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Crack propagation from a filled flaw in rocks considering the infill influences
Journal of Applied Geophysics 152 (2018) 137–149
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Crack propagation from a filled flaw in rocks considering the infill influences Xu Chang a,b, Yan Deng b, Zhenhua Li c,⁎, Shuren Wang a, C.A. Tang d a
Opening Laboratory for Deep Mine Construction, Henan Polytechnic University, Jiaozuo, China School of Civil Engineering, Henan Polytechnic University, Jiaozuo, China School of Energy Science and Engineering, Henan Polytechnic University, Jiaozuo, China d Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, China b c
a r t i c l e
i n f o
Article history: Received 18 March 2017 Received in revised form 1 February 2018 Accepted 16 March 2018 Available online 22 March 2018 Keywords: Crack propagation Filled flaw Infill strength Critical value
a b s t r a c t This study presents a numerical and experimental study of the cracking behaviour of rock specimen containing a single filled flaw under compression. The primary aim is to investigate the influences of infill on crack patterns, load-displacement response and specimen strength. The numerical code RFPA2D (Rock Failure Process Analysis) featured by the capability of modeling heterogeneous materials is employed to develop the numerical model, which is further calibrated by physical tests. The results indicate that there exists a critical infill strength which controls crack patterns for a given flaw inclination angle. For case of infill strength lower than the critical value, the secondary or anti-cracks are disappeared by increasing the infill strength. If the infill strength is greater than the critical value, the filled flaw has little influence on the cracking path and the specimen fails by an inclined crack, as if there is no flaw. The load-displacement responses show specimen stiffness increases by increasing infill strength until the infill strength reaches its critical value. The specimen strength increases by increasing the infill strength and almost keeps constant as the infill strength exceeds its critical value. © 2017 Elsevier B.V. All rights reserved.
1. Introduction It is well known that rock mass contains many flaws at various scales resulting from a variety of geological processes (Ali and Jakobsen, 2011; Wang et al., 2014; Behraftar et al., 2017; Wang et al., 2016). New cracks may initiate at or near the tips of these existing flaws and propagate in various directions under different stress state, sometimes coalescing with other cracks. Therefore, these existing flaws play critical roles in the overall behaviour of the rocks. Considerable experiments, analytical studies and numerical modeling have been conducted to investigate the cracking behaviour of rocks with existing flaws, including crack initiation, propagation and coalescence in past decades. Different artificial rock-like materials or natural rocks have been adopted to observe the cracking process. Hoek and Bieniawski (1965) early used glass materials to investigate the crack behaviour of brittle materials. Since the gypsum material can provide rapid prototyping during specimen preparation, the modeled gypsum has become the most popular method to observe the cracking process of rocks (Shen and Stephansson, 1995; Bobet, 2000; Sagong and Bobet, 2002). Cement mortar is other commonly adopted artificial material (Mughieda and Alzo'ubi, 2004; Wong and Chau, 1998). The cracking processes of natural rocks, such as granite (Miller and Einstein, 2008), ⁎ Corresponding author. E-mail address: [email protected] (Z. Li).
https://doi.org/10.1016/j.jappgeo.2018.03.018 0926-9851/© 2017 Elsevier B.V. All rights reserved.
marble (Jiefan et al., 1990; Li et al., 2005; Wang et al., 1987), etc., have been also extensively investigated. These experimental studies indicated that the cracks initiation from the existing flaws can be generally classified into two types: wing (or primary) cracks and secondary cracks. The wing cracks from tensile stress initiated at the tips of the flaws and propagated along a curvilinear path that eventually in the loading direction. However, the secondary cracks from shear stress emanated from the flaw tips and propagated in a different direction from the wing cracks. A great number of numerical simulations have also been performed to describe the crack process of rocks with existing flaws. The finite element method (FEM) has been widely adopted to study crack propagation problem. However, two issues are always encountered for the traditional FEM: remeshing and mesh distortion (Bobet and Einstein, 1998). Therefore, the numerical manifold method (NMM) (Wu and Wong, 2012) and extended finite element method (XFEM) (Belytschko and Black, 1999) are further developed. The NMM is a combination of the FE method and the discontinuous deformation analysis (DDA). It offers a unified framework for both continuous and discontinuous problems. The XFEM has proved to be an effective approach to overcome the difficulties with remeshing of FEM. The discrete element method (DEM) (Damjanac and Chundall, 2016) has proven to be another efficient modeling approach for crack propagation rocks. As a result, many recent works based on the DEM have been conducted to study cracking behaviour of rocks (Wang and Mora, 2008a, 2008b; Duriez et al., 2016). A numerical simulation code,
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RFPA2D (Rock Failure Process Analysis), was developed to investigate cracking process of rocks (Tang, 1997). It has successfully simulated the global failure of a rock specimen as well as local cracking at the flaw tips (Tang et al., 2001; Xu et al., 2013). The hybrid finite-discrete element method (FDEM) has also suggested as powerful tools to model the cracking behaviour of rocks, for example, ELFEN (Rockfield Software Ltd., 2004) and Y-Geo (Munjiza, 2004; Mahabadi et al., 2012). It should be noted that these existing studies on cracking behaviour from pre-assigned flaws are mostly focused on the specimens with open flaws. In fact, many natural discontinuities are often filled with finegrained materials. The infill can be naturally formed as a result of weathering or joint shearing. It can be also manually filled during the construction process such as grouting and shotcreting with the attempt to strengthen the cracked rocks. The infill is considered to reduce the stress concentration near the existing flaws, which in turn plays a critical role in the crack propagation. Pereira (1997) presented an experimental investigation of shear failure of specimens containing crack filled with sands. Papaliangas et al. (1993) also carried out a series of shear test to investigate the shear behaviour of infilled rock cracks. Indraratna et al. (2014) conducted a series of test to investigate the influences of crack infill on rock shear strength. Their results indicated that the specimen shear strength increased as decreasing the degree of saturation. Zhang et al. (2012) pointed that the crack infill had an important influence on the overall performances of rocks. These cited studies about crack propagation have led to a clear understanding of the cracking behaviour of rock materials under compression. However, the lack of attention to infill influences has limited the further understanding of the fracture mechanics of rocks. In the present contribution, the influences of infill on the crack propagation are therefore investigated based on the numerical code RFPA2D, which has been employed successfully to investigate the progressive rock failure. This study is organized in the following way. In Section 2, the numerical code is briefly introduced and the numerical models containing preexisting flaws are set up. In Section 3, the numerical model is validated by physical test and the numerical results about the crack initiation and propagation from a single pre-existing filled flaw are presented. A parametric study, including the flaw inclination angle and infill material parameters, is also conducted. 2. Basal principles of RFPA The numerical code, RFPA has been fully documented by Tang (1997). Therefore, the basal principles are briefly outlined here.
