On differences and comparisons of peridynamic differential operators and nonlocal differential operators

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.