A Unified Framework of Bond-Associated Peridynamic Material Correspondence Models

Abstract

This paper presents a unified framework for bond-associated peridynamic material correspondence models that were proposed to inherently address the issue of material instability or existence of zero-energy modes in the conventional correspondence formulation. The conventional formulation is well-known for having the issue of material instability due to the non-unique mapping between bond force density state and nonlocal deformation gradient. Several bond-associated models that employ bond-level deformation gradients address this issue in a very effectively and inherent manner. Although different approaches were taken to formulate bond-level deformation gradient so the bond-associated quantities can be captured more accurately, a detailed study finds a unified systematic framework exists for these models. It is the purpose of this paper to consolidate these approaches by providing a unified and systematic framework for bond-associated peridynamic correspondence models. Based on all the bond-associated deformation gradients proposed in the literature, a unified bond-associated deformation gradient is formulated. Assuming energy equivalence with the local continuum mechanics theory, the unified bond force density state is derived using the Fr\'echet derivative. Additionally, the properties of the formulated unified framework including linear momentum balance, angular momentum balance, and objectivity are thoroughly examined. This work serves as a valuable reference for the further development and application of bond-associated correspondence formulations in peridynamics.

Keywords: Peridynamics, Material correspondence formulation, Bond-associated model, Horizon, Influence function