Numerical Study of Fracture Propagation in Isotropic and Transverse Isotropic Rocks Using Coupled Peridynamic and FEM
Authors: Kaiwei Tian , Zeqi Zhu , Qian Sheng 
Published: September 2023

ABSTRACT
In this paper, we propose a new simulation method based on peridynamic for quasi-static fracture propagation in isotropic and transverse isotropic rocks. The new method is based on coupled peridynamic least square minimization and finite element method (PDLSM-FEM). The original isotropic elastic PDLSM-FEM is further extended to a transverse isotropic model, which can analyze the elastic deformation and fracture propagation of transverse isotropic rocks. The proposed model naturally employs transverse isotropic Hooke's law to calculate the internal force. To break the bond and characterize crack propagation, a new bond stretch failure criterion is also presented. The elastic deformation of a transverse isotropic plate with a hole and fracture propagation in a transverse isotropic rock under semi-circular bending (SCB) tests are simulated and compared to finite element method (FEM) results and experimental data, respectively. It is demonstrated that the presented model is capable of modeling transverse isotropic rock elastic deformation and fracture propagation in isotropic SCB specimens and transverse isotropic specimens subjected to SCB tests.

INTRODUCTION
There are many sedimentary rocks in nature, such as shale and coal, which should be regarded as transverse isotropic materials because of their interior transverse isotropic structures formed by geological deposition. Fracture propagation is important in the failure process of rocks and has a great influence on the stability of underground excavation, efficiency of hydraulic fracture and other geotechnical engineering applications (Xie et al., 2017). Given the distinctions in fracture propagation patterns between transverse isotropic and isotropic rocks, a thorough understanding of the inner mechanism of fracture propagation in isotropic rocks and transverse isotropic rocks is fundamental in rock engineering.
Several computational approaches, such as the extended finite element method (XFEM) (Motamedi, 2010) and phase field method (PFM) (Wang, 2020), have been developed in recent years to investigate crack propagation in isotropic and transverse isotropic materials. These approaches, however, are based on classical (local) continuum mechanics (CCM) and need extra processing strategies for dealing with fractures. Peridynamic (PD) theory (Silling, 2000) is a type of non-local theory, which could naturally simulate fracture propagation since its governing equations are integral equations rather than partial differential equations. Bond-based theory (Silling, 2000), ordinary state-based theory (Silling et al., 2007), and non-ordinary state-based theory (Warren et al., 2009) are the three main types of PD theory. Bond-based PD has a fixed Poisson's ratio, but the other two types have no restriction. The advantage of NOSB PD is evident, since it inherits PD's capability of dealing with cracks while permitting implementations of stress and strain the same as CCM. However, the traditional NOSB PD is computationally expensive and suffers from numerical oscillation. To overcome those limitations, coupled PD least squares minimization and FEM (PDLSM-FEM) (Liu, 2021a) model have been proposed. In light of these concerns, we use the PDLSM-FEM theory as the foundation in this paper for simulating elastic deformation and fractures propagation in isotropic and transverse isotropic rocks.