In this study, we follow the work of Tvergaard and Needleman (1995, 1997) and Needleman and Tvergaard (1998) and present the numerical implementation and initial applications of a non-local Gurson-Tvergaard-Needleman (GTN) model for explicit finite element (FE) analysis. The delocalization relates to the damage mechanism and is incorporated in terms of an integral condition on the rate of change of the porosity. To demonstrate the mesh independence during all stages of ductile damage and fracture, several material test specimens were simulated using different mesh sizes until full fracture occurred. For comparison purposes, the results are also obtained for the corresponding local GTN model in all cases. The effect of the material characteristic length on the ductile damage and fracture behavior and on the mesh sensitivity of the results is discussed. The numerical study shows that simulation results obtained in all stages of the ductile fracture process, including void growth, fracture initiation by coalescence and crack propagation all the way to a fully fractured specimen, are mesh independent for a certain mesh size ratio related to the material characteristic length, provided the non-local integral is evaluated on the current configuration. This ratio is unique for each individually simulated specimen as it depends on the spatial gradients of the porosity and the material parameters adopted for the problem at hand. It is shown that excessive averaging occurs at large deformations if the non-local integral is evaluated on the reference configuration, i.e., without updating the element interaction matrix resulting from the discretization of the non-local integral.

在这项研究中，我们遵循Tvergaard和Needleman（1995，1997）以及Needleman和Tvergaard（1998）的工作，并提出了非局部Gurson-Tvergaard-Needleman（GTN）模型用于显式有限元的数值实现和初始应用。 （FE）分析。离域化与损伤机理有关，并以孔隙率变化率的整体条件纳入。为了证明在延性破坏和断裂的所有阶段中的网格独立性，使用不同的网格尺寸模拟了几个材料试样，直到发生完全断裂。为了进行比较，还可以在所有情况下获得对应的本地GTN模型的结果。讨论了材料特征长度对延性破坏和断裂行为以及结果的网格敏感性的影响。数值研究表明，在延性断裂过程的所有阶段（包括空洞生长，通过聚结引发的裂纹以及一直延伸到完全断裂的试样的裂纹扩展）获得的模拟结果在与材料有关的特定网格尺寸比下均与网格无关。如果在当前配置上评估了非局部积分，则可以确定特征长度。该比率对于每个单独模拟的样本都是唯一的，因为它取决于孔隙率的空间梯度和解决当前问题所采用的材料参数。