Bond-based peridynamics does not converge to hyperelasticity as the horizon goes to zero

TL;DR

This paper investigates the connection between bond-based peridynamics and hyperelasticity, revealing that under certain physical assumptions, the former does not converge to the latter as the interaction horizon shrinks, limiting the applicability of hyperelastic models.

Contribution

The study demonstrates that bond-based peridynamics cannot generally recover hyperelastic models like Mooney-Rivlin through the zero-horizon limit, under frame-indifference and isotropy constraints.

Findings

Few hyperelastic functionals are Gamma4-limits of bond-based peridynamics.

Mooney-Rivlin materials are not obtainable as limits of bond-based peridynamics.

The Gamma4-limit of bond-based peridynamics is highly restricted under physical symmetries.

Abstract

Bond-based peridynamics is a nonlocal continuum model in Solid Mechanics in which the energy of a deformation is calculated through a double integral involving pairs of points in the reference and deformed configurations. It is known how to calculate the {\Gamma}-limit of this model when the horizon (maximum interaction distance between the particles) tends to zero, and the limit turns out to be a (local) vector variational problem defined in a Sobolev space, of the type appearing in (classical) hyperelasticity. In this paper, we impose frame-indifference and isotropy in the model and find that very few hyperelastic functionals are {\Gamma}-limits of the bond-based peridynamics model. In particular, Mooney-Rivlin materials are not recoverable through this limit procedure.