It is known that damage or inelastic softening can cause an ill-posed problem leading to localization and mesh-dependence in finite element simulations. In this paper, a nonlocal hardening variable, $\bar{\kappa }$, is introduced in a finite deformation Eulerian formulation of inelasticity with a rate-independent smooth elastic–inelastic transition. This nonlocal variable is defined over an Eulerian region of nonlocality, which is a sphere with radius equal to the characteristic length, $L_{\mathrm{c}}$, defined in the current deformed geometry of the material. Two models of this nonlocal hardening variable are explored. One model where $\bar{\kappa }$ is the minimum value of the local hardening $\kappa$ within the region of nonlocality, and another model where $\bar{\kappa }$ is the average of $\kappa$ in the same region. The influence of the nonlocal hardening variable is studied using an example of a plate that is loaded by a prescribed boundary displacement causing formation of a shear band. Predictions of the applied load vs. displacement curves and contour plots of the total distortional deformation of the plate and the hardening variable $\kappa$ are studied. The model based on the minimum value of $\kappa$ in the nonlocal region predicts mesh-independent post-peak response of the load vs. displacement curve. Also, it is shown that the characteristic material length, $L_{\mathrm{c}}$, controls the structure of the shear band developed in the plate.

众所周知，损伤或非弹性软化会导致不适定问题，从而导致有限元模拟中的局部化和网格依赖性。在本文中，一个非局部硬化变量，$\bar{\kappa }$, 被引入具有与速率无关的平滑弹性 - 非弹性过渡的非弹性的有限变形欧拉公式中。这个非局部变量是在一个非局部的欧拉区域上定义的，该区域是一个半径等于特征长度的球体，${\mathrm{大号}}_{\mathrm{C}}$，在材料的当前变形几何中定义。探索了这个非局部硬化变量的两个模型。一种型号$\bar{\kappa }$是局部硬化的最小值$\kappa$在非局部区域内，以及另一个模型$\bar{\kappa }$是平均值$\kappa$在同一地区。以板为例子研究了非局部硬化变量的影响，该板由规定的边界位移加载，导致剪切带的形成。施加载荷与位移曲线的预测以及板的总变形变形和硬化变量的等高线图$\kappa$被研究。基于最小值的模型$\kappa$在非局部区域预测负载与位移曲线的与网格无关的峰值后响应。此外，它表明特征材料长度，${\mathrm{大号}}_{\mathrm{C}}$，控制板中产生的剪切带的结构。