where Ed and E0 are elastic moduli for damaged and undamaged element, respectively. d is the damage variable. The evolution of the damage variable is also needed. If the element fails by tension, the maximum tensile strain criterion is used and the damage variable can be determined by:
d¼
8 > <
0 σ tr 1− εE > 0 : 1
εbεt0 εt0 ≤εbεtu ε ≥εtu
ð3Þ
where σtr is the residual tensile strength; εt0 and εtu are the strains for elastic limit (damage initiation) and ultimate limit (element completely damaged). If the element fails by shear, the accordingly damage variable can be described as: ( d¼
0 σ cr 1− εE0
ε ≤εc0 εNεc0
ð4Þ
It is assumed that the element would be damaged if the equivalent tensile strain ε reaches εt0. Therefore, the evolution damage variable in multi-axial stress states can be easily obtained by substituting the strain, ε in Eqs. (3) and (4) with the equivalent strain, ε, which is described as follow: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε ¼ − h−ε1 i2 þ h−ε2 i2 þ h−ε3 i2
ð5Þ
where ε1 ε2 and ε3 are three principal strains, and 〈〉 is a function defined as: hxi ¼
xx ≥ 0 0x b 0
ð6Þ
2.3. Crack formation No special element is adopted for the crack. The simulation of the crack initiation and propagation is just as the smeared crack model. The crack nucleation is simulated through complete damage of element while crack propagation is obtained by coalescence of multiple damaged elements. 3. Crack propagation from a filled flaw
2.1. Inhomogeneity assumption 3.1. Model set-up The numerical specimens are meshed by many elements with the same size, and the mechanical properties of these elements are statistically distributed by a Weibull function:
where v is the element property (elastic modulus or strength); v0 is the average of element property; r is the shape of the Weibull function and named as heterogeneity index.
As presented in Fig. 1, the numerical model containing a single flaw is set-up. The specimen has the dimensions of 60 mm (H) × 120 mm (L) and the flaw is located at the center of the specimen. The flaw has a length (2B) of 10 mm and a thickness (t) of 2 mm. The flaw inclination angle is denoted as β, which ranges from 0° to 90°. A mesh for the numerical model is also indicated in Fig. 1. The four-node isoparametric quadrilateral element with size of 1 × 1 mm is used for the rock and infill. The loading is applied through imposed displacement on the top edge.
2.2. Damage mechanics
3.2. Model validation
The stress-strain response of an element is linear elastic before damage occurs, and then followed by a softening branch. Two types of criterions are adopted as the damage thresholds: the Mohr-Coulomb criterion and maximum tensile stress criterion. The elastic modulus for a damaged element can be determined by Tang (1997).
3.2.1. Physical tests As an attempt to validate the numerical model, physical tests are conducted. It is difficult to obtain adequate rock specimens containing filled and unfilled cracks. Therefore, the most common artificial material, cement mortar is adopted in this paper. The silica-based sand with particle size less than 0.5 mm and Poland cement with strength level of 42.5 are used. The rubber powder is also added to the mixes
pðvÞ ¼
r r v r−1 v exp − v0 v0 v0
Ed ¼ ð1−dÞE0
ð1Þ
ð2Þ
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Fig. 2. Test specimens.
cracks follow. Following Duriez et al. (2016), these cracks are classified as wing cracks, secondary cracks and anti-cracks. These typical cracks are sketched in Fig. 5. To further verify the numerical model, the analytical solution is also adopted in this section. The analytical solution has been fully documented by many researchers (Erdogan and Sih, 1963; Horii and Nemat-Nasser, 1985; Lehner and Kachanov, 1996; Fan, 2003) and therefore briefly described here. For a crack shown in Fig. 6(a), the stress intensity factors can be described as:
Fig. 1. Model for specimens with unfilled and filled flaws.
to control the brittleness of the cement mortar. More details about the test specimen are summarized in Table 1. During the specimen preparation process, the cement mortar for the inclusion material was firstly casted in the mold with the dimension of 60 × 120 × 30 mm. Then the specimens were placed in a 28 °C room for curing for 28 days after removal from the mold. In the second stage, the infill materials were filled. Also, after curing for 28 days, the test specimens were polished by hand. Fig. 2 gives a general view of the test specimens. All specimens were tested under a universal testing machine with a capacity of 1500 kN. The applied load was controlled and measured by the electronic load transducer. The load was applied in small increment of 0.5 kN until failure and the strain readings and the deflection measurements were recorded during the whole loading process. As stated before that the randomness of the rocks is considered in the numerical model by the heterogeneity index r. Therefore, selection of the parameter, r, is a key task for accurate modeling the rock behaviour. The influences of the parameter r on the mechanical performances of specimen with one unfilled flaw are firstly discussed. The inclination angle is 30o. As presented in Fig. 3, the peak stress and the corresponding strain increase by increasing the value of r. It can be seen that the numerical stress-strain response agrees well with the test one for case of r = 3.3. It can be further obtained that the numerical fracture patterns and stress-strain responses are in close agreement with the test results, as presented in Fig. 4. Therefore, r = 3.3 is adopted in modeling the rock behaviour.
pffiffiffiffiffiffi KІ ¼ σ πB
ð7Þ
pffiffiffiffiffiffi KІІ ¼ τ πB
ð8Þ
where σ and τ are far-field normal stress and transverse compressive stress, respectively. 2B is the crack length. The normal stress σ can be described as: σ ¼
1 pffiffiffiffiffiffiffiffi σ T ρ=a−σ N 2
σN ¼
1 ½ðσ V þ σ H Þ þ ðσ V −σ H Þ cos2β 2
ð10Þ
σT ¼
1 ½ðσ V þ σ H Þ þ ðσ V −σ H Þ cos2β 2
ð11Þ
ð9Þ
where ρ is the radius of the crack tip. σV and σH are the stresses in vertical and horizontal directions, respectively.
3.2.2. Analytical verification Many experimental and numerical studies indicate that the cracks initiate and propagate in a certain sequence under compressive conditions. The wing cracks always firstly initiate and then the secondary
Table 1 Parameters for test specimens in this study. Parameters
Rock-like materials
Infill I
Infill II
Infill III
Elastic modulus (Gpa) Compressive strength (Mpa) Tensile strength Poisson ratio Density (g/cm3)
24.0 68.1 6.1 0.26 2.3
6.0 6.5 0.5 0.20 1.31
8.2 9.4 0.8 0.24 1.44
9.0 12.9 1.1 0.22 1.60
Fig. 3. Influences of index r on the stress-strain response of rock containing one unfilled flaw (β = 30°).
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Fig. 4. Comparison of cracking patterns and stress-strain responses between test and simulation.
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According to ∂KІ(θ)/∂θ = 0, the relationship between KІ, KІІ and θ0 can be obtained as (Erdogan and Sih, 1963; Fan, 2003): K І sinθ0 þ K ІІ ð3 cosθ0 −1Þ ¼ 0
ð14Þ
where θ0 is the wing crack initiation angle. According to Eqs. (7)–(14), the wing crack initiation angle can be obtained. Based on the physical tests, the radius of the crack tip ρ is 0.1 mm and the length of the crack (2B) is 10 mm. Other materials for the analytical solution are same as those listed in Table 1. Fig. 6(b) presents the numerical wing crack initiation angles compared with the analytical ones. It is indicated that the numerical wing crack initiation angles agree well with the analytical ones. As stated by Indraratna et al. (2014), different infill materials with various properties could be found in nature. To further investigate the infill material on the crack propagation, an infill factor related to the infill strength is defined as follows:
Fig. 5. Sketch for crack types.
