Application of the expanded distinct element method for the study of crack growth in rock-like materials under uniaxial compression

Abstract

The expanded distinct element method (EDEM) was used to investigate the crack growth in rock-like materials under uniaxial compression. The tensile-shear failure criterion and the Griffith failure criterion were implanted into the EDEM to determine the initiation and propagation of pre-existing cracks, respectively. Uniaxial compression experiments were also performed with the artificial rock-like samples to verify the validity of the EDEM. Simulation results indicated that the EDEM model with the tensile-shear failure criterion has strong capabilities for modeling the growth of pre-existing cracks, and model results have strong agreement with the failure and mechanical properties of experimental samples. The EDEM model with the Griffith failure criterion can only simulate the splitting failure of samples due to tensile stresses and is incapable of providing a comprehensive interpretation for the overall failure of rock masses. Research results demonstrated that sample failure primarily resulted from the growth of single cracks (in the form of tensile wing cracks and shear secondary cracks) and the coalescence of two cracks due to the growth of wing cracks in the rock bridge zone. Additionally, the inclination angle of the pre-existing crack clearly influences the final failure pattern of the samples.

Keywords

expanded distinct element method (EDEM) / crack growth / rock-like material / tensile-shear failure criterion / Griffith failure criterion / mechanical and failure behavior

Introduction

The design for rock-based structures such as dams, bridges and nuclear power plants requires a comprehensive understanding of the mechanical behavior of rock masses involved in their construction. The deformation and failure behavior of rock masses, which concerns the safety and stability of rock structures, is principally governed by the fractures from joints and cracks that exist in the host rock masses [1]. When subjected to external loads and disturbances, rock fractures can propagate and join with other neighboring fractures, inducing stress redistribution and localized stress concentration, which subsequently lead to large nonlinear deformations and the failure of rock masses [2]. Therefore, investigation of the growth mechanism of rock fractures and the effects on the mechanical and failure behavior of rock masses is a fundamental and promising issue in both academics and engineering.

To account for the failure mechanism of brittle materials under compression, several micromechanical models have been proposed [3-9]. These theoretical models describe the initiation and growth of tensile and shear cracks corresponding to the splitting and faulting failures of brittle materials, respectively, and provide a fundamental method for understanding the failure behavior of rock-like materials. However, most theoretical models focus on individual cracks and are incapable of dealing with complex situations such as propagation and coalescence of multiple cracks, as well as geotechnical problems with complex rock structures and boundary conditions.

To date, extensive experimental studies have been performed to investigate the growth and coalescence mechanisms of cracks in rock-like materials [10-16]. Based on the experimental phenomena, a common crack growth pattern has been proposed as: 1) the tensile wing cracks start at the tips of cracks and propagate stably along a curvilinear path in the direction of the maximum compressive stress; 2) the secondary cracks (generally described as shear cracks) initiate from the crack tips and grow coplanar (or quasi-coplanar) to the pre-existing crack or the opposite direction of the wing cracks [13]. These experimental studies indicate that the unstable growth of secondary cracks produces coalescence and is a major contributor to the overall failure of rock samples [14]. Different types of crack coalescence in rock-like materials, which are characteristic of particular crack geometries and stress conditions, have been reported in the literature [14-16].

Compared with theoretical and experimental research, numerical simulation can provide an effective, convenient and low-cost method to investigate the failure mechanism of rock-like materials. Numerical simulation has significant advantages such as representing the internal stress and deformations within the model and considering more complex problems. Currently, a number of numerical studies focusing on the crack growth problem have been performed with the finite element method (FEM) [17-21] and boundary element method (BEM) [22,23]. The FEM and BEM usually describe the crack propagation process with two approaches: the first describes the cracking process as the evolution of damage through the material (the degradation of mechanical properties of the surrounding elements), and the second simulates the crack growth by separating crack edges, which requires remeshing of the surrounding elements [17]. However, the first approach cannot strongly represent the displacement discontinuity characteristic of cracks, which is important for estimating the mechanical behavior of fractured rock masses. Additionally, the second approach has difficulty with model remeshing during each step of crack growth.

