤变量在单个积分点上趋于统一，不会发生损伤扩散。在单维环境中，评估两点之间的相互作用距离涉及计算积分，其中积分函数取决于它们之间的损伤场。德斯莫拉特等人。[9] 使用梯形规则进行积分（假设相邻积分点之间的损伤场是恒定的）。相比之下，Jirasek 和 Desmorat [19] 假设损伤场在附近的积分点之间是线性的。主要受提高积分精度的推动，该方法导致更渐进的损坏过程，并在损坏局部化时更容易跟踪结构响应。在这个贡献中，我们表明尽管有这些优点，但这样的选择会导致寄生虫损伤扩散问题，从而使标准梯形规则的使用更加相关。而且，一旦做出这个选择，