Analysis of stress and failure in rock specimens with closed and open flaws on the surface

Abstract

The influence of closed and open surface flaws on the stress distribution and failure in rock specimens is investigated. Heterogeneous finite element models are developed to simulate the compression tests on flawed rock specimens. The simulated specimens include those with closed flaws and those with open flaws on the surface. Systematic analyses are conducted to investigate the influences of the flaw inclination, friction coefficient and the confining stress on failure behavior. Numerical results show significant differences in the stress, displacement, and failure behavior of the closed and open flaws when they are subjected to pure compression; however, their behaviors under shear and tensile loads are similar. According to the results, when compression is the dominant mode of stress applied to the flaw surface, an open flaw may play a destressing role in the rock and relocate the stress concentration and failure zones. The presented results in this article suggest that failure at the rock surface may be managed in a favorable manner by fabricating open flaws on the rock surface. The insights gained from this research can be helpful in managing failure at the boundaries of rock structures.

Graphical abstract

Keywords

surface flaw / heterogeneity / circular hole / numerical modeling / relative displacement

1 Introduction

Rock failure at the excavation boundaries is a common problem in mining and civil projects. Usually at the surface of the rock excavations, various rock defects such as cracks, fractures, pores, and notches exist. These defects either may naturally exist due to geological processes, or might have resulted from excavation activities, or even have been created manually for some engineering purposes. Presence of defects at the surface of excavations can be critical to the stability of structures because defects act as stress raisers leading to local failures in the structures. Moreover, the surface of excavations is generally the location of the maximum differential stresses. Therefore, presence of defects at the excavation boundaries can contribute to surface instabilities in rock structures.

Studies have shown that presence of surface defects in structures such as bridges, dams, buildings, spacecrafts, industrial parts, etc., can result in tragic disasters. Therefore, usually special attention is given to the study of surface defects to prevent unexpected failures and instabilities of the structures. The failure behavior of surface defects in different materials such as metals, ceramics, polymers and alloys has been studied [1–4]. In geomechanics, most of the research has been devoted to the study of internal cracks under various loading conditions. Meanwhile, surface cracks have not been given sufficient attention. Analysis of failure in beams and half-disc rock specimens containing surface defects (notch) is usually conducted to study the rock tensile strength and fracture toughness. A limited number of investigations have been focused on studying the failure behavior of surface defects under shear and compressive loads. For example, Lu et al. [5] conducted an experiment to study the strength and failure in sandstone specimens containing a single surface flaw under compression. In that experiment, they changed the flaw dimensions, i.e. the length, depth and the angle, to investigate the fracturing mechanism of the specimens in the uniaxial compression test.

Because rocks are not transparent, it is difficult to observe the rock fracturing processes inside the specimens in physical experiments. As an alternative, advanced numerical models can be used to simulate the rock fracturing process [6–15]. Numerical models have been developed to study the surface defects under different loading conditions. Jiang et al. [16] used PFC^2D models to study the failure mechanism and acoustic emission characteristics of rock specimens, each with a crack on the surface, under the uniaxial compression. They showed that the crack length and position have important roles to play on the strength of specimens. In another numerical study, Qian et al. [17] used RFPA^3D models to study dynamic fracture in rock specimens containing a surface flaw with different dip angles. In that research, they recognized some characteristics and differences in rock failure process under dynamic and static loading conditions. The studies mentioned above, and many other similar studies, have shown the importance of surface defects and their contribution to rock failure. However, more research is needed to understand the surface crack contribution to the rock failure under different loading conditions.

In rock masses, defects and discontinuities are either closed or open. In laboratory experiments, the failure behavior of closed flaws [18–24] and open flaws [25–29] under different loading conditions has been studied. Previous studies have shown that the relative normal and tangential displacements of rock discontinuity surfaces cause damage initiation and propagation in rocks. In an open flaw, three relative displacements between the surfaces of the flaw are possible: opening, closure, and slip. However, in a closed flaw the closure is not possible, because there is nil or negligible distance between the two surfaces of the flaw to allow any further relative normal displacement under compression. Therefore, the failure behavior of the closed and open flaws under different loading conditions may be different [23,30–32]. The differences in the failure behavior of the closed and open defects under different loading conditions have been investigated in some studies. For example, Park and Bobet [23] studied the cracking behavior in specimens containing closed and open flaws. They identified eight types of crack coalescence behavior which were common in both the closed and open flaws. They concluded that the main difference between the closed and open flaws failure behavior was the crack initiation stress, which was higher for the closed flaw. In another work, Pan et al. [30] studied the failure behavior of specimens with infilled and open flaws under uniaxial compression. They used different infilling materials to change the relative displacement of the flaw surface at a given load. In that study, the specimen with an open flaw showed the maximum failure potential. These examples show that the closed and open defects may behave differently in some circumstances. Hence, closed and open defects should not be treated in the same way for engineering purposes.

