Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics

Highlights

• Stress-based failure criteria are implemented in NOSB-PD.

• The effect of flaw length on the propagation of cracks is researched.

• The effect of ligament angle on the coalescence of cracks is studied.

• The effect of the confining stresses on the coalescence of flaws is investigated.

Abstract

The maximum tensile stress criterion and the Mohr-Coulomb criterion are incorporated into the extended non-ordinary state-based peridynamics (NOSB-PD) to simulate the initiation, propagation and coalescence of the pre-existing flaws in rocks subjected to compressive loads. Wing cracks, oblique secondary cracks, quasi-coplanar secondary cracks and anti-wing cracks can be modeled and distinguished using the proposed numerical method. In the present study, a four-point beam in bending with two notches as a benchmark example is firstly conducted to verify the ability, accuracy and numerical convergence of the proposed numerical method. Then, the numerical samples of rock materials containing the one single pre-existing flaw with various lengths under uniaxial compression are modeled. Four significant factors, i.e. the axial stress versus axial strain curves, the peak strength, the ultimate failure mode and crack coalescence process, are obtained from the present numerical simulation. The effect of the flaw length on the propagation of cracks is investigated. Next, sandstone samples containing three pre-existing flaws with different ligament angles under uniaxial compression are also simulated. The effect of ligament angle on the propagation and coalescence of cracks is studied. Finally, rock-like samples containing two parallel pre-existing flaws subjected to biaxial compressive loads with confining stresses of 2.5, 5.0, 7.5 and 10.0 MPa are simulated. The effect of the confining stresses on the initiation, propagation and coalescence of flaws is investigated. The present numerical results are in good agreement with the previous experimental ones.

Introduction

In rock engineering, rock masses are discontinuous media containing flaws, joints and faults, etc., which play a significant role in the strength, deformability and failure behaviors of rock materials. The propagation and coalescence of cracks have an important influence on the mechanical behaviors of rocks and the stability of the rock engineering.

In order to understand the mechanism of crack propagation and coalescence in the rock masses, numerous experimental [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and theoretical efforts [16], [17], [18], [19], [20], [21], [22] were devoted to investigating the initiation, propagation and coalescence of flaws in the past decades. Although there are many differences in the crack coalescence modes, there are also some common characteristics of initiation, propagation and coalescence of flaws. Experimental studies related to crack propagation and coalescence are mainly concentrated on the specimens containing flaws subjected to uniaxial compressive loads. Only a few biaxial tests have been done [1], [2], [23], [24]. Theoretical studies of crack initiation, propagation and coalescence were mainly focused on the crack initiation criteria and the direction of crack propagation. Most crack initiation criteria are based on Linear Elastic Fracture Mechanics (LFEM) and Material Strength (MS). The results obtained from experiments and theoretical studies have served as valuable references for the numerical simulation.

Numerical simulations prove necessary to study crack propagation and coalescence in rock materials due to the mechanical and geometrical complexity of most problems. In the past decade, a number of numerical techniques are available to model crack propagation process under compressive loads.

The finite element method (FEM) has been applied to investigate crack propagation and coalescence in the rock specimens subjected to compressive loads [25], [26]. In the finite element-based method, singular crack-tip elements are frequently encountered [27]. However, the FEM remains limiting because it requires that the finite element meshes coincide with cracks. This limitation makes the task of generating the mesh very difficult. Recently, Areias and Rabczuk [28] summarized the computational techniques on effective remeshing, which can be classified as discrete or continuum-based methods. Some researchers developed full and localized rezoning and remeshing approaches [29], [30], [31], [32], local displacement enrichments [27], [33], [34], [35], clique overlaps [36], [37], element erosion [38], smeared band procedures [39], viscous-regularized techniques [40], gradient and nonlocal continua [41] and phase-field models based on decoupled optimization with sensitivity analysis [42], [43] to model crack propagation and coalescence.

