Unveiling Inhenrent Feature of Peridynamics: The “Trade-off Balance” Law Between Material Symmetry and Poisson’s Ratio

Abstract

The material correspondence formulation plays an essential role in connecting the Classical Continuum Mechanics and Peridynamics. In this paper, we analyze the material correspondence formulation in both the generalized two-parameter bond-based and state-based Peridynamics with a particular emphasis on the material symmetry principles. We discover a “trade-off balance” law between material symmetry and Poisson’s ratio in Peridynamics. Specifically, in the generalized two-parameter bond-based Peridynamics, the Poisson’s ratio limitation is eliminated, but the symmetry of the homogenized fourth-order material tensor in this model differs from that in Classical Continuum Mechanics. This asymmetry in the material tensor leads to energy incompatibility between the bond-based Peridynamics and Classical Continuum Mechanics. Furthermore, it can be proved that this incompatible energy has an upper bound and approaches zero as the characteristic length of the non-local interaction domain vanishes. In the case of the state-based Peridynamics, the symmetry of material tensors aligns with Classical Continuum Mechanics. However, the material correspondence formulation imposes a lower bound constraint on the Poisson’s ratio for the state-based Peridynamics. Inspired by this ‘trade-off balance’ law in Peridynamics, we propose a novel continuum model that maintains symmetry consistency. The proposed model integrates local and non-local energy into a single energy functional. By employing the Hamilton’s variational principle, we derive the governing equations with exact force boundary conditions. Unlike Peridynamics, the proposed model exerts the force boundary on the outer surface of the solids. We demonstrate that the proposed model is asymptotically compatible with Classical Continuum Mechanics. Wave dispersion analysis shows that the proposed model does not exhibit zero-energy mode oscillations.