In recent years, two types of nonlocal differential operators and their theories and formulations have been proposed and used in numerical modeling and computations, particularly simulations of material and structural failures, such as fracture and crack propagations in solids. Since the differences of these nonlocal operators are subtle, and they often cause confusion and misunderstandings. The first type of nonlocal differential operators is derived from the Taylor series expansion of nonlocal interpolation, e.g., Bergel and Li (Comput Mech 58(2):351–370, 2016), Madenci et al. (Comput Methods Appl Mech Eng 304:408–451, 2016) and Ren et al. (Comput Methods Appl Mech Eng 358:112621, 2020). The second type of nonlocal operators is based on the nonlocal operator theory in peridynamic theory, which is a class of antisymmetric nonlocal operators stemming from the nonlocal balance laws, e.g., Gunzburger and Lehoucq (Multiscale Model Simul 8(5):1581–1598, 2010) and Du et al. (SIAM Rev 54(4):667–696, 2012; Math Models Methods Appl Sci 23(03):493–540, 2013a). In this work, a comparative study is conducted for evaluating the computational performances of these two types of nonlocal differential operators. It is found that the first type of nonlocal differential operators can yield convergent results in both uniform and non-uniform particle distributions. In contrast, the second type of nonlocal differential operators can only converge in uniform particle distributions. Specifically, we have evaluated the performance of the two types of nonlocal differential operators in three crack propagation simulation examples. The results show that the second type of nonlocal differential operators is more suitable to deal with complex crack branching patterns than the first type of nonlocal differential operators, and the simulation results obtained by using the second type of nonlocal differential operators have better agreement with experimental observations. For modelings of simple crack growth and conventional elastic deformation problems, both nonlocal operators can provide good results in simulations.

近年来，已经提出了两种类型的非局部微分算子及其理论和公式，并将其用于数值建模和计算，特别是材料和结构故障的模拟，例如固体中的断裂和裂纹扩展。由于这些非本地操作符的差异是微妙的，它们经常引起混淆和误解。第一类非局部微分算子源自非局部插值的泰勒级数展开，例如 Bergel 和 Li (Comput Mech 58(2):351–370, 2016)、Madenci 等人。(Comput Methods Appl Mech Eng 304:408–451, 2016) 和 Ren 等人。（计算方法应用机械工程 358：112621，2020）。第二种非局域算子基于近场动力学理论中的非局域算子理论，这是一类源自非局域平衡定律的反对称非局域算子，例如 Gunzburger 和 Lehoucq (Multiscale Model Simul 8(5):1581–1598, 2010) 和 Du 等人。(SIAM Rev 54(4):667–696, 2012; Math Models Methods Appl Sci 23(03):493–540, 2013a)。在这项工作中，对这两种非局部微分算子的计算性能进行了比较研究。发现第一类非局部微分算子可以在均匀和非均匀粒子分布中产生收敛结果。相比之下，第二类非局部微分算子只能收敛于均匀的粒子分布。具体而言，我们在三个裂纹扩展模拟示例中评估了两种类型的非局部微分算子的性能。结果表明，第二类非局部微分算子比第一类非局部微分算子更适合处理复杂的裂纹分支模式，使用第二类非局部微分算子得到的模拟结果与实验观察结果吻合较好。 . 对于简单裂纹扩展和常规弹性变形问题的建模，两种非局部算子都可以提供良好的模拟结果。