In this study, heterogeneous Abaqus models are developed to study the rock failure in the specimens with closed and open flaws on the surface. Introduction of the heterogeneity using Python scripting into Abaqus models is explained in Section 2. In the developed heterogeneous models, the mechanical properties of the different materials are assigned to the elements to represent the mineral composition of a typical granitic rock. In Section 3, compression tests on rock specimen with two different geometries are simulated and the contribution of the closed and open surface flaws to the rock failure is discussed. A parametric study is conducted to understand the effect of the flaw inclination, friction, and the confining stress, on the failure behavior of the flaws. A discussion on the results is given in Section 4.

2 Heterogeneous Abaqus model

Advanced computational models can simulate different scientific and engineering processes. In geomechanics, continuum and discontinuum models are widely used to solve routine and sophisticated engineering problems. From a micro-scale point of view, a rock is made up of minerals, voids, and fractures. This means that basically rock is a heterogeneous medium. However, a rock can be treated as an equivalent homogeneous medium if no significant discontinuity exists inside. Homogeneous models are usually efficient in capturing the response of rock to mechanical loads. For example, analysis of stress and displacement around excavations can be achieved using homogeneous models. However, in problems in which the rock heterogeneity plays a key role in the processes, homogeneous models are not efficient; instead, appropriate heterogeneous models should be utilized.

Formation of concentrated local stresses is a key function of rock heterogeneity. In reality, when rock is loaded the stresses are distributed around micro-defects such as pores, cracks and grain boundaries, which results in local tensile stresses even if the rock as a whole is under compression [33,34]. Previous studies have shown that the rock failure process is mainly initiated as a result of tensile micro-cracking, progressing to dominant shear fractures at a later stage of the failure process, when a sufficient amount of tensile micro-cracks are generated [35–39]. Thus, models which can capture tensile splitting are needed to simulate the rock fracturing realistically. Because homogeneous models are not capable of capturing the axial splitting, they cannot be used to fully simulate the rock failure process [40,41] and heterogeneous models are required.

2.1 Introduction of material heterogeneity into Abaqus models

Abaqus is a general-purpose finite element analysis package developed by Dassault Systèmes and has a wide range of applications in different engineering disciplines [42]. In the last three decades, Abaqus has been used as a strong analysis tool for simulation of complex systems in geomechanics [15,43–45]. In this research, an explicit Abaqus code is utilized to study the stress and failure of rock. In explicit numerical models, the convergence problem is eliminated; therefore, these models are efficient for simulating plasticity. The Abaqus package only includes homogeneous models, but there is a possibility to build heterogeneous models using Python scripting. Because the main purpose of this study is investigation of rock failure behavior, material heterogeneity is added into Abaqus models to make it suitable to simulate the fracturing processes realistically.

Rock material heterogeneity can be introduced into numerical models with two approaches. In the first approach, the physical and mechanical properties of each element are assigned randomly following an appropriate statistical distribution function such as Weibull or Gaussian distribution [41,46]. In such an approach, many material properties following a distribution function are generated within the model which may increase the computation time. Such a procedure is suitable for simulation of large-scale problems. In the second approach, the rock textural properties such as voids and mineral content are considered in the modeling. In this latter approach, because the number of rock constitutive materials is limited, the computation time is less. The latter approach is mainly suitable for simulation of laboratory-scale problems because modeling of rock texture for large-scale problems is not computationally cost-effective.

In this study, the second approach is used to introduce material heterogeneity into the models. For this purpose, the properties of each mineral are assigned randomly to elements within the model domain according to the percentage of each mineral in the rock. In the models with many elements, it is too difficult to implement the above-mentioned approach to introduce material heterogeneity into the models using the Abaqus Graphical User Interface. In Abaqus, Python scripts can be used to automate repetitious tasks, change parameters of a model as part of an optimization study, and extract information from outputs [47]. In this study, a Python script is used to build the heterogeneous model.

