Coupling of non‐ordinary state‐based peridynamics and finite element method with reduced boundary effect

Abstract

In this study, we propose a new numerical technique to couple non-ordinary state-based peridynamics (NOSB-PD) and the finite element method (FEM), and improve the scheme by implementing an effective boundary imposition method and a stabilization method. This coupling scheme takes the mutual advantages of peridynamics and the FEM to solve fracture problems without the imposition of additional criteria, yielding enhanced computational efficiency. In addition, the coupling model brings the reduction of the boundary effect using the boundary imposition method. To combine the peridynamics and FEM, the model is partitioned into the peridynamic subregion and finite element subregion. Subsequently, the two subregions are bridged by interface elements, where peridynamic nodes are embedded. Two types of coupling schemes are developed and the boundary effect of the peridynamic subregion is analyzed in each coupling approach. Moreover, stabilization of the coupling method is implemented to control the zero-energy mode inherent in NOSB-PD to simulate fracture problems. The proposed methodology is verified by solving several quasi-static problems in the one-and two-dimensional domains, and fracture problems are solved. The crack paths predicted by the proposed coupling method are in good agreement with the results of the exact solution and the experiments. K E Y W O R D Sboundary effect, crack propagation, finite element method, non-ordinary state-based peridynamics INTRODUCTIONThe finite element method (FEM), which requires continuous displacement fields to calculate stress fields, is the most popular technique for numerical analysis. 1 However, the displacement fields are usually discontinuous if a crack tip or crack surfaces exist. A remeshing technique might be required to prevent discontinuous fields within meshes, 2,3 notably in nonlinear analysis. In the framework of the FEM, the extended FEM (XFEM) was proposed to deal with the spatial derivatives on either the crack tip or crack surfaces. 4,5 In the XFEM, tip enrichment functions based on analytical solutions are used to describe crack tip fields. 6 Moreover, to circumvent the expensive mesh generation and the problems that arise in the treatment of discontinuities, various meshless techniques, such as smoothed particle hydrodynamics (SPH), 7 finite spheres, 8 and the element-free Galerkin method (EFGM), 9 have been proposed. The SPH method is a nonlocal meshless technique that calculates the field quantities at each node using a smoothing function and is used extensively to solve fluid dynamics and fractures. 10,11 The finite sphere method replaces finite elements with overlapping spheres and the

Explicit implementation of the non-local operator method: a non-local dynamic formulation for elasticity solid

“…Various PD models have been proposed over the years, such as bond-based peridynamic (BB-PD) models, state-based peridynamic (SB-PD) models, hybrid models leveraging PD and CCM, and analytical PD solutions. These PD models have been used successfully to tackle a variety of multi-physical issues, such as heat conduction (Bobaru and Duangpanya, 2010;Oterkus et al, 2014), fluid transport (Katiyar et al, 2014;Gao and Oterkus, 2019;Katiyar et al, 2020), material corrosion (Chen and Bobaru, 2015;De Meo et al, 2016;Jafarzadeh et al, 2018), etc., and PD also has been expanded in a variety of directions, such as dual-horizon PD (Ren et al, 2016(Ren et al, , 2017Rabczuk and Zhuang, 2021), PD plate/shell theory (Chowdhury et al, 2016;Zhang et al, 2021;Dorduncu et al, 2021), phase field based PD damage model (Roy et al, 2017(Roy et al, , 2021, wave dispersion analysis in PD (Gu et al, 2016;Butt et al, 2017;Zhang et al, 2019), weak form of PD (Madenci et al, 2018), PD least squares minimisation (Madenci et al, 2019b,a), coupling of FEM and ordinary state-based (OSB)/non-ordinary state-based (NOSB) PD (Yaghoobi and Chorzepa, 2018;Ni et al, 2021;Jin et al, 2021;Liu et al, 2021), higher-order PD (Yang et al, 2021a(Yang et al, , 2021b(Yang et al, , 2021c, etc. In comparison to the local theory, the non-local theory not only has a superior numerical well-posedness, but it also resembles the real physical process better due to its inherent length scale.…”