Multiscale Analysis of Heterogeneous Media in the Peridynamic Formulation

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.

Peridynamic Theory of Solid Mechanics

“…This work also presented a comparison between the local and peridynamic theories for linear elastic solids. Alali and Lipton (2009), Zhou (2010a, 2010b), and Weckner (2007a, 2007b) establish various existence and uniqueness results for the linear peridynamic balance of momentum. These papers also draw equiva lences with the weak solution of the classical equations of linear elasticity, and show, in a precise sense, the well-posedness of the peridynamic equations in the limit as the nonlocality vanishes.…”

“…Multiscale analysis of a fiber-reinforced composite in the limit of small fiber diameter is treated by Alali and Lipton (2009) for different types of assumed limiting behavior of the constituent materials and their interfaces. These authors also investigate the homogenized models resulting from alternative ways of coupling the peridynamic horizon to the geometrical length scales naturally present in the material during this limiting process.…”

Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media

“…A more explicit account of heterogeneities has also been considered within the peridynamic model at the computational meshing level, see for example [21]. On the other hand, in the peridynamics framework, bottom up multiscale approaches are given in [2,25]. Here local field interaction between heterogeneities are modeled directly.…”

“…Here local field interaction between heterogeneities are modeled directly. The work in [2] provides a systematic and rigorous method for developing multiscale approximations to solutions of bond based peridynamic problems in fiber reinforced media. The approach delivers homogenized dynamics and strong approximation using correctors.…”

Variational theory and domain decomposition for nonlocal problems

“…Consequently, nonlocal models are also useful for multiscale modeling. Recent examples of nonlocal multiscale modeling include the upscaling of molecular dynamics to nonlocal continuum mechanics [23], and development of a rigorous multiscale method for the analysis of fiber-reinforced composites capable of resolving dynamics at structural length scales as well as the length scales of the reinforcing fibers [24]. Progress towards a nonlocal calculus is reported in [25].…”