2.2 Rock material modeling

The procedure explained in Section 2.1 is implemented to simulate heterogeneous rock specimens for the study. Figure 1 shows a rectangular heterogeneous Abaqus model with the width of 100 mm and height of 200 mm including 20000 elements and four different materials. The simulated rock is composed of three minerals including 70% feldspar, 24% quartz, 5% biotite, as well as 1% void. This composition represents the mineral composition of granitic rocks such as quartz-syenite. It should be noted that in this study, the geometrical heterogeneity of rock grains is not considered. Due to the limitations in preparation of a standard test specimen, evaluation of the mechanical properties of minerals is not as straightforward as it is for rocks. However, some mechanical properties of minerals such as the Young’s modulus, uniaxial compressive and tensile strengths have been explored and reported in the geological literature [48–51]. Table 1 lists the ranges for some mechanical properties of feldspar, quartz, and biotite. In this study, an elasto-plastic Mohr−Coulomb softening model is used to model the strength behavior of the rock. Because the Mohr−Coulomb softening parameters of minerals are not known, in this study these parameters are adjusted in a way to represent the strength of the minerals. The adjusted parameters for defining the elastic and plastic behaviors of the materials are presented inTable 2.

Unconfined and confined compression tests as well as uniaxial tensile test simulations are conducted to evaluate the mechanical properties of the simulated heterogeneous rock specimen. Figure 2 shows the failure pattern (indicated by the maximum principal plastic strain) of the specimen in an unconfined compression test. The figure shows that the developed heterogeneous model can capture axial splitting, which is essential for realistic simulation of rock failure. In the developed heterogeneous model, because the materials are assigned to each element randomly, the rock properties resulted from different simulation-runs may be different; however, the overall model response is similar. Figure 3 shows the results of the unconfined compression test on the given rock for ten different simulations. Basic descriptive analysis for results of different unconfined compression test simulations shown in Table 3 provides the average values of 248.0 MPa, 53.3 GPa, and 0.2 for uniaxial compressive strength (σ_c), Young’s modulus (E), and Poisson’s ratio (υ), respectively. The calculated coefficients of variation (CV) show very low variability around the mean values. For the heterogeneous model, a friction angle of 60° and a cohesion of 33.4 MPa are calculated. Also, the uniaxial tensile strength of 11.1 MPa is calculated. The textural and mechanical properties of the simulated rock represent typical properties of a granitic rock. In the next section, the developed heterogeneous model is used to study the rock failure in the specimens with closed and open flaws on the surface.

3 Numerical modeling

In different rock features such as slopes, mine pillars, wells, tunnels, caverns, etc., closed and/or open defects may exist on the rock surface. In the laboratory, specimens with different geometries are used to study the rock behavior under different circumstances. In this section, the failure behavior of closed and open flaws in two types of laboratory specimens under compression is simulated. The first specimen is a rectangular prism with a single flaw on its surface and the second specimen is a square prism with a central hole and two co-planar flaws on the rock surface within the hole. The general stress state in the specimens with consistent geometries under uniaxial and biaxial compression is two dimensional (2-D). Therefore, 2-D plane-stress models are used for simulation of laboratory compression tests in this study.

3.1 Rock specimen with a single flaw on the external surface

3.1.1 Model setup

In some features such as mine pillars and large caverns, rock is subjected to uniaxial compressive loads. Studying the role of surface defects on the stability of this type of feature is helpful in preventing unexpected failures. The in situ rock failure process under uniaxial compressive loads can be reproduced in the laboratory. In this section, uniaxial compression tests are simulated to study the failure behavior of closed and open surface flaws. A rectangular specimen with a width of 100 mm and a height of 200 mm is modeled using the heterogeneous Abaqus model described in Section 2.1. A flaw with a length of 30 mm and an inclination of θ is added to the midlevel of the specimen surface on the left side. In the models with an open flaw, the distance between the two faces of the flaw is 1.0 mm. The geometry of the model is illustrated in Fig. 4. A rigid contact with a friction coefficient (μ) of 0.5 is used to define the normal and the tangential behaviors of the flaw. To simulate the uniaxial compression test, the bottom of the specimen is fixed in the maximum stress direction (y-direction) and the other direction is free (roller constraint), and a constant velocity of 0.04 m/s is applied directly to the top of the specimen to apply the compressive load. The rock material properties presented in Section 2.2 are used for simulations.

