From its early days the SPH method has been criticised for its shortcomings namely tensile instability and consistency. Without thorough understanding of the method attempts were made to make the classical SPH method consistent and stable which resulted in the local and Total Lagrangian forms of SPH similar to the finite element method. In this paper we derived and analysed a consistent nonlocal SPH which has similarity with Bazant’s imbricate continuum. In addition, the paper provides comparison and discussion of different SPH forms including: Classical SPH, Nonlocal, Local and Mixed SPH. The partition of unity approach was used to define the following two mixed forms: Local–Nonlocal and Local–Classical SPH. These mixed forms were intended for modelling of physical processes characterised with local and nonlocal effects (local and nonlocal constitutive equations), e.g. progressive damage and failure. The stabilising effect of the Local form on the Classical SPH, which is inherently unstable (tensile instability), are also illustrated. The stability analysis, presented in appendices A and B, demonstrate stability of the continuous and discrete form of the nonlocal SPH based on Eulerian kernels for elastic continuum.

从早期开始，SPH 方法就因其拉伸不稳定性和一致性等缺点而受到批评。在没有彻底理解该方法的情况下，试图使经典 SPH 方法一致和稳定，从而导致 SPH 的局部和总拉格朗日形式类似于有限元方法。在本文中，我们推导出并分析了一致的非局部 SPH，它与 Bazant 的覆瓦状连续体具有相似性。此外，本文还提供了不同 SPH 形式的比较和讨论，包括：Classical SPH、Nonlocal、Local 和 Mixed SPH。统一划分方法用于定义以下两种混合形式：本地-非本地和本地-经典 SPH。这些混合形式旨在模拟具有局部和非局部效应（局部和非局部本构方程）的物理过程，例如渐进式损坏和失效。还说明了局部形式对固有不稳定（拉伸不稳定性）的经典 SPH 的稳定作用。附录 A 和 B 中提供的稳定性分析证明了基于弹性连续体的欧拉核的非局部 SPH 的连续和离散形式的稳定性。