The transverse compressive stress, τ can be described as: 1 τ ¼ ðσ V −σ H Þ sin2β 2
ð12Þ
For a composite mode crack shown in Fig. 6(a), the tensile stress in hoop direction can be described as:
σθ
θ cos 2 ¼ pffiffiffiffiffiffiffiffi ½K І ð1 þ cosθÞ−3K ІІ sinθ 2 2πr
s¼
fi fw
ð15Þ
where fi is the infill strength; unfilled flaw is adopted for fi = 0; fw is the rock strength. 3.3. Crack propagation behaviour
ð13Þ
Fig. 6. Comparison of wing crack initiation angles between analytical solution and simulation (a) sketch for unfilled crack; (b) comparison of wing crack initiation angles.
In the following discussion, the default material properties for rock are same as those listed in Table 1 unless otherwise specified. The specimen containing a single flaw with various inclination angles and infill strengths is modeled. And the crack propagation processes are presented in Figs. 7–11. For β = 0°, the propagating processes for the filled crack with different infills are presented in Fig. 7. For the unfilled crack (Fig. 7a), two cracks initiate near middle of the flaw and propagate towards to the top and bottom sides of the specimen in the loading direction, which is very similar to the existing results (Jiefan et al., 1990; Wang et al., 1987). Note that these cracks in this paper do not exactly initiate the middle of the flaw just due to randomly distributed elements in the numerical model. Two cracks initiate at the flaws tips successively and propagate in the loading direction by further increasing the external load. As the strength factor increases to 0.1 (as shown in Fig. 7b), the cracking process is similar to that of s = 0.0. Two cracks initiate near the middle of the flaw and then propagate in the loading direction and then two cracks initiate at the flaw tips and propagate in the loading direction. As the infill factor increases to s = 0.2, only two cracks initiate near the middle of the filled flaw and no crack initiates near or at the filled flaw tips (As shown in Fig. 7c). As the infill factor further reaches s = 0.5, the filled flaw seems to have little effect on cracks that no local cracks can be found near the filled flaw (As shown in Fig. 7d). The crack initiates at a random location the specimen fails by a dominated inclined crack, which is similar to the fracture mode of the specimen without any flaws. The cracking processes for specimens containing one crack with β = 30 ° are presented in Fig. 8. For the specimen with unfilled flaw (Fig. 8a), two wing cracks firstly initiate near the flaw tips and then propagate in the loading direction, and then followed by two secondary wing cracks. For case of s = 0.1 (Fig. 8b), secondary cracks can be observed after the formation of two wing cracks. As s further increases to 0.3, only wing crack can be observed (Fig. 8c). The filled flaw also has little effect on cracks as the value for s increases to 0.5. The crack initiates at a random location to form a dominated inclined crack, as shown in Fig. 8d. The crack initiation location and wing crack initiation angle (α) are two important indices for describing the fracture patterns. According to Fig. 8, all wing cracks initiate near the tips of the filled flaws when the infill factor is lower than 0.2. However, for case of s = 0.3 (Fig. 8c), only wing crack initiates near the middle of the flaw.
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Fig. 7. Cracking process for specimen containing a single flaw of β = 0° with various infill strengths.
According to the final fracture patterns in Fig. 8, the wing crack angle decreases obviously as the infill factor increases to s = 0.3. Therefore, it can be concluded that the increasing of infill factor reduces the wing crack angle until the infill factor reaches a critical value. For the unfilled flaw with β = 45°, the wing cracks firstly initiate near the flaw tips and then propagate in the loading direction, and then followed by anti-wing and secondary cracks (Fig. 9a). For infill factor between s = 0.1 (Fig. 9b) and s = 0.4 (Fig. 9c), the initiation and propagation of the wing cracks are similar to those from the unfilled flaw. However, the anti-wing cracks are not observed, which is in conformance with the conclusion by Zhang et al. (2017). It can be further
seen that the wing crack angle obviously decreases as the infill factor increases to s = 0.5. Also, as the infill factor, s increases 0.7 (Fig. 9d), the filled flaw has little effect on cracking behaviour of the specimen, as if there is no flaw. Fig. 10 gives the cracking processes for β = 60°. For the unfilled flaw, the wing cracks and the secondary cracks can be observed (Fig. 10a). However, for the infill factor between s = 0.1 (Fig. 10b) and s = 0.5 (Fig. 10c), only wing cracks can initiate and propagate from the filled flaw. As the infill factor, s increases to 0.7, the filled flaw has little influence on the crack initiation and propagation (Fig. 10d). Fig. 11 shows the cracking process for β = 75°. It can be seen that the filled flaw has
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Fig. 8. Cracking process for specimen containing a single flaw of β = 30° with various infill strengths.
little effect on the crack initiation and propagation even at a low level of s = 0.1. 4. Critical infill strength Fig. 12 gives the plot of initiation angle of wing crack versus the flaw inclination angle. It can be seen that the initiation angle of wing crack decreases as increasing the flaw inclination angle. Zhang et al. (2017) pointed that initiation angles of wing crack for unfilled flaws are larger than those for filled flaws. However, the infill strength is not considered. In this paper, the influences of the infill strength on the initiation angle
of wing crack are presented in Fig. 13. According to Figs. 7–11, the wing crack may disappear if the infill strength exceeds the critical value. Therefore, only cases for infill strength lower than the critical value are included in Fig. 13. It can be further stated that the initiation angle of wing crack decreases as the infill factor increases at the same flaw inclination angle. This trend will stop as the infill strength increases to a certain value, after which the filled flaw has little effect on the cracking behaviour of the specimen. As mentioned above, there exists a critical value for the infill strength that controls the crack pattern of specimen containing a filled flaw. If the infill strength is less than the critical value, the filled flaw
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Fig. 9. Cracking process for specimen containing a single flaw of β = 45° with various infill strengths.
can greatly influence the crack pattern. If the infill strength is greater than the critical value, the filled flaw has little influence on the cracking path and the specimen fails in a similar pattern to that without any flaws. Also, the critical value for the infill strength is dependent on the flaw inclination angle. The critical infill strengths for each flaw angle are presented in Fig. 14. It is indicated that the critical infill strength increases as the flaw angle increases to about 50°, and after that the critical value decreases as further increasing the flaw angle.
