On the stability of the generalized, finite deformation correspondence model of peridynamics

Abstract

A class of peridynamic material models known as constitutive correspondence models provide a bridge between classical continuum mechanics and peridynamics. These models are useful because they allow well-established local constitutive theories to be used within the nonlocal framework of peridynamics. A recent finite deformation correspondence theory (Foster and Xu, 2018) was developed and reported to improve stability properties of the original correspondence model . This paper presents a stability analysis that indicates the reported advantages of the new theory were overestimated. Homogeneous deformations are analyzed and shown to exibit unstable material behavior at the continuum level. Additionally, the effects of a particle discretization on the stability of the model are reported. Numerical examples demonstrate the large errors induced by the unstable behavior. Stabilization strategies and practical applications of the new finite deformation model are discussed.

A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

“…The bond-associated model, without the higher-order corrections, is denoted as BA-PD, which is equivalent to the linear BA-RK-PD. The following numerical examples are provided for a threefold goal: (1) to test the convergence of the different variants of the correspondence model, (2) to demonstrate the unstable behavior of RK-PD and GMLS-PD and that it can be fixed through the bond-associative modeling approach, and (3) to study the robustness obtained by a combination of the bond-associative approach and higher-order corrections.…”

“…Despite offering convenience in utilizing classical constitutive relations, PD correspondence materials have been found problematic in practice as they involve instability issues. The unstable behavior appears as zero-energy mode oscillations in meshless peridynamic simulations and has been largely attributed to the cancellation of the non-uniform parts of deformation, which is allowed within the integration (averaging) technique [2,7,13,31,36]. The instabilities can lead to large errors in practical applications [16].…”

A Unified, Stable, and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part II: Wave Propagation and Enforcement of Stress Boundary Conditions

“…Both of these algorithms can compute non-local gradients with n th -order accuracy. We showed that the higher-order corrections cannot eliminate the wellknown issue of numerical instability in correspondence PD theory, which manifest itself as zero-energy mode oscillations in simulations that involve inhomogeneous deformations [3,8,17,24,27], and may lead to large prediction errors in practical applications [16]. To provide stability for the higher-order framework, we developed a continuum-based, bondassociated model, inspired by [4,9,10], which takes into account non-uniform deformations and maintains stability without additional stabilization mechanisms employed.…”

Crack nucleation in brittle and quasi-brittle materials: A peridynamic analysis

“…Peridynamics was introduced as a nonlocal form of continuum mechanics by Silling in 2000 [15] for modeling fracture. Since then, it has been extended to a variety of other problems in which domain changes/discontinuities are part of the problem [5,6,8,[25][26][27][28][29][30][31][32][33][34][35]. In this theory, each material point is connected through peridynamic bonds to other points within a certain neighborhood region called "the horizon".…”