Possible causes of numerical oscillations in non-ordinary state-based peridynamics and a bond-associated higher-order stabilized model

Abstract

The peridynamic correspondence material model (PD CMM), generally regarded as a non-ordinary state-based peridynamic (NOSB PD) model, is attractive because of its capability to incorporate existing constitutive relations for material models. This study focuses on the mitigation of the numerical oscillations in the NOSB PD model. It compares the similarities and differences of smoothed particle hydrodynamics (SPH), corrected-SPH (CSPH), reproducing kernel particle method (RKPM), gradient-RKPM (G-RKPM) and NOSB PD based on their deformation gradient tensor and motion equations in the kernel integral form and their completeness and computational complexity. Inspired by the comparison and the peridynamic differential operator (PDDO), this study introduces a higher-order representation of the nonlocal deformation gradient and the force density vector by including the effect of higher-order terms in the Taylor series expansion (TSE) in order to improve the numerical accuracy and reduce the numerical oscillations. The numerical oscillations possibly arise from (1) the non-unique mapping between deformation states and force states via converting the point-associated variables into the bond force vector in each bond within a horizon, and (2) the violation of kinematic constraint condition for each bond under an arbitrary deformation state due to the point-associated nonlocal deformation gradient tensor. Therefore, a bond-associated higher-order NOSB PD model is adopted and numerically demonstrated to be effective in improving the accuracy and completely removing the oscillations. The bond-associated force vector state eliminates the concern of non-unique mapping from a deformation state to a force state. Also, the two bond-associated force vectors in a bond are equal and opposite, but not parallel to the bond direction. It can be viewed as a combination of the bond-based PD and the original NOSB PD. Finally, an implicit solver for both higher-order NOSB PD and bond-associated higher-order NOSB PD is presented for the solution of governing equations.