5. Mechanical behaviour The typical stress-strain curves for specimens with unfilled and filled flaws under uniaxial compression are plotted in Fig. 15, where the flaw inclination angles are β = 0° (Fig. 15(a)), β = 30° (Fig. 15(b)), β = 60° (Fig. 15(c)) and β = 90° (Fig. 15(d)). It can be seen from Fig. 15 that the specimens with filled flaw have higher stiffness than the specimen with unfilled flaw before peak stress at
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Fig. 10. Cracking process for specimen containing a single flaw of β = 60° with various infill strengths.
a given flaw inclination angle. It is further indicated that the peak stress and the corresponding strain increase by increasing the infill strength. However, if the infill strength exceeds the critical value (s = 0.2 for β = 0°, s = 0.4 for β = 30°, s = 0.5 for β = 60° and s = 0.1 for β = 90°) the infill strengths seem to have little influence on the stress-strain curves. The differences in specimen stiffness and peak stress are very small after the critical value. This means that the flaw infill strength
controls not only the crack pattern of a specimen but also its mechanical behaviour. As an attempt to further explore the influences of infill strength on the mechanical behavior, the stress level at which the wing crack initiates is shown in Fig. 16, where the flaw inclination angle is β = 30. It can be seen that the wing crack initiates at stress level of 11.3 MPa for s = 0.0. For s = 0.1, the wing crack initiates at 23.1 MPa. As the infill
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Fig. 11. Cracking process for specimen containing a single flaw of β = 75° with various infill strengths.
strength. It is clear that the infill strength contributes more to specimen strength for a specimen with lower flaw inclination angle.
strength factor further increases to s = 0.4, the wing crack initiates at 37.6 MPa. Once the wing crack initiates, the specimen stiffness gradually decreases. As a result, the specimen with lower infill strength factor can reach peak stress earlier. It can be therefore concluded that the infill can delay the initiation of the wing crack if the infill strength is lower than the critical value. A higher infill strength factor can lead to a higher stress level at which the wing crack initiate. If the infill strength exceeds the critical value, the wing crack cannot be found. This means that the infill can prevent the initiation of the wing crack. The plot of specimen strength versus the infill strength for each flaw inclination angle is presented in Fig. 17. It can be seen from the figure that the strength for specimen containing a flaw increases as the infill strength increases. And the specimen strength keeps almost constant as the infill strength exceeds its critical value. For β = 0°, the specimen strength increases from 31 MPa to 59.5 MPa as the infill strength approaches its critical value (s = 0.3). The specimen strength increases from 44 MPa to 60 MPa for case of β = 90° as increasing the infill
The primary aim of this study is to investigate the influences of filled flaws on cracking behaviour of rock-like materials. Therefore, influences of the infill heterogeneity are discussed in detail. Two typical cases are discussed: β = 30° and β = 60°. According to the discussed mentioned above, the infill has little influence on the stress-strain response if the infill strength exceeds its critical value. Therefore, only infill strength lower than its critical value are included in this section. Fig. 18 gives the influence of heterogeneity index r for infill on stress-strain response at the inclination angle of β = 30°, where four values for r are adopted: r = 1.0, r = 3.0, r = 5.0, r = 7.0. It can be seen from Fig. 18 that the peak stress and the corresponding strain increase by increasing the value of r. The peak stress increases from 53 MPa to 60 MPa as r increases from 1.0
Fig. 12. Influences of flaw inclination angle on wing crack initiation angle.
Fig. 13. Influence of infill strength factor on wing crack initiation angles.
6. Influences of heterogeneity index r for infill
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Fig. 14. Critical infill strength factors for different flaw inclination angles. Fig. 16. Influences of infill strength factors on wing crack initiation (β = 30°).
to 7.0 for case of s = 0.1. However, for case of s = 0.4, the peak stress increases from 54 MPa to 67 MPa. For β = 60° (Fig. 19), the peak stress increases from 47 MPa to 52 MPa as r increases from 1.0 to 7.0 for case of s = 0.1. For case of s = 0.5, the peak stress increases from 50 MPa to 64 MPa. One can conclude that the mechanical response of the specimen containing a filled flaw is more sensitive to heterogeneity of the infill for case of higher infill factor.
7. Conclusions and discussion This study focuses on the influences of infill strength on the crack behaviour and mechanical performances of the rock specimen containing one filled flaw. The numerical code, RFPA (Tang, 1997) was used to model behaviour of rock specimen containing one flaw. The important
Fig. 15. Typical stress-strain response for specimen with various infill strength factors.
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Fig. 17. Evolution of specimen strength with various infill strength factors.
contribution in the current work is a thorough analysis of the effect of infill strength on the cracking behaviours of rocks. An extensive study based on the numerical model leads to the following conclusions: (1) There exists a critical value for the infill strength which controls the crack propagation and the crack pattern. If the infill strength is higher than critical value, the filled flaw has little influence on the crack propagation from the flaw. The critical infill strength is
Fig. 19. Influences of heterogeneity index for infill on the stress-strain response (β = 60°).
related to the flaw inclination angle and reaches its peak value at β = 50°. Further analysis indicates that the crack inclination angle decreases as the infill strength increases to the critical value. (2) The stress-strain response shows that the specimen with a filled flaw has higher stiffness than that with an unfilled flaw before peak stress. The peak stress and the corresponding strain increase as the infill strength increases. If the infill strength reaches its critical value, the infill strength has little influence on the stress-strain response. The infill can delay the initiation of wing crack and a higher infill strength can lead to a higher stress level at which wing crack initiates. (3) The flaw infill strength controls not only the crack pattern of a specimen but also its strength. The specimen strength increases as the infill strength increases. And the specimen strength keeps almost constant as the infill strength exceeds its critical value.
Fig. 18. Influences of heterogeneity index for infill on the stress-strain response (β = 30°).
In RFPA code, the heterogeneity index is introduced to reflect the random nature of rock-like materials by the method that mechanical properties of the elements are assumed to follow the statistical Weibull distribution. However, it is a very difficult work to select the heterogeneity index since there is no standard to determine it. In this paper, the adopted heterogeneity index is determined by limited test specimens and should be further validated by more specimens.
