Peridynamic theory and coupling with finite element method for discontinuities analysis - Liu, Qibang (Kansas State University, 2022)
Peridynamics (PD) is a nonlocal continuum theory and is better suited for failure analysis of solids compared with conventional continuum mechanics. However, published bond-based and state-based PD theories and numerical implementations suffer from some drawbacks, including the need for surface correction and volume correction and computational inefficiency. The research presented in this dissertation focuses on advancing the theory of PD, overcoming existing drawbacks of PD, and applying PD in the deformation and failure analysis of solids. To remove the drawback of the need for a surface correction and a volume correction, a revised non-ordinary state-based PD (RNOSBPD) theory is proposed in this study. For spherical interaction domains, RNOSBPD recovers the original non-ordinary state-based PD (NOSBPD). However, for interaction domains of arbitrary shapes, RNOSBPD remains valid, eliminating the need for a surface or volume correction. Compared with the finite element method (FEM), PD is computationally expensive because of the nature of nonlocal interactions. A new approach for coupling PD with FEM based on the weighted residual method (WRM) is proposed. This new coupling approach is straightforward, and special techniques such as overlapping regions, interface elements, or fictitious nodes published in the literature for coupling PD with FEM are not needed. Failure analysis frequently involves the investigation of crack propagation. A quasi-static simulation method using our coupled PD-FEM is pioneered to study crack propagation efficiently. The failure criterion in the simulation is based on the maximum circumferential tensile stress. A new numerical method is proposed to evaluate the stress intensity factors using the interaction integral based on the J-integral and to determine the onset of crack propagation and direction of crack growth. Since it is impractical to store a full global stiffness matrix and to implicitly solve PD in 3D, a new PD model for 3D problems is proposed and presented, in which the global stiffness matrix is stored in a Compressed Sparse Row (CSR) format based on Message Passing Interface (MPI), and the system equations are solved by parallel computing by a Parallel Direct Sparse Solver (PDSS), also based on MPI. To further improve the computational efficiency of PD, an Enriched Finite Element Method using Peridynamics (EFEM-PD) is proposed, following a similar concept of element enrichment of extended finite element method (XFEM). In EFEM-PD, the FEM model is enriched adaptively with PD equations in regions containing cracks. An adaptive algorithm is implemented with the dual goals of meshing regions with cracks using PD equations while minimizing PD interactions to maximize computational efficiency.
Keywords: Peridynamics; FEM; Fracture; Least square minimization; Surface correction