Failure Mechanism of Rock Specimens with a Notched Hole Under Compression - A Numerical Study

Discontinuities are natural structures that exist in rocks and can affect the stability of rock structures. In this article, the influence of notch presence on the failure evolution around a hole in compressed rock specimens is investigated numerically. Firstly, the uniaxial compressive test on a rock specimen with a circular hole is modeled and the failure evolution in the specimen is simulated. In a separate model, notches are created at the surface of the hole. Results show that when the notches are created in the model, failure zone around the hole is transferred to a distance away from the surface of the hole. In addition, a parametric study is carried out to investigate the influence of the notch length and the confining pressure on the fracturing behavior of the specimen. Numerical results presented in this article indicate that the presence of notches at the surface of the hole and their dimensions can affect the fracturing mechanism of the specimen. In some cases, the failure at the boundary of the hole is prevented when the notches of certain dimensions are added to the hole. The insights gained from this numerical study may be helpful to control the failure around underground excavations.


Introduction
Discontinuity is an inherent characteristic of rocks and rock-like materials. Generally, rocks include defects and discontinuities at various scales. Different types of defect exist in rocks such as faults, bedding planes, pores, grain boundaries, joints, holes, cracks, and notches.
These natural defects originate from the various geological processes acting on the rocks.
Moreover, defects in rock may be created artificially for some engineering purposes. Whether the defects are formed naturally or created artificially, they affect the load-bearing capacity of the rock (Brace 1961;Cai et al. 2004;Howarth and Rowlands 1987;Kemeny 2005). Defects in rock masses are stress raisers and play a key role in the rock fracturing processes.
Therefore, understanding the role of different types of defect on the mechanical behavior of rocks under various loading conditions is an essential task for any rock engineering design.
Cracks and notches are two types of planar defects that may exist naturally in rock masses or may be created artificially in different structures such as dams, tunnels, and wells. The definition of a "crack" refers to a defect that is generated during the loading process and a "notch" or a "flaw" refers to a prefabricated defect in the solids. Cracks and notches show a similar behavior under tensile loads, however, their behavior under compressive and shear stresses are different. When a crack is subjected to pure compression, the two faces of the crack are pushed together and the free space between them closes due to the penetration of the molecules on each face into the other face. In this condition, the presence of the crack has a little or no effect on the strength of the solid. Thus, the crack behavior under pure compression (negative Mode-I) is not crucial to the stability of a structure. In contrast, a notch under compression can affect the stability of a solid depending on the distance between the two faces of the notch and its geometry, In rock structures, rocks are subjected to different loading conditions. Damage initiation and propagation from a pre-existing defect in rock depends on the loading condition. In theory, three basic modes of crack extension are postulated: opening mode (Mode-I), shearing mode (Mode-II), and tearing mode (Mode-III). Basic modes of crack extension are sketched in  (Gehle and Kutter 2003;Kulatilake et al. 2001;Le et al. 2018;Lee and Jeon 2011;Lee and Ravichandran 2003;Manouchehrian et al. 2014;Park and Bobet 2010;Wang et al. 2011;Wong and Einstein 2009;Wong and Chau 1998;Wu et al. 2019c;Yang et al. 2013;Zhang et al. 2009;Zhuang et al. 2014). These studies revealed many fracturing characteristics of rocks and led to the improvement of safety in rock structures.

Figure 1. Basic modes of crack extension
In the studies related to failure mechanism of defected rock specimens (including cracks, flaws, notches, etc.), tensile and shear loads are usually more applied while the pure compression is neglected. In rock structures such as mine pillars, tunnel walls, and slopes, the existing defects at the surface of the excavations are subjected mainly to pure compression.
Hence, the influence of presence of a defect on the failure evolution of rocks under compression is important. The fracturing behavior of defects under pure compression has been studied in a limited number of investigations (Berto et al. 2013;Bura et al. 2019;Torabi et al. 2015). For example, Berto et al. (2013) used polycrystalline graphite to study the brittle failure of specimens with V-shaped notches under pure compression. They observed and explained the failure evolution from the notch tips. In a similar experiment, Bura et al. (2019) studied the fracturing process in polymethyl methacrylate (PMMA) specimens under compression which were weakened by V and U-shaped edge notches. In their research, fracture evolution during the notch closure was observed. Torabi et al. (2015)  blunt V-shaped notches under compression. These studies showed that the failure behavior of notches under compression has some specific characteristics.
In this research, a numerical study is conducted to understand how the failure mechanism of specimens with a circular hole would change if notches exist at the surface of the hole. In Section 2, the stress distribution around circular holes and cracks under Mode-I loading condition is described. In Section 3, bonded particle models are built to study the failure mechanism of a specimen with a notched hole under compression. In Section 4, a parametric study is conducted to investigate the influence of the notch length and the confining pressure on the failure behavior of the specimen.