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Acknowledgements The work described in this paper was fully supported by National Natural Science Foundation of China (No. 51674100), Science and Technology Project of Henan Province (No. 162102210064), the Program for Science &Technology Innovation Talents Universities of Henan Province (Project No. 18HASTIT017) and the Fundamental Research Funds for the Universities of Henan Province (Project No. NSFRF140206). The authors also appreciate the comments from the reviewers. References Ali, A., Jakobsen, M., 2011. Seismic characterization of reservoirs with multiple fracture sets using velocity and attenuation anisotropy data. J. Appl. Geophys. 75, 590–620. Behraftar, S., Torres, S.A.G., Scheuermann, A., Williams, D.J., Marques, E.A.G., Avarzaman, H.J.A., 2017. Calibration methodology to obtain material parameters for the representation of fracture mechanics based on discrete element simulations. Comput. Geotech. 81, 31–38. Belytschko, T., Black, T., 1999. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45 (5), 601–620. Bobet, A., 2000. The initiation of secondary cracks in compression. Eng. Fract. Mech. 66, 187–219. Bobet, A., Einstein, H.H., 1998. Numerical Modeling of Fracture Coalescence in a Model Rock Mat. Damjanac, B., Chundall, P., 2016. Application of distinct element methods to simulation of hydraulic fracturing in naturally fractured reservoirs. Comput. Geotech. 71, 283–294 (Int. J. Fracture 92, 221-252). Duriez, J., Scholtès, L., Donzé, F.-V., 2016. Micromechanics of wing crack propagation for different flaw properties. Eng. Fract. Mech. 153, 378–398. Erdogan, F., Sih, G., 1963. On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 85 (4), 519–527. Fan, T., 2003. Fracture Theory Foundation. Science Press, Beijing. Hoek, E., Bieniawski, Z.T., 1965. Brittle fracture propagation in rock under compression. Int. J. Fract. 1, 137–155. Horii, H., Nemat-Nasser, S., 1985. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J. Geophys. Res. 90 (B4), 3105–3125. Indraratna, B., Premadasa, W., Brown, E.T., Gens, A., Heitor, A., 2014. Shear strength of rock joints influenced by compacted infill. Int. J. Rock Mech. Min. Sci. 70, 296–307. Jiefan, H., Ganglin, C., Yonghong, Z., Ren, W., 1990. An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics 175, 184–269. Lehner, F., Kachanov, M., 1996. On modeling of “winged” cracks forming under compression. Int. J. Fract. 77, 65–75. Li, Y.P., Chen, L.Z., Wang, Y.H., 2005. Experimental research on pre-cracked marble under compression. Int. J. Solids Struct. 42, 2505–2516. Mahabadi, O.K., Lisjak, A., Munjiza, A., Grasselli, G., 2012. Y-Geo: new combined finite-discrete element numerical code for geomechanical applications. Int. J. Geomech. 12, 676e88.
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Miller, J.T., Einstein, H.H., 2008. Crack coalescence tests on granite. Proceedings of 42nd US Rock Mechanics Symposium, ARMA 08-162, San Francisco. Mughieda, O., Alzo'ubi, A.K., 2004. Fracture mechanisms of offset rock joints-a laboratory investigation. Geotech. Geol. Eng. 22, 545–562. Munjiza, A., 2004. The Combined Finite-discrete Element Method. John Wiley & Sons Ltd, Chichester, UK. Papaliangas, T., Hencher, S.R., Lumsden, A.C., et al., 1993. The effect of friction fill thickness on the shear strength of rock discontinuities. Int. J. Rock Min. Sci. Geomech. Abstracts 30 (2), 81–91. Pereira, J.P., 1997. Rolling friction and shear behavior of rock discontinuities filled with sand. Int. J. Rock Mech. Min. Sci. 34 (3–4), 241–244. Rockfield Software Ltd, 2004. ELFEN 2D/3D Numerical Modelling Package. Rockfield Software Ltd., Swansea, UK. Sagong, M., Bobet, A., 2002. Coalescence of multiple flaws in a rock-model material in uniaxial compression. Int. J. Rock. Mech. Min. Sci. 39 (2), 229–241. Shen, B., Stephansson, O., 1995. Coalescence of fractures under shear stresses in experiments. J. Geophys. Res 100 (B4), 5975–5990. Tang, C.A., 1997. Numerical simulation of progressive rock failure and associated seismicity. Int. J. Rock Mech. Min. Sci. 34, 249–261. Tang, C.A., Lin, P., Wong, R.H.C., Chau, K.T., 2001. Analysis of crack coalescence in rock-like materials containing three flaws-part two: numerical approach. Int. J. Rock Mech. Min. Sci. 38, 925–939. Wang, Y., Mora, P., 2008a. Modeling wing crack extension: implications for the ingredients of discrete element model. Pure Appl. Geophys. 165 (3–4), 609–620. Wang, Y., Mora, P., 2008b. Modeling wing crack extension: implications for the ingredients of discrete element model. Pure Appl. Geophys. 165 (3–4), 609–620. Wang, R., Zhao, Y., Chen, Y., Yan, H., Yin, Y.Q., Yao, C.Y., Zhang, H., 1987. Experimental and finite simulation of X-shear fractures from a crack in marble. Tectonophysics 144, 141–150. Wang, L.L., Bornert, M., Héripré, E., Yang, D.S., Chanchole, S., 2014. Irreversible deformation and damage in argillaceous rocks induced by wetting/drying. J. Appl. Geophys. 107, 108–118. Wang, S.Y., Sloan, S.W., Sheng, D.C., Tang, C.A., 2016. 3D numerical analysis of crack propagation of heterogeneous notched rock under uniaxial tension. Tectonophysics 677, 45–67. Wong, R.H.C., Chau, K.T., 1998. Crack coalescence in a rock-like material containing two cracks. Int. J. Rock Mech. Min. Sci. 35, 147–164. Wu, Z., Wong, L.N.Y., 2012. Frictional crack initiation and propagation analysis using the numerical manifold method. Comput. Geotech. 39, 38–53. Xu, T., Ranjith, P.G., Wasantha, Zhao, J., et al., 2013. Influence of the geometry of partiallyspanning joints on mechanical properties of rock in uniaxial compression. Eng. Geol. 167, 134–147. Zhang, B., Li, S.C., Zhang, D.F., et al., 2012. Uniaxial compression mechanical propertytest, fracture and damage analysis of similar material of jointed rock mass with filled cracks. Rock Soil Mech. 33 (6), 1647–1652. Zhang, X.P., Zhang, Q., Wu, S.C., 2017. Acoustic emission characteristics of rock-like materials containing a single flaw under different compressive loading rates. Comput. Geotech. 83, 83–97.
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Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
Crack propagation from a filled flaw in rocks considering the infill influences Xu Chang a,b, Yan Deng b, Zhenhua Li c,⁎, Shuren Wang a, C.A. Tang d a
Opening Laboratory for Deep Mine Construction, Henan Polytechnic University, Jiaozuo, China School of Civil Engineering, Henan Polytechnic University, Jiaozuo, China School of Energy Science and Engineering, Henan Polytechnic University, Jiaozuo, China d Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, China b c
a r t i c l e
i n f o
Article history: Received 18 March 2017 Received in revised form 1 February 2018 Accepted 16 March 2018 Available online 22 March 2018 Keywords: Crack propagation Filled flaw Infill strength Critical value
a b s t r a c t This study presents a numerical and experimental study of the cracking behaviour of rock specimen containing a single filled flaw under compression. The primary aim is to investigate the influences of infill on crack patterns, load-displacement response and specimen strength. The numerical code RFPA2D (Rock Failure Process Analysis) featured by the capability of modeling heterogeneous materials is employed to develop the numerical model, which is further calibrated by physical tests. The results indicate that there exists a critical infill strength which controls crack patterns for a given flaw inclination angle. For case of infill strength lower than the critical value, the secondary or anti-cracks are disappeared by increasing the infill strength. If the infill strength is greater than the critical value, the filled flaw has little influence on the cracking path and the specimen fails by an inclined crack, as if there is no flaw. The load-displacement responses show specimen stiffness increases by increasing infill strength until the infill strength reaches its critical value. The specimen strength increases by increasing the infill strength and almost keeps constant as the infill strength exceeds its critical value. © 2017 Elsevier B.V. All rights reserved.