An alternative from FEM and BEM, the distinct element method (DEM), has better capabilities for representing the displacement discontinuity behavior of fractured rock masses, due to inherent advantages in simulating the separation and slipping of fractures. Jiang [1] and Nakagawa [24] developed an expanded distinct element method (EDEM) based on the DEM UDEC code to investigate the cracking of large-scale rock structures due to tensile and shear failures. In this method, a mass of potential cracks with high strength is pre-defined in the regions where new cracks will likely be generated. When the tensile-shear failure criterion is satisfied, the potential cracks are converted to real cracks and assigned proper mechanical properties. The EDEM provides a novel way to address the failure problems of rock masses, which overcomes the difficulty of model remeshing without sacrificing the displacement discontinuity feature of rock fractures.

In this study, the EDEM based on UDEC code is applied to investigate the mechanisms of crack growth and coalescence in rock-like materials under uniaxial compression. In addition to the tensile-shear failure criterion adopted in literature [1], the Griffith failure criterion has also been implanted into the EDEM to determine the initiation and propagation of pre-existing cracks and to analyze the failure patterns of fractured rock-like samples. For the purpose of verification, uniaxial compression tests with rock-like gypsum samples containing single and two-parallel cracks were performed. The crack growth patterns and the mechanical behavior of fractured samples obtained from EDEM simulations with two different failure criteria were compared with the experimental results to verify the EDEM and to investigate the crack growth pattern under different failure criteria. Based on experimental and numerical results, the mechanisms of crack growth and coalescence in rock-like materials under compression were characterized.

Description of EDEM

The DEM UDEC code represents the rock mass as an assembly of discrete blocks and the fractures as interfaces between blocks. This code can realistically model the mechanical behavior (compression, slipping, and separation) and geometrical properties (orientation, gap, and spacing) of fractures [25]. However, the current DEM is incapable of simulating the propagation and coalescence processes of existing fractures in the host rock masses, which play an important role in the mechanical and deformational behavior of fractured rock masses. To simulate the cracking and failure of rock masses, the EDEM was developed within the UDEC framework by introducing specific rock mass failure criteria into the DEM, which is accompanied by the processes of crack generation, propagation and coalescence. Figure 1 shows a flowchart of the EDEM. The realization of crack growth in numerical models mainly falls into four procedures: 1) definition of potential cracks in the model; 2) stress calculation for all contacts; 3) safety/failure judgment for all contacts; and 4) conversion of potential cracks into real cracks.

To overcome the difficulty of mesh rebuilding during the process of crack growth, a large number of potential cracks with bonding strength that is equivalent to the rock matrix are first defined in the entire model or in the regions where new cracks are possibly generated. These potential cracks can be converted into real cracks when the failure criteria are satisfied. The potential cracks represent the potential paths for the initiation of new cracks and the propagation of existing cracks. Because cracks can theoretically be generated in any direction and position in the model due to increased external stress, the proper distribution of potential cracks (position and direction) is a crucial factor for performing an accurate simulation of rock cracking. In the EDEM, a special treatment is performed by introducing six sets of gapped joints into the model, which separate the intact rock matrix into an assembly of hexagonal blocks, as shown in Fig. 2. The interfaces between blocks represent potential cracks, which have a large bonding strength and act like the rock matrix before failure happens. If the dimension of hexagonal blocks in the model is small enough, the new cracks can be generated anywhere and can propagate in any direction. Compared with triangular and quadrangular blocks, a hexagonal block with the same size can provide more propagation directions for potential cracks that may initiate from the six vertexes of the block. Additionally, the hexagonal block reduces the number of acute-angle meshes and effectively avoids the overlap between blocks, thereby improving the accuracy and stability of numerical simulations.