The extended finite element method (XFEM) [27], [44] is a methodology for modelling cracks of arbitrary geometry without re-meshing effort. Some researchers applied XFEM to simulate the crack propagation and coalescence under compressive loads [45]. However, for some more complicated problems, such as compressive and dynamic problems, the definition of the enrichment functions can be very difficult. Moreover, without contact algorithm, it will be very challenging for these methods to consider the crack, slipping and separation along the flaw direction [46], [47].

A meshfree concept of XFEM was proposed by Ventura et al. [48] for linear elastic fracture mechanics. The advantage of this meshfree method over XFEM is higher smoothness, non-local interpolation character and higher order continuity which results in a better stress distribution around the crack tip which is important for the propagation of the crack. Different methods have been presented to model cracks in FEM and meshfree method [49], [50], [51], [52]. A ‘cracking particle’ model (CPM) has been developed in meshfree methods by Rabczuk and Belytschko [53], [54], where discontinuities are introduced at the particle position. The major advantages of these methods are their robustness and ease in implementation. However, for certain classes of problems, more accurate methods are needed. Therefore, the extended element Galerkin method (XEFG), which is a kind of meshfree method, was proposed by Rabczuk and Zi [55] for crack propagation problems. The XEFG is based on the introduction of a discontinuity in displacement field after the material looses stability. The cohesive model can be used in the XEFG that solely depends on the discontinuous part of the displacement. Moreover, Rabczuk et al. [56] incorporated a three-dimensional crack tracking algorithm into XFEG method for cohesive cracks. They found that the meshfree methods have a large advantage over FEM based techniques such as XFEM for crack propagation problems involving large deformation and fragmentation.

Smoothed particle hydrodynamics (SPH) is a special meshfree method proposed by Lucy [57] and Gingold and Monaghan [58] to solve astrophysical problems in three-dimensional open space, and has been extensively studied and extended to dynamics response with material strength as well as dynamic fluid flows with large deformations. Although SPH has successfully simulated the initiation, propagation and coalescence of the flaws [59], inherent fracture mode of processing by defining influence only within the interaction domain of the basis function [60], [61] is not well to simulate initiation and propagation and coalescence of the flaws. Moreover, the governing equations in SPH are in a spatial derivative form, which makes this numerical method inefficient.

Discontinuous deformation analysis (DDA) proposed by Shi and Goodman [62] has been used to model failures in intact rock by using artificial joints with discrete blocks [63], [64]. Fracturing has been simulated by converting artificial joints into real joints once a failure criterion is met. However, the predicted crack trajectory is dependent on the advance block discretization configuration [65].

Numerical manifold method (NMM) proposed by Shi [66] is a numerical method combined the FEM and the DDA, which has been usually applied to analyze the processes of crack propagation and coalescence in rock masses [7], [11], [12], [67], [68]. However, the crack tips are constrained to stop at the edges of the element, which reduces the accuracy if a crack tip happens to stop inside the element.

Particle flow code (PFC) proposed by Potyondy and Cundall [69] is a special numerical method of the Discrete Element Method (DEM) proposed by Cundall and Strack [70], which becomes a popular mesh-free method to analyze the fracturing behaviors of rocks. Some researchers applied PFC to simulate the propagation and coalescence of cracks by simply breaking the bonds when the interaction force between two distinct elements overcomes the tensile or shear strength [69], [71], [72], [73], [74]. However, as mentioned in works by Donze et al. [75], the following limitations remain: (1) Fracture is closely related to the size of elements, and that is so called size effect. (2) Cross effect exists because of the difference between the size and shape of elements with real grains. (3) In order to establish the relationship between the local and macroscopic constitutive laws, data obtained from classical geomechanical tests which may be impractical are used.

Peridynamic (PD) theory is a nonlocal formulation of continuum mechanics was first proposed by Silling [76] to solve discontinuous problems such as cracks. In contrast with the conventional numerical methods, PD governing equation of motion is in an integral-differential form without any spatial derivative. PD theory [77], [78], [79] can be classified into three categories: bond-based PD (BB-PD), ordinary state-based PD (OSB-PD) and non-ordinary state-based PD (NOSB-PD).