3.1.2 Simulation results

Crack initiation and propagation from a pre-existing defect in rock depend on the loading condition. In a rock specimen under a given boundary condition, with a change in the flaw orientation, the loading condition around the flaw changes. From previous studies it is well understood that the inclination of discontinuities in rock is an important factor affecting rock mechanical behaviors [52–54]. For example, laboratory experiments have shown that the strength of jointed rock specimens in the triaxial test is governed by the orientation of the joints [52].

A systematic study is conducted to understand the influence of the flaw inclination (θ) on the stress distribution and rock failure in the flawed specimens. The flaw inclination is varied by selecting θ = 0°, 30°, 45°, and 60°. The relative normal and tangential displacements between the flaw surfaces at the left surface of the specimen at strain (ε_y) of 0.25% are shown in Fig. 5. Hereafter, the “relative displacement” corresponds to the flaw tip at the surface of the specimen. In the specimens with the closed flaw, the relative normal displacement between the flaw surfaces (closure) with any inclination angle is zero (Fig. 5(a)). Also, in the model with θ = 0°, the relative tangential displacement between the flaw surfaces (slip) is zero (Fig. 5(b)). It means that when θ = 0°, no relative displacement between the flaw surfaces happens. However, by increasing the flaw inclination, the relative tangential displacement between the flaw surfaces (slip) increases. In an open flaw, a closure of 0.19 mm happens when θ = 0° (Fig. 5(a)). By increasing the flaw inclination to 60° the closure reduces to 0.13 mm. As is the case for the closed flaw, the amount of slip in the open flaw increases from 0 to 0.056 mm (Fig. 5(b)).

The stress–strain curves obtained from models with different flaw inclinations are shown in Fig. 6. In this figure, due to the relative displacement between the flaw surfaces, the slope of the stress-strain curves of specimens with open flaws is lower than that of specimens with closed flaws and this difference decreases as the flaw inclination increases. In this figure, the specimen with a closed flaw inclined at θ = 0° and the specimen with an open flaw inclined at θ = 0° have the maximum and minimum uniaxial compressive strengths, respectively. The relation between the flaw inclination and the uniaxial compressive strength of the specimens is shown in Fig. 7. The figure shows that in the model with a closed flaw, when θ = 0° the strength of the specimen is 236.5 MPa. According to the statistical analysis presented in Table 3, the strength of intact specimen is in a range between 232.0 and 260.7 MPa. Thus, the strength of the specimen with the flaw inclination of θ = 0° (closed flaw) is in the range of the intact rock specimen’s strength. It means that the closed flaw does not influence the strength of the specimen. In contrast, the open flaw with θ = 0° has a significant influence on the strength of the specimen and decreases it to 141.9 MPa. By increasing the flaw inclination angle, the strength of the specimens increases, which is in agreement with the laboratory experiment observations [5,23,30,55,56]. The modeling results indicate that the strength of specimens with closed flaw is higher than that in the models with open flaw. This difference is significant when θ = 0° and is negligible when the flaw is inclined. The relative normal displacement of the flaw surfaces is the reason of this difference (Fig. 5(a)).

The distribution of σ_y at the strain level of ε_y = 0.25% in different models is shown in Fig. 8. In the models, when θ = 0°, the flaw is subjected to the pure compression. In this condition, the stress distribution in the specimen with the closed flaw is uniform which means that the flaw does not play a stress raising role in the specimen (Fig. 8(a)). In the model with the open flaw the stress is non-uniform (Fig. 8(b)). In this case, the free space between the flaw surfaces prevents the rock at the top and bottom of the flaw from being loaded. Thus, the specimen surface at the left side bears less compression. Figure 8(b) shows that the stress is concentrated in a small area at the flaw tips. By increasing the flaw inclination angle, the shear stress acts on the flaw and creates a mixed mode loading condition. In the closed flaw, the friction may resist the shear stresses. For example, when θ = 30°, the friction between the flaw surfaces does not allow a significant slip to occur and as a result, a minor stress disturbance exists around the flaw (Fig. 8(a)). The figure shows that as the flaw inclination angle increases, the influence of the closed flaw on the stress distribution increases because the shear stress causes more tangential displacement of the flaw surfaces.