1. Introduction It is well known that rock mass contains many flaws at various scales resulting from a variety of geological processes (Ali and Jakobsen, 2011; Wang et al., 2014; Behraftar et al., 2017; Wang et al., 2016). New cracks may initiate at or near the tips of these existing flaws and propagate in various directions under different stress state, sometimes coalescing with other cracks. Therefore, these existing flaws play critical roles in the overall behaviour of the rocks. Considerable experiments, analytical studies and numerical modeling have been conducted to investigate the cracking behaviour of rocks with existing flaws, including crack initiation, propagation and coalescence in past decades. Different artificial rock-like materials or natural rocks have been adopted to observe the cracking process. Hoek and Bieniawski (1965) early used glass materials to investigate the crack behaviour of brittle materials. Since the gypsum material can provide rapid prototyping during specimen preparation, the modeled gypsum has become the most popular method to observe the cracking process of rocks (Shen and Stephansson, 1995; Bobet, 2000; Sagong and Bobet, 2002). Cement mortar is other commonly adopted artificial material (Mughieda and Alzo'ubi, 2004; Wong and Chau, 1998). The cracking processes of natural rocks, such as granite (Miller and Einstein, 2008), ⁎ Corresponding author. E-mail address: [email protected] (Z. Li).
https://doi.org/10.1016/j.jappgeo.2018.03.018 0926-9851/© 2017 Elsevier B.V. All rights reserved.
marble (Jiefan et al., 1990; Li et al., 2005; Wang et al., 1987), etc., have been also extensively investigated. These experimental studies indicated that the cracks initiation from the existing flaws can be generally classified into two types: wing (or primary) cracks and secondary cracks. The wing cracks from tensile stress initiated at the tips of the flaws and propagated along a curvilinear path that eventually in the loading direction. However, the secondary cracks from shear stress emanated from the flaw tips and propagated in a different direction from the wing cracks. A great number of numerical simulations have also been performed to describe the crack process of rocks with existing flaws. The finite element method (FEM) has been widely adopted to study crack propagation problem. However, two issues are always encountered for the traditional FEM: remeshing and mesh distortion (Bobet and Einstein, 1998). Therefore, the numerical manifold method (NMM) (Wu and Wong, 2012) and extended finite element method (XFEM) (Belytschko and Black, 1999) are further developed. The NMM is a combination of the FE method and the discontinuous deformation analysis (DDA). It offers a unified framework for both continuous and discontinuous problems. The XFEM has proved to be an effective approach to overcome the difficulties with remeshing of FEM. The discrete element method (DEM) (Damjanac and Chundall, 2016) has proven to be another efficient modeling approach for crack propagation rocks. As a result, many recent works based on the DEM have been conducted to study cracking behaviour of rocks (Wang and Mora, 2008a, 2008b; Duriez et al., 2016). A numerical simulation code,
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RFPA2D (Rock Failure Process Analysis), was developed to investigate cracking process of rocks (Tang, 1997). It has successfully simulated the global failure of a rock specimen as well as local cracking at the flaw tips (Tang et al., 2001; Xu et al., 2013). The hybrid finite-discrete element method (FDEM) has also suggested as powerful tools to model the cracking behaviour of rocks, for example, ELFEN (Rockfield Software Ltd., 2004) and Y-Geo (Munjiza, 2004; Mahabadi et al., 2012). It should be noted that these existing studies on cracking behaviour from pre-assigned flaws are mostly focused on the specimens with open flaws. In fact, many natural discontinuities are often filled with finegrained materials. The infill can be naturally formed as a result of weathering or joint shearing. It can be also manually filled during the construction process such as grouting and shotcreting with the attempt to strengthen the cracked rocks. The infill is considered to reduce the stress concentration near the existing flaws, which in turn plays a critical role in the crack propagation. Pereira (1997) presented an experimental investigation of shear failure of specimens containing crack filled with sands. Papaliangas et al. (1993) also carried out a series of shear test to investigate the shear behaviour of infilled rock cracks. Indraratna et al. (2014) conducted a series of test to investigate the influences of crack infill on rock shear strength. Their results indicated that the specimen shear strength increased as decreasing the degree of saturation. Zhang et al. (2012) pointed that the crack infill had an important influence on the overall performances of rocks. These cited studies about crack propagation have led to a clear understanding of the cracking behaviour of rock materials under compression. However, the lack of attention to infill influences has limited the further understanding of the fracture mechanics of rocks. In the present contribution, the influences of infill on the crack propagation are therefore investigated based on the numerical code RFPA2D, which has been employed successfully to investigate the progressive rock failure. This study is organized in the following way. In Section 2, the numerical code is briefly introduced and the numerical models containing preexisting flaws are set up. In Section 3, the numerical model is validated by physical test and the numerical results about the crack initiation and propagation from a single pre-existing filled flaw are presented. A parametric study, including the flaw inclination angle and infill material parameters, is also conducted. 2. Basal principles of RFPA The numerical code, RFPA has been fully documented by Tang (1997). Therefore, the basal principles are briefly outlined here.
where Ed and E0 are elastic moduli for damaged and undamaged element, respectively. d is the damage variable. The evolution of the damage variable is also needed. If the element fails by tension, the maximum tensile strain criterion is used and the damage variable can be determined by:
d¼
8 > <
0 σ tr 1− εE > 0 : 1
εbεt0 εt0 ≤εbεtu ε ≥εtu
ð3Þ
where σtr is the residual tensile strength; εt0 and εtu are the strains for elastic limit (damage initiation) and ultimate limit (element completely damaged). If the element fails by shear, the accordingly damage variable can be described as: ( d¼
0 σ cr 1− εE0
ε ≤εc0 εNεc0
ð4Þ
It is assumed that the element would be damaged if the equivalent tensile strain ε reaches εt0. Therefore, the evolution damage variable in multi-axial stress states can be easily obtained by substituting the strain, ε in Eqs. (3) and (4) with the equivalent strain, ε, which is described as follow: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε ¼ − h−ε1 i2 þ h−ε2 i2 þ h−ε3 i2
ð5Þ
where ε1 ε2 and ε3 are three principal strains, and 〈〉 is a function defined as: hxi ¼
xx ≥ 0 0x b 0
ð6Þ
2.3. Crack formation No special element is adopted for the crack. The simulation of the crack initiation and propagation is just as the smeared crack model. The crack nucleation is simulated through complete damage of element while crack propagation is obtained by coalescence of multiple damaged elements. 3. Crack propagation from a filled flaw
2.1. Inhomogeneity assumption 3.1. Model set-up The numerical specimens are meshed by many elements with the same size, and the mechanical properties of these elements are statistically distributed by a Weibull function:
where v is the element property (elastic modulus or strength); v0 is the average of element property; r is the shape of the Weibull function and named as heterogeneity index.
