The basic feature of the peridynamics [introduced by Silling (2000)] considered is a continuum description of material behavior as the integrated nonlocal force interactions between infinitesimal material points. A heterogeneous bar of the periodic structure of constituents with peridynamic mechanical properties is analyzed. One introduces the volumetric periodic boundary conditions (PBCs) at the interaction boundary of a representative unit cell (UC), whose local limit implies the known locally elastic PBCs. This permits us to generalize the classical computational homogenization approach to its counterpart in peridynamic micromechanics (PM). Alternative to the finite element methods (FEM) for solving computational homogenization problems are the fast Fourier transforms (FFTs) methods developed in local micromechanics (LM). The Lippmann−Schwinger (L−S) equation-based approach of the FFT method in the LM is generalized to the PM counterpart. Instead of one convolution kernel in the L−S equation, we use three convolution kernels corresponding to the properties of the matrix, inclusions, and interaction interface. The Eshelby tensor in LM depending on the inclusion shape is replaced by PM counterparts depending on the inclusion size and interaction interface (although the Eshelby concept of homogeneous eigenfields does no work in PM). The mentioned tensors are estimated one time (as in LM) in a frequency domain (also by the FFT method). Numerical examples for 1-D peridynamic inhomogeneous bar are considered. Computational complexities O (N log₂ N) of the FFT methods are the same in both LM and PM.

中文翻译：

---

周期结构周动力棒的快速傅里叶变换方法

近场动力学[Silling (2000) 提出]的基本特征是对材料行为的连续描述，即无穷小材料点之间的集成非局部力相互作用。分析了具有近场动力学力学特性的成分周期结构的异质棒。在代表性晶胞 (UC) 的相互作用边界处引入体积周期性边界条件 (PBC)，其局部极限意味着已知的局部弹性 PBC。这使我们能够将经典的计算均质化方法推广到近场动力学微力学（PM）中的对应方法。用于解决计算均匀化问题的有限元方法 (FEM) 的替代方案是局部微力学 (LM) 中开发的快速傅立叶变换 (FFT) 方法。LM 中基于 Lippmann−Schwinger (LS) 方程的 FFT 方法被推广到 PM 对应方法。我们使用与矩阵、包含物和交互界面的属性相对应的三个卷积核，而不是 L−S 方程中的一个卷积核。LM 中取决于夹杂物形状的埃谢尔比张量被 PM 对应物取代，具体取决于夹杂物大小和相互作用界面（尽管齐次本征场的埃谢尔比概念在 PM 中不起作用）。上述张量在频域（也通过 FFT 方法）估计一次（如在 LM 中）。考虑一维近场动力学非均匀杆的数值例子。LM 和 PM 中 FFT 方法的计算复杂度 O (N log ₂ N) 相同。