On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models

Journal Article · 15 August 2016 · Computer Methods in Applied Mechanics and Engineering

DOI: https://doi.org/10.1016/j.cma.2016.07.039 · OSTI ID:1297917

Seleson, Pablo  ^[1]; Du, Qiang ^[2]; Parks, Michael L. ^[3]

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1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
2. Columbia Univ., New York, NY (United States)
3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

The peridynamic theory of solid mechanics is a nonlocal reformulation of the classical continuum mechanics theory. At the continuum level, it has been demonstrated that classical (local) elasticity is a special case of peridynamics. Such a connection between these theories has not been extensively explored at the discrete level. This paper investigates the consistency between nearest-neighbor discretizations of linear elastic peridynamic models and finite difference discretizations of the Navier–Cauchy equation of classical elasticity. While nearest-neighbor discretizations in peridynamics have been numerically observed to present grid-dependent crack paths or spurious microcracks, this paper focuses on a different, analytical aspect of such discretizations. We demonstrate that, even in the absence of cracks, such discretizations may be problematic unless a proper selection of weights is used. Specifically, we demonstrate that using the standard meshfree approach in peridynamics, nearest-neighbor discretizations do not reduce, in general, to discretizations of corresponding classical models. We study nodal-based quadratures for the discretization of peridynamic models, and we derive quadrature weights that result in consistency between nearest-neighbor discretizations of peridynamic models and discretized classical models. The quadrature weights that lead to such consistency are, however, model-/discretization-dependent. We motivate the choice of those quadrature weights through a quadratic approximation of displacement fields. The stability of nearest-neighbor peridynamic schemes is demonstrated through a Fourier mode analysis. Finally, an approach based on a normalization of peridynamic constitutive constants at the discrete level is explored. This approach results in the desired consistency for one-dimensional models, but does not work in higher dimensions. The results of the work presented in this paper suggest that even though nearest-neighbor discretizations should be avoided in peridynamic simulations involving cracks, such discretizations are viable, for example for verification or validation purposes, in problems characterized by smooth deformations. Furthermore, we demonstrate that better quadrature rules in peridynamics can be obtained based on the functional form of solutions.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: ORNL LDRD Director's R&D; ORNL work for others; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
• Grant/Contract Number: AC04-94AL85000; AC05-00OR22725
• OSTI ID: 1297917
• Report Number(s): SAND--2016-7606J; PII: S0045782516308210
• Journal Information: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering; ISSN 0045-7825
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English