Stress state around circular holes and cracks under Mode-I loading condition
Prior to any excavation, rock mass is subjected to lithostatic stress field. Upon excavation, the stresses around the excavation are redistributed to reach an equilibrium state. The stress state after excavation depends on the geometry of the opening in addition to the rock material properties, and boundary and loading conditions. Generally, the geometry of openings is designed in a way to minimize the stress concentration at the boundaries of the excavation to prevent the consequent instabilities. Among the basic geometrical shapes, a hole with a circular cross-section has the minimum stress concentration factor at its boundary in a homogeneous, continuous and isotropic medium (Sharma 2011 where σr, σθ and τrθ are radial, tangential and shear stresses, respectively. It should be noted that the stress state around an opening is not dependent on the mechanical properties of the material. By placing r = a in Eqs.1 through 3, the stress state at a circular hole boundary in an infinite plate is = 0 (4) = 0 (6) According to Eq.5, in a stress field in which K<1, the maximum principal stress is concentrated at the side walls ( = 0, ) and the minimum principal stress is located at the top ( = 2 ) and bottom ( = 3 2 ) of the hole boundary. The fracture mechanics provides solutions for calculation of the stress and displacement in the solids in which cracks exist. Generally, the fracture mechanics provides accurate solutions for brittle elastic materials. For a given crack, its extension may be accomplished in one or more modes (Figure 1). The near crack tip (r<<l) stresses under opening mode loads (Mode-I) can be calculated using Irwin's concept of the stress intensity factors, which characterize the strength of the singularity at a crack tip (Figure 3) (Anderson 2017), as follows: where b is the half-length of the crack. Eqs.7 through 9 suggest that the maximum stress concentration happens at the tips of the crack.  Advanced computational models empower scientists and engineers to build more complex and more detailed models for solving problems. Presence of notches at the surface of a hole ( Figure 4c) and its influence on the stress distribution around the hole can be simulated using advanced computational models. In a recent study, the authors used FEM models to study the stress distribution around a notched circular tunnel under compression. They showed that when notches are created in the tunnel, the adjacent rock at the sides of the notches is destressed and the stress concentration zones are transferred to a distance away from the wall ( Figure 5). In that study, they concluded that the presence of notches at a tunnel wall might be capable of changing the failure pattern at the wall in a favorable manner (Manouchehrian and Kulatilake 2021). In the present study, bonded particle models are used to simulate the fracturing process around a notched hole.  Peng et al. 2017;Schöpfer et al. 2009;Sukumar et al. 2000;Tan et al. 1998;Wu et al. 2019a;Wu et al. 2019b). In this section, simulation of failure evolution in rock specimens with a notched hole is discussed.