In the EDEM model, each hexagonal block is connected to adjacent blocks via point contacts located at the potential cracks (Fig. 2). The contact can be considered as a boundary condition, which applies external forces to each block. A number of triangular zones (meshes) are generated in each block to investigate the internal stress and strain. The zones in a block divide a potential crack into several sections, which are connected via contacts. The principal stresses for a contact are estimated by using the average stress of four surrounding zones with the shortest distances from the centroids of zones to the contact (e.g., Z_A1 & Z_A2 in Block A and Z_B1 & Z_B2 in Block B are used to calculate the stress tensor of the middle contact located in the potential crack, as shown in Fig. 2). Assuming the average stress tensor for four surrounding zones is σ_x, t_xy, σ_y and σ_z (under plane strain conditions), the principal stresses at the corresponding contact can be obtained in the form of Eqs. (1), (2) and (3). Then, the order of the three principal stresses is rearranged according to their values to ensure σ₁≥σ₂≥σ₃.

(1){\sigma }_{\mathrm{1}}={\displaystyle \frac{{\sigma }_x+{\sigma }_y}{\mathrm{2}}}+\sqrt{{\left({\displaystyle \frac{{\sigma }_x-{\sigma }_y}{\mathrm{2}}}\right)}^{\mathrm{2}}+{\tau }^{\mathrm{2}}{}_{xy},}

(2){\sigma }_{\mathrm{2}}={\displaystyle \frac{{\sigma }_x+{\sigma }_y}{\mathrm{2}}}-\sqrt{{\left({\displaystyle \frac{{\sigma }_x-{\sigma }_y}{\mathrm{2}}}\right)}^{\mathrm{2}}+{\tau }^{\mathrm{2}}{}_{xy}},

(3){\sigma }_{\mathrm{3}}={\sigma }_z.

To evaluate the safety/failure status of contacts, two types of failure criteria are adopted in the EDEM: the tensile-shear failure criterion [1] and the Griffith failure criterion [26]. The tensile-shear failure criterion has been widely adopted in rock mechanics and engineering practices because it comprehensively describes the failure behavior of rock mass when subjected to tensile and shear stresses, as expressed in Eqs. (4) and (5), where symbols c, ϕ and σ_t denote the cohesion, the internal friction angle and the tensile strength of the rock or rock-like materials, respectively.

(4){\sigma }_{\mathrm{3}}=-{\sigma }_{\mathrm{t}}(tensile\; failure),

(5){\sigma }_{\mathrm{1}}={\displaystyle \frac{(\mathrm{1}+\sin \phi ){\sigma }_{\mathrm{3}}}{\mathrm{1}-\sin \phi }}+{\displaystyle \frac{\mathrm{2}c\cos \phi }{\mathrm{1}-\sin \phi }}(shear\; failure).

The Griffith failure criterion, proposed by Griffith (1924), indicates that in brittle materials, fractures initiate when the tensile strength is exceeded by the concentrated tensile stress, which is induced at the ends of microscopic cracks. The Griffith failure criterion was developed based on energy from a series of experiments with brittle materials such as glass and can be expressed in terms of principal stresses, as shown in Eqs. (6) and (7).

(6){\sigma }_{\mathrm{3}}=-{\sigma }_{\mathrm{t}},\mathrm{i}\mathrm{f}{\sigma }_{\mathrm{1}}+\mathrm{3}{\sigma }_{\mathrm{3}}\ge \mathrm{0},

(7){({\sigma }_{\mathrm{1}}-{\sigma }_{\mathrm{3}})}^{\mathrm{2}}-\mathrm{8}{\sigma }_{\mathrm{t}}({\sigma }_{\mathrm{1}}+{\sigma }_{\mathrm{3}})=\mathrm{0},\mathrm{i}\mathrm{f}{\sigma }_{\mathrm{1}}+\mathrm{3}{\sigma }_{\mathrm{3}}\le \mathrm{0.}

At each calculation step, the principal stresses at all contacts that are located on potential cracks are calculated and substituted into the expressions of the tensile-shear failure criterion or the Griffith failure criterion. For a low stress level, the failure of rock-like materials cannot occur, and the numerical simulation will move to the next loading step. When the failure criterion is satisfied for a contact due to increased stress, the sections of a potential crack, which are connected to that contact, are converted to real cracks with the proper mechanical properties, representing the initiation and propagation of cracks. Through the above-mentioned procedures, the failure process for rock-like materials under various boundary conditions can be simulated.