The failure evolution in the specimens at different strain levels is illustrated in Fig. 9. The figure indicates different failure behaviors of the closed and open flaws in the models with θ = 0° (Fig. 9(a)), and similar behaviors in the models with inclined flaw (Fig. 9(b)–9(d)). The figure shows that in the specimen with the closed flaw and θ = 0°, micro fractures are formed randomly in the specimen and the flaw does not result in any failure localization in the specimen. In contrast, in the model with the open flaw, the fracturing process starts from the flaw tip and then propagates toward the external boundaries of the specimens to create a major failure plane (Fig. 9(a)). The figure shows that in the models with inclined flaw, i.e., θ = 30°, 45°, and 60°, the failure mainly nucleates from the flaw tip. At this location, wing cracks appear first and then, with increase of the strain, the secondary cracks are formed. Results show that as the flaw inclination angle increases, closed and open flaws tend to show increasingly similar behaviors.

When the shear stress acts on the closed flaw, the friction resists the slip. Thus, the friction coefficient of the flaw surfaces can influence the closed flaw failure behavior. Specimens including a closed flaw with θ = 30° and different friction coefficients of μ = 0, 0.5 and 1.0 are modeled to investigate the influence of friction coefficient on the failure behavior of the closed flaw. Figure 10 shows the maximum principal stress in the specimens at ε_y = 0.25%. In the model with μ = 0, the stress field in the specimen is non-uniform because there is no friction to resist the shear forces and as a result relative tangential displacement between the flaw surfaces occurs. By increasing the friction coefficient, the frictional resistance prevents the flaw slip and consequently the effect of closed flaw on the stress localization becomes less. In Fig. 11, the failure pattern in the models with different μ values at the strain level of ε_y = 0.38% is presented. The figure shows that, when μ = 0, the flaw slip creates a tensile wing crack at the tip of the flaw. At μ = 0.5, a smaller wing crack is formed at the flaw tip because less slip is allowed. In the model with μ = 1, no localized failure occurs at the flaw tip.

3.2 Rock specimen with two flaws on the internal surface

3.2.1 Model setup

Upon excavation of any opening in a rock mass, the in situ stresses are redistributed around the opening. In the laboratory, specimens with different cavity geometries can be used to study the failure around the excavations [57–59]. In this section, the uniaxial and biaxial compression tests are simulated to study the failure behavior of closed and open surface flaws in specimens with a central hole and two flaws at the surface of the hole. A 200 mm × 200 mm square specimen with a central hole is modeled using the heterogeneous Abaqus model explained in Section 2.1. The radius of the hole is 20 mm. This is a common specimen type to study the failure around circular openings [57,60]. On the perimeter of the hole, two co-planar flaws with a length of 30 mm and perpendicular to the hole surface are added. The flaws are located on the perimeter of the hole at the location of β and β + 180°. In the models containing the open flaws, the distance between two surfaces of the flaws is 1.0 mm. The geometry of the model is illustrated in Fig. 12. A rigid contact with a friction coefficient of μ = 0.5 is assigned to the flaw surfaces. To simulate the uniaxial compression test, a roller constraint is used to fix the bottom of the specimen in the maximum stress direction (y-direction) and the specimen is free to deform in the other direction (x-direction). A constant velocity of 0.04 m/s is applied directly to the top end to apply load to the specimen in y-direction. The same end boundary conditions are applied to the specimens in the biaxial compression test simulation. The rock material properties presented in Section 2.2 are used for the simulations.

3.2.2 Simulation results

The stress state around a circular opening at different locations is different; thus the influence of surface flaws at different locations on the perimeter of a hole may be different. Specimens with two co-planar flaws positioned at different locations on the perimeter of the hole (β) are simulated. The inclination of the flaws is varied by selecting β = 0°, 45°, and 90°. The relative displacements of the flaw surfaces at the strain level of ε_y = 0.11% are listed in Fig. 13. The figure shows that when β = 0°, no relative displacement (closure and slip) between the closed flaws occurs. In the open flaws, a closure of 0.2 mm happens; however, the relative tangential displacement is zero. In the models with β = 45°, a relative tangential displacement of 0.11 mm occurs in the closed flaw while the normal displacement is zero. For β = 45°, the normal and tangential displacements in the open flaw are 0.12 and 0.08 mm, respectively. The results show that the slip in the closed flaw is slightly smaller, because the frictional forces resist the slip. When β = 90°, the flaws are only subjected to tension, hence the relative normal displacement between the flaw surfaces are in the form of opening. The opening mode of displacement is possible in the closed flaws. Figure 13(a) shows a similar relative normal displacement (opening) of 0.04 mm between the flaw surfaces for the closed and open flaws. The value of slip in both types of flaws is zero.