Model setup
In this study, the bonded particle models are developed to simulate the fracturing process in rock specimens. In the bonded particle models, the cracking process is simply simulated by breakage of bonds between particles. This is a simple approach for simulation of cracking in comparison to some other methods in which the crack evolution is simulated by solving complex mathematical equations related to fracture mechanics. For implementation of the bonded particle models, Particle Flow Code (PFC) (Itasca 2004) is utilized. Because the purpose of this study is to simulate the fracturing behavior of rectangular prismatic specimens, the problem can be simplified to a plane strain problem. Therefore, in this study the twodimensional version of the code (PFC 2D ) is used. For this purpose, usually the unconfined and confined compressive tests as well as the Brazilian test are simulated. In the calibration process, different micro-parameter values are varied to calibrate the model to get the desired macro-parameter values of the material. The procedures given by Yang et al. (2015) and Mehranpour and Kulatilake (2017) were used as guidance to perform the calibration process. The calibrated micro-parameter values and the used macro-parameter values for the simulated rock specimens in this study are listed in Table 1 and Table 2, respectively. The macro-parameter values of the simulated rock represent typical mechanical properties of soft rocks such as shale and marl and rock-like materials such as concrete. In this study a rectangular rock specimen with a width of 100 mm and a height of 200 mm is simulated including the mechanical properties listed in Table 2. The geometry of the models is shown in Figure 6. The simulated specimen includes a rectangular assembly of 18892 circular particles. The radius of the particles follows a Gaussian distribution function with the minimum and the maximum radii of 0.4 mm and 0.66 mm, respectively. After reproduction of the rectangular assembly, the circular hole and notches are created by removing the particles at the appropriate locations. The circular hole with a radius of R = 10 mm is created at the center of the specimen (See Figure 6). The notches are created by removing the particles in a narrow strip with a width of 1.5 mm. This approach is appropriate for simulation of open cracks. Simulation of closed and filled cracks can be done by assigning zero tensile strength to the bonds of neighbor particles in a straight line or by using smooth joint models (Mehranpour and Kulatilake 2017). The particle assembly of a notched hole with the length of L = 10 mm is illustrated in Figure 7. The pushing forces to the specimen's ends are applied by moving two parallel rigid walls toward each other in the direction of the loading. In this paper, the models containing a central circular hole is referenced as "the circular hole" (Figure 6a) and the models which include notches at the surface of the hole is referenced as "the notched hole" (Figure 6b). It should be noted that in this study, one particle assembly is used in all models. In this way, only the influences of the geometry and the boundary condition changes are considered.

Failure evolution in the circular hole and the notched hole
In this section, models with and without notches are built to study the effect of notches on the failure behavior of rock specimens under compression. Firstly, the uniaxial compression test on a rectangular specimen with a circular hole is simulated (Figure 6a). The stress-strain curve for this specimen is plotted in Figure 8. In this figure, the stress-strain curve obtained from the uniaxial compressive test on the intact rock specimen (i.e. with no hole and no notches) is also plotted. The figure shows that the specimen with a central circular hole reaches a maximum stress of 15.65 MPa before it fails. This is 17.4% less than that in the intact specimen in which the maximum uniaxial strength is 18.95 MPa.  Figure 9a shows the failure evolution in the specimen with the circular hole. In this figure, the failure patterns at the 75% peak stress, 90% peak stress, peak stress and 80% post-peak stress are illustrated. The black and the yellow lines in Figure 9 represent the tensile and the shear cracks, respectively. When the specimen is loaded at 75% peak stress, a few cracks are nucleated at the left side of the hole (θ = 180°) where the tangential stress is concentrated (see Eq. 5). In addition, tensile cracks are formed at the bottom of the hole (θ = 270°). Also, some minor cracks appear in the specimen which are mainly distributed around the hole. At 90% of the peak stress level, more fractures are formed around the hole and localized failure zones appear at the two sides of the hole (θ = 0° and 180°). At the peak stress, the cracks are coalesced and formed into obvious failure planes. Figure 10a shows a close view of the failure zone at the left side of the hole at the peak stress level. In this figure, the rock at the surface of the hole is crushed because it is the location of the maximum tangential stress around the circular hole (Eq. 5). By applying more strain to the model and bringing it to the 80% of the peak stress (post-peak), the number of cracks is increased, and the failure planes become wider.  (Figure 11a). When a notch is created, the free space provided by the notch destroys this closed ring of rock ( Figure 11b) and destress the adjacent rock (see Figure 5b). It explains the integrity of the rock that exists at the sides of the notch in Figure 10b. However, at high stresses the notch may get closed and the rocks at the notch sides may get reloaded (Manouchehrian and Kulatilake 2021). The failure patterns in the models with different notch lengths at 80% post-peak stress level are illustrated in Figure 12. The figure shows that when the length of the notch is increased, the tensile failure at the bottom of the hole is longer. In the models with longer notches, the free space resulted from the notches is larger; therefore, a higher number of particles can move towards the free space. As a result, more tensile stress is induced at the top (θ = 90°) and bottom of the hole (θ = 270°). Figure 12 indicates that in the models with different notch lengths, cracks are mainly formed at the tips of the notches. In the models with longer notches, the failure zone is far away from the hole wall. When the length of the notch is decreased, the short notch lengths. The failure patterns given in Figure 13 indicate that it is possible to select L/R ratios including a minimum value, which depends on the in-situ stresses, rock material properties as well as the boundary and loading conditions, to prevent failure occurring at the wall of the hole. From the numerical results presented in this section, it can be concluded that the influence of the notches on the fracturing mechanism around a hole is a function of the notch dimensions.