The BB-PD theory is based on the assumption of pairwise interactions of the same magnitude, which results in being restricted to a Poisson’s ratio of 1/4 for isotropic problems. Recently, Liu and Hong [80] proposed a force compensation scheme adding additional forces on each pairwise force between particles to model materials that have Poisson’s ratio other than 1/4. Moreover, a BB-PD code was developed by Liu and Hong [81] for 3D simulation for brittle and ductile solids. Ha et al. [82] simulated the fracturing patterns of rock-like materials in compression using the coupling FEM and BB-PD method with the force compensation scheme. Additionally, a new contact algorithm is implemented for the interaction of PD regime and finite elements for the effective impact fracture analysis by Lee et al. [83].

The state-based PD (SB-PD) theory is based on the concept of PD states that are infinite dimensional arrays containing information about PD interactions. The OSB-PD method is restricted to only central force between interacting PD particles. However, in the NOSB-PD method the forces in the bonds are represented with conventional strain and stress tensors which allow using general constitutive models, and the bonds are able to carry stresses in all directions which is much more realistic for modelling a continuum. Additionally, compared to BB-PD and OSB-PD, the NOSB-PD has the advantages in the aspect of modelling general non-linear anisotropic materials. Moreover, some researchers have elucidated on some properties of PD with respect to other meshfree method, such as Smooth Particle Hydrodynamics (SPH) in Total-Lagrangian formulation by Ganzenmüller [84] and the Reproducing Kernal Particle Method (RKPM) by Bessa et al. [85]. They concluded that SB-PD is equivalent to RKPM if nodal integration is adopted, and the SB-PD is much faster compared to RKPM using cell based integration techniques [85]. Meanwhile, the discretizations of PD directly arrive at correct equations which conserve linear and angular momentum. These features can only be obtained in SPH by assuming ad hoc corrections such as explicit symmetrization [84].

Although some researchers applied the PD theory to solve the fracture problems [86], [87], [88], [89], [90], [91], these studies mainly focused on the pre-existing cracks under tension loads. In the present study, the maximum tensile stress criterion and Mohr-Coulomb failure criterion are implemented into NOSB-PD to simulate the initiation, propagation and coalescence of flaws in rock specimens under compression. In the extended NOSB-PD, the maximum tensile stress criterion is applied to determine the tensile failure of the bond between interacting particles. While Mohr-Coulomb failure criterion is employed to define the shear failure of the bond between interacting particles. Different representative crack types such as wing cracks, oblique secondary cracks, quasi-coplanar secondary cracks and anti-wing cracks can be modeled using the proposed numerical method.

In this paper, the extended NOSB-PD is applied to investigate the initiation, propagation and coalescence of cracks in the rock specimens under compressive loads and reveal the complex mechanism of the coalescence of cracks. Firstly, a benchmark example is presented to prove the accuracy and numerical convergence of the proposed numerical method. Then, the crack propagation and coalescence modes in rock samples containing one single pre-existing flaw or three pre-existing flaws under uniaxial compression are conducted using the proposed numerical method. The effects of the length of flaw and the ligament angle between flaws on crack propagation and coalescence are studied. Next, the numerical samples of the sandstone containing two parallel pre-existing flaws subjected to various biaxial compressive loads are simulated. The effect of confining stress on the initiation, propagation and coalescence of flaws and crack coalescence modes is analyzed. Finally, some characteristics of crack propagation and coalescence under biaxial compression are summarized. Compared with the present numerical and previous experimental results, it can be proved that the proposed numerical method is efficient and robust to determine the trajectory of crack propagation and modes of crack coalescences under different geometric and loading conditions.

This paper is organized as follows: The extended NOSB-PD is briefly described in Section 2. A benchmark example is presented in Section 3. Rock samples containing one single pre-existing flaw or three pre-existing flaws subjected to uniaxial compressive loads are simulated in Section 4. Numerical samples of the sandstone containing two parallel pre-existing flaws under biaxial compression are modeled in Section 5. The conclusions are drawn in Section 6.