The relation between the strength of the specimen and the flaw inclination is summarized in Fig. 14. This figure shows a general upward trend in the strength of the specimens, as β increases. However, the model with closed flaws and β = 0° is an exception which does not follow the trend. Because no relative displacement (closure and slip) exists between the closed flaw surfaces, the flaw does not influence the strength of the specimen.

The distribution of σ_y in the specimens at ε_y = 0.11% is shown in Fig. 15. This figure indicates that when the flaws are subjected to the pure compression, i.e., β = 0°, the closed flaws do not influence the stress state around the hole. However, the open flaws cause the rock adjacent to the flaw to stay unloaded. In fact, the free space provided by the open flaw prevents the rock from being loaded. In this condition, the open flaws play a local destressing role in the specimen. The figure shows that when the flaw inclination is increased, the stress-fields in the models with closed and open flaws are similar.

The failure evolution in the specimens at different strain levels is illustrated in Fig. 16. The figure shows different failure patterns in the models with closed and open flaws under pure compression (β = 0°); on the other hand, similar failure patterns can be observed in the models for which the flaws are subjected to shear and tensile loads (β = 45° and 90°). When β = 0°, the surfaces of the closed flaws are not allowed to have any relative displacements, but in the open flaws the relative normal displacement is allowed. Therefore, in the model with the closed flaws, first, the tensile fractures appear at the top and bottom of the hole, and then the fractures are developed around the hole (Fig. 16(a)). In the model with the open flaw, tensile fractures occur at the perimeter of the hole, but due to the destressing effect of the open flaws, the fractures are localized at the flaw tip and no crushed zone is created at the right and left of the hole. In the models with β = 45°, the flaws are subjected to a mixed loading condition in which the shear and compressive loads act on the flaws at the same time. The failure behaviors of the closed and open flaws in this condition are similar. In Fig. 16(b), first the wing cracks (tensile) initiate at the tips of the flaws and the secondary cracks (shear) nucleate at the flaw tips as the strain increases. This is similar to the laboratory test observations of Refs. [20,23]. The secondary cracks propagate toward the specimen external boundaries in the direction of the maximum principal stress. Also, some fractures turn toward the internal surface of the specimen at the hole surface. In the models with β = 90°, the flaws are subjected to pure tension, and shear stress does not exist. In this condition, the relative normal displacement between the surfaces of the flaws is in the form of opening. Figure 16(c) indicates that the failure behaviors of the closed and open flaws under pure tension are the same. First, the tensile fractures occur at the hole surface and then propagate toward the external boundaries in the direction of the maximum principal stress.

In underground excavations, the rocks are confined. Thus, in practical engineering, it is always important to evaluate the rock behavior under confined loading conditions. A parametric study is conducted to understand the influence of the confining stress on the failure behavior of rock with closed and open flaws. The confinement, σ₃ (the minimum principal stress), is varied by applying 0, 20, and 40 MPa in the horizontal direction (x-direction). The numerical results presented above show that the behaviors of the closed and open flaws under pure compression (β = 0°) are significantly different. Hence, for investigation of confinement influence on the failure behavior, the specimens with flaw inclination of β = 0° are considered. The closed form solution of the stress state around a circular hole in an infinite plate is [61]:

(1){\sigma }_{\text{r}}=0,

(2){\sigma }_{\beta }={\sigma }_1+{\sigma }_3+2\left({\sigma }_1-{\sigma }_3\right)\text{cos}2\beta ,

(3){\tau }_{\mathrm{r}\beta }=0,

where σ_r, σ_β, and τ_rβ are the radial, tangential and shear stresses, respectively. In a stress-field in which σ₁ > σ₃, the maximum tangential stress is at β = 0° and 180°, while the minimum tangential stress is at β = 90° and 270° (see Fig. 12). Hence, the failure generally happens at these locations. The failure pattern in the models with different confining stress is shown in Fig. 17. The figure shows that when no confining stress is applied to the specimens, in the specimens with closed and open flaws, the tensile cracks are created at the top (β = 90°) and the bottom (β = 270°) of the hole. By increasing the confining stress, shorter tensile cracks are formed at the top and bottom of the hole because the confining stresses prevent the crack opening. In the models with the closed flaws, the rock at the right (β = 0°) and the left (β = 180°) sides of the hole are fractured because these are the locations of the maximum tangential stress (Eq. (2)). Results show that increase in the confinement results in the enlargement of the crushed zone at the right and left sides of the hole (Fig. 17(a)). In the models with the open flaws, the surfaces of the flaws can have normal displacement. As shown in Fig. 15, the flaws play a destressing role in the specimen and transfer the stress concentration zones to the tips of the flaw. Because of this, the rock at the surface of the hole at the right and left sides is not damaged and the failure is localized at the flaw tips. In Fig. 17(b), the increase in the confining stress does not produce any failures at the right and left sides of the hole surface; note that those are the locations of the maximum tangential stress in a flawless hole.