Experimental preparation and numerical models

Preparation of verification experiments

To verify the validity and accuracy of the EDEM in modeling the mechanical and failure behavior of fractured rock masses, uniaxial compression experiments on rock-like samples with pre-existing cracks were performed, and the failure patterns and mechanical properties of samples were obtained.

The rock-like material used in the verification experiments is a mixture of gypsum (Rock-1 type produced by the Tokuyama Dental Corporation), sand, water and retarder with a mixing ratio by weight of 1∶3∶0.4∶0.04. The mechanical properties of this rock-like material were measured by performing unconfined and triaxial compression tests and the Brazilian test on intact samples, with the results shown in Table 1. This rock-like material was developed to model highly weathered sandstone with low strength.

Four types of gypsum samples with the same dimension of 50 × 100 × 20 mm (width × height × thickness) were processed. During casting samples, Teflon sheets with a length of 28 mm were embedded into the geometric center of the samples to model non-frictional penetrating cracks (two-dimensional cracks). Three types of samples contain single cracks, and the inclination of the crack to the axial direction-α was set as 30°, 45° and 60° to investigate the growth patterns of inclined cracks with low, moderate and high inclination angles, respectively. The samples with single cracks were tested to verify the growth patterns of single cracks and the mechanical properties of the samples obtained from the EDEM simulations. Another type of sample containing two vertically aligned parallel cracks with an inclination (α) of 45° and a crack spacing (s) of 14 mm was processed and tested to verify the coalescence pattern of two cracks through comparison with the numerical results. All samples were uniaxially loaded by a servo-controlled testing device in a displacement controlled loading mode at a rate of 0.25 mm/min.

To obtain the mechanical properties of pre-existing cracks and the newly generated cracks during compression tests, which are important parameters for the EDEM simulations, the normal loading tests and direct shear tests for gypsum samples containing single fractures were performed by using the test approaches described in [27]. The test results are listed in Table 2.

Numerical models

Corresponding to the experimental samples, four types of EDEM models with the same dimension (W × H_S = 50 × 100 mm), physico-mechanical properties of rock-like material and cracks (see Tables 1 and 2), and crack distribution (position, dip angle and spacing) were established, as shown in Fig. 3. A loading board (W × H_L = 50 × 10 mm.) with a high strength and stiffness was set above the models for applying uniaxial loads. To accurately simulate the propagation and coalescence of cracks, a large number of hexagonal blocks were created throughout the entire model with a side length of L = 1 mm. The interfaces between hexagonal blocks, which were pre-defined as potential cracks, represent the potential growth paths of cracks. When a certain failure criterion (the tensile-shear failure criterion or the Griffith failure criterion) is satisfied, the potential cracks can be converted into real cracks by changing their mechanical properties. Initially, the mechanical properties of the potential cracks should be chosen carefully based on calibrations to ensure that the model behaves identically to the intact rock samples before the cracking occurs. In this study, the properties of potential cracks were set as k_n (normal stiffness) = 8 × 10⁶ MPa/m; k_s (shear stiffness) = 8 × 10⁶ MPa/m; c (cohesion) = 100 MPa; ϕ (Internal friction angle) = 80°; and σ_t (tensile strength) = 100 MPa. After the hexagonal blocks were created, a mass of triangular zones (meshes) with the maximum side length of 1 mm were also generated in the interior of the blocks to estimate the stress and deformation in the rock matrix. The internal zones divide potential cracks into smaller sections. Therefore, the cracking process can be executed section by section, which improves the accuracy of simulations.

In the numerical models, the Mohr-Coulomb elements were adopted to represent the elastic/plastic deformation behavior of the rock matrix, and the Coulomb slip model was used to describe the deformation behavior of cracks. During simulations, compressive stress was uniaxially applied to the upper surface of the loading board, and the vertical displacement on the bottom of the rock sample was fixed.

