Deriving peridynamic influence functions for one-dimensional elastic materials with periodic microstructure

TL;DR

This paper derives a physically-informed influence function for one-dimensional peridynamic models with periodic microstructure, improving the accuracy of dynamic wave and fracture simulations by linking microstructural properties to the influence function.

Contribution

It introduces a method to derive influence functions from microstructural mechanics, enhancing the physical realism of peridynamic models for elastic materials.

Findings

Derived influence functions better predict wave dynamics than standard models.

Homogenized equations link microstructure to nonlocal behavior.

Improved accuracy in displacement and wave propagation simulations.

Abstract

The influence function in peridynamic material models has a large effect on the dynamic behavior of elastic waves and in turn can greatly effect dynamic simulations of fracture propagation and material failure. Typically, the influence functions that are used in peridynamic models are selected for their numerical properties without regard to physical considerations. In this work, we present a method of deriving the peridynamic influence function for a one-dimensional initial/boundary value problem in a material with periodic microstructure. Starting with the linear local elastodynamic equation of motion in the microscale, we first use polynomial anzatzes to approximate microstructural displacements and then derive the homogenized nonlocal dynamic equation of motion for the macroscopic displacements; which, is easily reformulated as linear peridyamic equation with a discrete influence function. The shape and localization of the discrete influence function is completely determined by microstructural mechanical properties and length scales. By comparison with a highly resolved microstructural finite element model and the standard linear peridynamic model with a linearly decaying influence function, we demonstrate that the influence function derived from microstructural considerations is more accurate in predicting time dependent displacements and wave dynamics.