4 Discussion

In recent decades, constructions of underground spaces in different mining, civil, and military projects have increased in number. In the ground, the gravitational and tectonic forces create a lithostatic stress condition in which the three principal stresses act on the rock. Excavation of underground openings disturbs the in situ stresses and results in redistribution of stress around the opening. In deep underground, where the in situ stresses are naturally high, excavation activities are often accompanied by rock failure. A large difference between the maximum and minimum in situ principal stresses creates a large tangential stress at the excavation boundary and usually results in localized failure. The V-shaped failures and breakouts in circular tunnels and boreholes are the outcome of a large difference between the maximum and minimum in situ principal stresses. For example, the V-shaped failure in the mine-by test tunnel at an underground research laboratory in Canada [62] resulted from the high tangential stress at the excavation surface. Rockburst is another type of localized rock failure which occurs in mines and tunnels excavated in highly stressed grounds. Case histories of rockburst have documented many tragic events that resulted in injuries, fatalities and destruction of equipment [63,64]. The multiple rockbursts that occurred during the construction of the Jinping II hydropower station in China [64] are examples of violent rock failure caused by high differential stresses at the excavation boundaries.

The presented numerical results show that closed and open defects may have different effects in the stability of structures. Therefore, their contribution to rock failure should be considered separately. Usually, the fractures and cracks are considered as closed defects and some cavities such as notches and cuts are considered as open defects. When an open defect is positioned in rock in a way that the compression is the dominant mode of loading applied to the defect surfaces, it may play a destressing role in the rock and redistribute the stress concentration and failure zones. This destressing attribute of the open defects might be helpful in destressing the rock at the boundaries of underground excavations in order to manage the rock failure. For example, the strainburst in deep excavations might be controlled if some notches are cut in the locations of the maximum tangential stresses at the surface of the opening. It can be a conceptual solution for strainburst control in mines and tunnels. More numerical and experimental studies are needed to investigate the feasibility of applying flaws for failure management around excavations.

5 Conclusions

Rock failure at the boundaries of rock structures is a common problem in many excavations. Presence of defects at the surface of excavations can be critical to the stability of the structures because defects are stress raisers which cause local failures in the structures. In this article, the stress and failure in the rock specimens with closed and open flaws on the surface were studied using heterogeneous Abaqus models. First, the material heterogeneity was introduced into Abaqus models using Python scripting to enhance its suitability for simulation of rock failure. For this purpose, the four different properties (feldspar, quartz, biotite, and void) were assigned to the elements within the model domain. The compression test simulation results showed that the developed heterogeneous model could capture the axial splitting which is a key feature required for a realistic rock failure simulation.

Next, the developed heterogeneous models were used to simulate compression tests on the flawed rock specimens. Systematic analyses were conducted to investigate the influences of flaw inclination, friction, and confining stress on the failure behavior of closed and open surface flaws. When the flaws were perpendicular to the direction of the maximum principal stress, a pure compression acted on the surfaces of the flaws. Numerical results showed significant differences in the stress, displacement, and failure behavior of the closed and open flaws when they were subjected to pure compression. In this condition, the presence of the closed flaws did not influence the strength, stress state or failure in the specimens. The open flaws resulted in a significant stress decrease in the adjacent rock and transferred the failure zone to the flaw tips. By increasing the flaw inclination, the closed and open flaws tended to show more similar behaviors.

The presented numerical results in this article showed that in some conditions, the open flaws may play a destressing role in the rock and relocate the stress concentration and failure zones. This destressing attribute of the open flaws might be feasible for destressing the rock at the boundaries of underground excavations to mitigate the rock failure risk. More numerical and experimental studies are needed to investigate the feasibility of using flaws for failure management around excavations.

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