Influence of confinement on the failure mechanism
In an underground rock structure, the rock mass is usually subjected to confined loading conditions. Therefore, evaluation of rock behavior under confinement is an essential task for any rock engineering design. In this section, some models are built to study the failure  Next, the influence of the confinement on the failure mechanism in the specimens with a notched hole (Figure 6b) is studied. Two notches with the length of L = 5 mm (L/R ratio of 0.50) are created in the model. In Figure 14b, when the confinement is zero, the failures are nucleated from the notch tips. Some minor tensile fractures are formed at the bottom of the hole (θ = 270°), where the tensile stress is maximum. At higher confinements, this tensile cracking does not appear. When the confining pressure is increased to 2 MPa, a few cracks turned back toward the surface of the hole (θ = 135° and 315°). However, a larger portion of the volume of the rock at the sides of the notches is remained unfractured. In the models with higher confinements, the damage to the rock at the wall of the hole has increased. Figure 14b shows that when the confinement is 6 MPa, cracks coalesced and created crushed zones around the hole. Figure 15 indicates that the strength of the specimen in the notched hole models are slightly smaller than that in the circular hole model. However, at 6 MPa the difference between the two afore-mentioned strengths becomes negligible. Figure 14b and

Conclusions
Discontinuity is an intrinsic characteristic of rocks which may exist in rocks in different shapes and sizes. The failure behavior of defects such as cracks and notches in rocks and brittle materials under different loading conditions has been studied in the previous rock mechanics and fracture mechanics studies. However, the failure behavior of defects under pure compression (negative Mode-I) is neglected in the previous studies. In this research, a numerical study was conducted to understand how the failure mechanism of specimens containing a circular hole would change if notches exist at the surface of the hole. Firstly, the uniaxial compressive test on a rock specimen with a circular hole was modeled and the failure evolution in the specimen was simulated. In a separate model, two notches were created at the surface of the hole. Results showed that when the notches were created in the model, failure zones around the hole were transferred to the tips of the notches and the rock at the sides of the notches remained unfractured.
Next, a parametric study was carried out to investigate the influence of the notch length on the fracturing behavior of the specimen. It was found that the influence of the notches on the fracturing mechanism around a hole is a function of the notch dimensions. Moreover, the influence of the confinement on the failure mechanism in the specimens with a circular hole was investigated. Results showed that the notches affect the failure mechanism at confinements up to 4 MPa. Under higher confining pressures, the notches do not play any meaningful role in fracturing mechanism.
The numerical modeling results presented in this study showed that the presence of defects (notches in this study) at the surface of a hole can affect the fracturing mechanism of the rock around that hole. In some cases, the failure at the boundary of the hole was prevented when the notches above a certain L/R ratio were created in the model. It implies that the notches might be useful to control the failures at the underground excavation boundaries. It is important to point out that the minimum L/R ratio to prevent failure at the boundary of the hole depends on the in-situ stresses, material properties as well as the boundary and loading conditions. Therefore, numerical modeling should be performed incorporating the sitespecific data to obtain the minimum L/R ratio. In addition, the validity of the numerical results should be examined in practice through physical modeling. In future, we plan to conduct some laboratory experiments to verify the reliability of the numerical results. Figure 1 Basic modes of crack extension  The tangential stress distribution around (a) a regular tunnel and (b) a notched tunnel The stress-strain curves obtained from PFC2d models Illustration of the load bearing closed ring around a circular hole Figure 12 The failure patterns in the models with different notch lengths (80% post-peak stress)

Figure 13
Failure pattern and displacement vector distribution around the notched hole with L = 2.5 mm (L/R ratio of 0.25) Figure 14 Failure patterns in the (a) circular hole and (b) notched hole models at 80% post-peak stress level for different con ning pressures Figure 15 The peak strength at different con ning pressures in the two models