Experimental and numerical results

Samples with single cracks

To verify the validity of the EDEM, a comparative study was performed between the experimental and numerical results that mainly involved failure statuses and the mechanical properties of the samples. Additionally, different growth patterns of cracks under different failure criteria were investigated. The numerical results can provide an explicit interpretation for the experimental phenomena, improving understanding of the mechanisms of crack growth.

Figure 4 shows the failure statuses of different samples containing single cracks obtained from experiments and the EDEM simulations. The experimental results (Fig. 4(a)) indicate that the overall fracture of the samples resulted from the growth of pre-existing cracks. With the increase of external loads, newly generated cracks (including wing cracks and secondary cracks) initiated from the tips of pre-existing cracks and grew along curvilinear paths in the direction of the maximum compressive stress. Experiments demonstrate that the crack inclination angle has an obvious influence on the failure patterns of the samples. For cracks with a low inclination angle (α = 30°), no obvious wing cracks were observed, and the secondary cracks grew approximately along the initial plane of pre-existing cracks to the sample ends, leading to the overall fracture. For a moderate crack angle (α = 45°), the combined growth of wing cracks and secondary cracks caused the sample failure. After the wing cracks initiated, the secondary cracks started their propagation from the crack tips, and the regions of shear failure are clearly confirmed at the crack tips. Additionally, an anti-wing crack was observed in this case, which initiated from the same position as the wing cracks but grew in the opposite direction. The phenomenon of anti-wing crack growth has been reported in [28]; however, its growth mechanism is still not well understood. For a high crack angle (α = 60°), the growth of wing cracks made a major contribution to the sample failure. Although the secondary cracks did not sufficiently propagate, they affected the growth direction of the wing crack, leading to newly generated cracks that propagated along the path between those of the wing crack and the secondary crack.

The failure of models obtained from the EDEM simulations with the tensile-shear failure criterion is shown in Fig. 4(b). Results from using the EDEM with the tensile-shear failure criterion agree well with the experimental phenomena. The wing cracks were generated due to the concentrated tensile stress around the pre-existing crack, while the secondary cracks were produced by the shear stress. For the low-angle crack (α = 30°), the tensile wing cracks stopped their propagation before their length reached that of the pre-existing crack. The sufficient growth of secondary cracks formed a macro shear plane and led to the model failure. The secondary cracks restrained the growth of wing cracks during the simulation. During the propagation of secondary cracks, a mass of small tensile cracks were produced along the path of the secondary crack, which grew in the axial direction. These small cracks may be caused by the tensile stress generated from the frictional force along the shear-failure plane. Therefore, the macro crack observed in the experiment may be comprised of a macro shear crack and a mass of small tensile cracks. For the case of α = 45°, both the tensile wing cracks and the shear secondary cracks can grow sufficiently. An anti-wing crack was produced at the upper tip of the pre-existing crack, which is in strong agreement with the experimental phenomenon. The numerical results indicate that the anti-wing crack is a type of tensile crack that splits the model. For α = 60°, the tensile wing cracks grew more sufficiently than the case of α = 45°, while the shear secondary cracks grew less sufficiently. In the upper-right region of the model, the secondary crack connected to the wing crack and changed the propagation direction of the macro fracture (from the crack tip to the top right corner of sample).

Figure 4(c) shows the failure of the EDEM models with the Griffith failure criterion. The Griffith failure criterion only considers the fracture of brittle material due to the concentrated tensile stress around the pre-existing crack. Therefore, only tensile wing cracks were generated in these numerical models. These wing cracks initiated from the crack tips, grew in the direction perpendicular to the initial crack plane, and then changed to the direction of the maximum compressive stress. In addition to the tensile cracks, two tensile failure regions can be clearly confirmed, which are located at both sides of the pre-existing crack. The EDEM model with the Griffith failure criterion is capable of simulating the splitting failure of brittle materials and strongly agrees with the theoretical results predicted by the micromechanical models [6,7,29] and certain experimental results for rock materials with high brittleness [6,7]. For some rock-like materials with low and moderate brittleness (e.g., the gypsum material used in this study), the Griffith failure criterion cannot represent the shear failure of materials, which is an important mechanism for evaluating the overall material failure. Therefore, the Griffith criterion may be incapable of providing a comprehensive understanding of the failure of rock masses under compression.