As presented in Fig. 1, the numerical model containing a single flaw is set-up. The specimen has the dimensions of 60 mm (H) × 120 mm (L) and the flaw is located at the center of the specimen. The flaw has a length (2B) of 10 mm and a thickness (t) of 2 mm. The flaw inclination angle is denoted as β, which ranges from 0° to 90°. A mesh for the numerical model is also indicated in Fig. 1. The four-node isoparametric quadrilateral element with size of 1 × 1 mm is used for the rock and infill. The loading is applied through imposed displacement on the top edge.
2.2. Damage mechanics
3.2. Model validation
The stress-strain response of an element is linear elastic before damage occurs, and then followed by a softening branch. Two types of criterions are adopted as the damage thresholds: the Mohr-Coulomb criterion and maximum tensile stress criterion. The elastic modulus for a damaged element can be determined by Tang (1997).
3.2.1. Physical tests As an attempt to validate the numerical model, physical tests are conducted. It is difficult to obtain adequate rock specimens containing filled and unfilled cracks. Therefore, the most common artificial material, cement mortar is adopted in this paper. The silica-based sand with particle size less than 0.5 mm and Poland cement with strength level of 42.5 are used. The rubber powder is also added to the mixes
pðvÞ ¼
r r v r−1 v exp − v0 v0 v0
Ed ¼ ð1−dÞE0
ð1Þ
ð2Þ
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Fig. 2. Test specimens.
cracks follow. Following Duriez et al. (2016), these cracks are classified as wing cracks, secondary cracks and anti-cracks. These typical cracks are sketched in Fig. 5. To further verify the numerical model, the analytical solution is also adopted in this section. The analytical solution has been fully documented by many researchers (Erdogan and Sih, 1963; Horii and Nemat-Nasser, 1985; Lehner and Kachanov, 1996; Fan, 2003) and therefore briefly described here. For a crack shown in Fig. 6(a), the stress intensity factors can be described as:
Fig. 1. Model for specimens with unfilled and filled flaws.
to control the brittleness of the cement mortar. More details about the test specimen are summarized in Table 1. During the specimen preparation process, the cement mortar for the inclusion material was firstly casted in the mold with the dimension of 60 × 120 × 30 mm. Then the specimens were placed in a 28 °C room for curing for 28 days after removal from the mold. In the second stage, the infill materials were filled. Also, after curing for 28 days, the test specimens were polished by hand. Fig. 2 gives a general view of the test specimens. All specimens were tested under a universal testing machine with a capacity of 1500 kN. The applied load was controlled and measured by the electronic load transducer. The load was applied in small increment of 0.5 kN until failure and the strain readings and the deflection measurements were recorded during the whole loading process. As stated before that the randomness of the rocks is considered in the numerical model by the heterogeneity index r. Therefore, selection of the parameter, r, is a key task for accurate modeling the rock behaviour. The influences of the parameter r on the mechanical performances of specimen with one unfilled flaw are firstly discussed. The inclination angle is 30o. As presented in Fig. 3, the peak stress and the corresponding strain increase by increasing the value of r. It can be seen that the numerical stress-strain response agrees well with the test one for case of r = 3.3. It can be further obtained that the numerical fracture patterns and stress-strain responses are in close agreement with the test results, as presented in Fig. 4. Therefore, r = 3.3 is adopted in modeling the rock behaviour.
pffiffiffiffiffiffi KІ ¼ σ πB
ð7Þ
pffiffiffiffiffiffi KІІ ¼ τ πB
ð8Þ
where σ and τ are far-field normal stress and transverse compressive stress, respectively. 2B is the crack length. The normal stress σ can be described as: σ ¼
1 pffiffiffiffiffiffiffiffi σ T ρ=a−σ N 2
σN ¼
1 ½ðσ V þ σ H Þ þ ðσ V −σ H Þ cos2β 2
ð10Þ
σT ¼
1 ½ðσ V þ σ H Þ þ ðσ V −σ H Þ cos2β 2
ð11Þ
ð9Þ
where ρ is the radius of the crack tip. σV and σH are the stresses in vertical and horizontal directions, respectively.
3.2.2. Analytical verification Many experimental and numerical studies indicate that the cracks initiate and propagate in a certain sequence under compressive conditions. The wing cracks always firstly initiate and then the secondary
Table 1 Parameters for test specimens in this study. Parameters
Rock-like materials
Infill I
Infill II
Infill III
Elastic modulus (Gpa) Compressive strength (Mpa) Tensile strength Poisson ratio Density (g/cm3)
24.0 68.1 6.1 0.26 2.3
6.0 6.5 0.5 0.20 1.31
8.2 9.4 0.8 0.24 1.44
9.0 12.9 1.1 0.22 1.60
Fig. 3. Influences of index r on the stress-strain response of rock containing one unfilled flaw (β = 30°).
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Fig. 4. Comparison of cracking patterns and stress-strain responses between test and simulation.
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According to ∂KІ(θ)/∂θ = 0, the relationship between KІ, KІІ and θ0 can be obtained as (Erdogan and Sih, 1963; Fan, 2003): K І sinθ0 þ K ІІ ð3 cosθ0 −1Þ ¼ 0
ð14Þ
where θ0 is the wing crack initiation angle. According to Eqs. (7)–(14), the wing crack initiation angle can be obtained. Based on the physical tests, the radius of the crack tip ρ is 0.1 mm and the length of the crack (2B) is 10 mm. Other materials for the analytical solution are same as those listed in Table 1. Fig. 6(b) presents the numerical wing crack initiation angles compared with the analytical ones. It is indicated that the numerical wing crack initiation angles agree well with the analytical ones. As stated by Indraratna et al. (2014), different infill materials with various properties could be found in nature. To further investigate the infill material on the crack propagation, an infill factor related to the infill strength is defined as follows:
Fig. 5. Sketch for crack types.
The transverse compressive stress, τ can be described as: 1 τ ¼ ðσ V −σ H Þ sin2β 2
ð12Þ
For a composite mode crack shown in Fig. 6(a), the tensile stress in hoop direction can be described as:
σθ
θ cos 2 ¼ pffiffiffiffiffiffiffiffi ½K І ð1 þ cosθÞ−3K ІІ sinθ 2 2πr
s¼
fi fw
ð15Þ
where fi is the infill strength; unfilled flaw is adopted for fi = 0; fw is the rock strength. 3.3. Crack propagation behaviour
ð13Þ
Fig. 6. Comparison of wing crack initiation angles between analytical solution and simulation (a) sketch for unfilled crack; (b) comparison of wing crack initiation angles.
