Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation

TL;DR

This paper introduces a multiscale methodology for analyzing the dynamics of heterogeneous media within the peridynamic framework, enabling simultaneous resolution of microstructural and macroscopic behaviors through a novel two-scale evolution equation.

Contribution

It presents a new two-scale evolution equation that couples microscopic and macroscopic dynamics, improving the modeling of heterogeneous materials in peridynamics.

Findings

The two-scale evolution equation accurately approximates deformation in heterogeneous media.

The method captures the interplay between microstructure and overall material response.

The approach provides a strong link between local forces and homogenized deformation.

Abstract

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation