Multiscale Dynamics 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.

Mathematical analysis for the peridynamic nonlocal continuum theory

“…Unlike discussions in the literature [1,10], the above convergence theorem does not require extra assumptions on the regularities of the solutions to the PD model beyond those already established in this work and remains valid for weak solutions. Moreover, using the properties stated in the Corollary 2.8, one may establish a general convergence in distribution sense.…”

“…being the components of z, and similarly, max(z 2 ) = max{z For kernel functions σ such that |x| 2 /σ(|x|) ∈ L 1 (B δ (0)), there are already some results in [1,10] showing that −L δ is a bounded linear operator from…”

“…Some results on the existence and uniqueness of L 2 solutions of the PD models associated with bounded integral PD operators have been given in [8][9][10]. Though much of the focus of [1] is on developing homogenization theory for the PD model, some existence and uniqueness results are also provided, again for bounded integral PD operators. In [12], a nonlocal vector calculus was developed which also provided a rigorous framework for studying the boundary value problems of the nonlocal peridynamic models.…”

Peridynamic State-Based Models and the Embedded-Atom Model

“…See [27] for a multiscale PD model of fiber-reinforced composites. PD has also been applied to concurrent multiscale modeling at the discrete level in [28,29] and at the continuum level in [30][31][32].…”

A Coarsening Method for Linear Peridynamics

“…A significant generalization of the linear peridynamic theory was presented in [9], in which it is shown that any Poisson ratio, as well as various other material response, can be represented by a more general choice of C than (1). Alali and Lipton [1] considered the problem of homogenization within linear peridynamics. They obtained relations governing the displacement field within a periodic microstructure in the limit of small length scales.…”