Sample with two parallel cracks

The sample containing two parallel cracks was tested to investigate the crack coalescence of pre-existing cracks. Figure 6 shows the failure statuses of the sample obtained from this experiment and the EDEM simulations. The experimental result (see Fig. 6(a)) reveals that the overall fracture of the sample resulted from the combined propagation of the initial cracks. Initially, two pre-existing cracks propagated separately with the growth of wing cracks. The upward-propagating wing crack from the lower initial crack and the downward-propagating wing crack from the upper initial crack penetrated the rock bridge zone gradually until finally the two pre-existing cracks were connected. Simultaneously, the secondary cracks propagated along the direction quasi-coplanar to the pre-existing crack until they reached the sample boundary.

A similar failure pattern was obtained from the EDEM model with the tensile-shear failure criterion (see Fig. 6(b)). The numerical results indicate that the combined propagation of tensile wing cracks and shear secondary cracks led to the coalescence of pre-existing cracks, which can also be observed in the experimental phenomenon. The shear secondary cracks made a major contribution to the model failure.

For the EDEM model with the Griffith failure criterion, only the tensile wing cracks were generated, which caused model failure (Fig. 6(c)). Similar to the cases of single cracks, the regions of tensile failure clearly conformed around the pre-existing cracks. In the rock bridge zone, the rock-like material was severely damaged due to the interaction of tensile failure regions.

Figure 7 shows the mechanical properties of the sample containing two parallel cracks, which were obtained from the experiment and the EDEM simulations. Samples with two cracks had a lower strength and elastic modulus than the samples with single cracks, due to the coalescence and interaction of pre-existing cracks. The EDEM model with the tensile-shear failure criterion has strong capabilities for representing the mechanical behavior of the fractured samples. Relatively, the EDEM model with the Griffith criterion overestimates the strength of the sample and thus cannot simulate the shear failure of the sample.

Conclusions

In this study, the expanded distinct element method (EDEM), developed with UDEC code, was used to investigate the crack propagation and coalescence in rock-like materials under uniaxial compression. The tensile-shear failure criterion and the Griffith failure criterion were implanted into the EDEM to determine the initiation and propagation of pre-existing cracks, respectively. To verify the validity of the EDEM, uniaxial compression tests on gypsum samples with pre-existing cracks were performed.

Through comparing the experimental and numerical results, the EDEM model with the tensile-shear failure criterion showed strong capabilities for modeling the growth and coalescence of pre-existing cracks, and the modeled results were in close agreement with the failure and mechanical properties of samples obtained from experiments. Therefore, EDEM is an effective method for investigating the mechanical and failure behaviors of rock masses.

The EDEM model with the Griffith failure criterion can only simulate the splitting failure of samples in the form of tensile wing cracks, corresponding well to the theoretical results described by certain micromechanical models. However, this model is incapable of representing the shear failure of samples and cannot provide a comprehensive explanation for the overall failure of rock masses.

The experimental and numerical results indicate that for samples with single cracks, the growth of tensile wing cracks and shear secondary cracks leads to the overall failure. The inclination angle of the pre-existing crack has an obvious effect on the failure pattern of the sample as follows: for a low-inclination crack (α = 30°), the growth of wing cracks is constrained by the secondary crack; for a moderate-inclination crack (α = 45°), both the wing cracks and secondary cracks can grow sufficiently, and for a high-inclination crack (α = 60°), the tensile wing cracks are a major contributor to the overall failure. For a sample with two cracks, the coalescence of pre-existing cracks is induced by the growth of wing cracks in the rock bridge zone, and the continuous propagation of secondary cracks causes the fracture of the sample.

This study verified the validity of the EDEM by simulating the cracking process of rock masses and investigated the growth patterns of cracks under different failure criteria. Future studies will be performed to investigate the coalescence mechanism of multiple cracks, considering the influences of the crack’s geometric characteristics as well as the mechanical properties of the rock and the cracks on the growth and coalescence patterns of cracks.

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