In the following discussion, the default material properties for rock are same as those listed in Table 1 unless otherwise specified. The specimen containing a single flaw with various inclination angles and infill strengths is modeled. And the crack propagation processes are presented in Figs. 7–11. For β = 0°, the propagating processes for the filled crack with different infills are presented in Fig. 7. For the unfilled crack (Fig. 7a), two cracks initiate near middle of the flaw and propagate towards to the top and bottom sides of the specimen in the loading direction, which is very similar to the existing results (Jiefan et al., 1990; Wang et al., 1987). Note that these cracks in this paper do not exactly initiate the middle of the flaw just due to randomly distributed elements in the numerical model. Two cracks initiate at the flaws tips successively and propagate in the loading direction by further increasing the external load. As the strength factor increases to 0.1 (as shown in Fig. 7b), the cracking process is similar to that of s = 0.0. Two cracks initiate near the middle of the flaw and then propagate in the loading direction and then two cracks initiate at the flaw tips and propagate in the loading direction. As the infill factor increases to s = 0.2, only two cracks initiate near the middle of the filled flaw and no crack initiates near or at the filled flaw tips (As shown in Fig. 7c). As the infill factor further reaches s = 0.5, the filled flaw seems to have little effect on cracks that no local cracks can be found near the filled flaw (As shown in Fig. 7d). The crack initiates at a random location the specimen fails by a dominated inclined crack, which is similar to the fracture mode of the specimen without any flaws. The cracking processes for specimens containing one crack with β = 30 ° are presented in Fig. 8. For the specimen with unfilled flaw (Fig. 8a), two wing cracks firstly initiate near the flaw tips and then propagate in the loading direction, and then followed by two secondary wing cracks. For case of s = 0.1 (Fig. 8b), secondary cracks can be observed after the formation of two wing cracks. As s further increases to 0.3, only wing crack can be observed (Fig. 8c). The filled flaw also has little effect on cracks as the value for s increases to 0.5. The crack initiates at a random location to form a dominated inclined crack, as shown in Fig. 8d. The crack initiation location and wing crack initiation angle (α) are two important indices for describing the fracture patterns. According to Fig. 8, all wing cracks initiate near the tips of the filled flaws when the infill factor is lower than 0.2. However, for case of s = 0.3 (Fig. 8c), only wing crack initiates near the middle of the flaw.
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Fig. 7. Cracking process for specimen containing a single flaw of β = 0° with various infill strengths.
According to the final fracture patterns in Fig. 8, the wing crack angle decreases obviously as the infill factor increases to s = 0.3. Therefore, it can be concluded that the increasing of infill factor reduces the wing crack angle until the infill factor reaches a critical value. For the unfilled flaw with β = 45°, the wing cracks firstly initiate near the flaw tips and then propagate in the loading direction, and then followed by anti-wing and secondary cracks (Fig. 9a). For infill factor between s = 0.1 (Fig. 9b) and s = 0.4 (Fig. 9c), the initiation and propagation of the wing cracks are similar to those from the unfilled flaw. However, the anti-wing cracks are not observed, which is in conformance with the conclusion by Zhang et al. (2017). It can be further
seen that the wing crack angle obviously decreases as the infill factor increases to s = 0.5. Also, as the infill factor, s increases 0.7 (Fig. 9d), the filled flaw has little effect on cracking behaviour of the specimen, as if there is no flaw. Fig. 10 gives the cracking processes for β = 60°. For the unfilled flaw, the wing cracks and the secondary cracks can be observed (Fig. 10a). However, for the infill factor between s = 0.1 (Fig. 10b) and s = 0.5 (Fig. 10c), only wing cracks can initiate and propagate from the filled flaw. As the infill factor, s increases to 0.7, the filled flaw has little influence on the crack initiation and propagation (Fig. 10d). Fig. 11 shows the cracking process for β = 75°. It can be seen that the filled flaw has
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Fig. 8. Cracking process for specimen containing a single flaw of β = 30° with various infill strengths.
little effect on the crack initiation and propagation even at a low level of s = 0.1. 4. Critical infill strength Fig. 12 gives the plot of initiation angle of wing crack versus the flaw inclination angle. It can be seen that the initiation angle of wing crack decreases as increasing the flaw inclination angle. Zhang et al. (2017) pointed that initiation angles of wing crack for unfilled flaws are larger than those for filled flaws. However, the infill strength is not considered. In this paper, the influences of the infill strength on the initiation angle
of wing crack are presented in Fig. 13. According to Figs. 7–11, the wing crack may disappear if the infill strength exceeds the critical value. Therefore, only cases for infill strength lower than the critical value are included in Fig. 13. It can be further stated that the initiation angle of wing crack decreases as the infill factor increases at the same flaw inclination angle. This trend will stop as the infill strength increases to a certain value, after which the filled flaw has little effect on the cracking behaviour of the specimen. As mentioned above, there exists a critical value for the infill strength that controls the crack pattern of specimen containing a filled flaw. If the infill strength is less than the critical value, the filled flaw
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Fig. 9. Cracking process for specimen containing a single flaw of β = 45° with various infill strengths.
can greatly influence the crack pattern. If the infill strength is greater than the critical value, the filled flaw has little influence on the cracking path and the specimen fails in a similar pattern to that without any flaws. Also, the critical value for the infill strength is dependent on the flaw inclination angle. The critical infill strengths for each flaw angle are presented in Fig. 14. It is indicated that the critical infill strength increases as the flaw angle increases to about 50°, and after that the critical value decreases as further increasing the flaw angle.
5. Mechanical behaviour The typical stress-strain curves for specimens with unfilled and filled flaws under uniaxial compression are plotted in Fig. 15, where the flaw inclination angles are β = 0° (Fig. 15(a)), β = 30° (Fig. 15(b)), β = 60° (Fig. 15(c)) and β = 90° (Fig. 15(d)). It can be seen from Fig. 15 that the specimens with filled flaw have higher stiffness than the specimen with unfilled flaw before peak stress at
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Fig. 10. Cracking process for specimen containing a single flaw of β = 60° with various infill strengths.
a given flaw inclination angle. It is further indicated that the peak stress and the corresponding strain increase by increasing the infill strength. However, if the infill strength exceeds the critical value (s = 0.2 for β = 0°, s = 0.4 for β = 30°, s = 0.5 for β = 60° and s = 0.1 for β = 90°) the infill strengths seem to have little influence on the stress-strain curves. The differences in specimen stiffness and peak stress are very small after the critical value. This means that the flaw infill strength
controls not only the crack pattern of a specimen but also its mechanical behaviour. As an attempt to further explore the influences of infill strength on the mechanical behavior, the stress level at which the wing crack initiates is shown in Fig. 16, where the flaw inclination angle is β = 30. It can be seen that the wing crack initiates at stress level of 11.3 MPa for s = 0.0. For s = 0.1, the wing crack initiates at 23.1 MPa. As the infill
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Fig. 11. Cracking process for specimen containing a single flaw of β = 75° with various infill strengths.
strength. It is clear that the infill strength contributes more to specimen strength for a specimen with lower flaw inclination angle.
strength factor further increases to s = 0.4, the wing crack initiates at 37.6 MPa. Once the wing crack initiates, the specimen stiffness gradually decreases. As a result, the specimen with lower infill strength factor can reach peak stress earlier. It can be therefore concluded that the infill can delay the initiation of the wing crack if the infill strength is lower than the critical value. A higher infill strength factor can lead to a higher stress level at which the wing crack initiate. If the infill strength exceeds the critical value, the wing crack cannot be found. This means that the infill can prevent the initiation of the wing crack. The plot of specimen strength versus the infill strength for each flaw inclination angle is presented in Fig. 17. It can be seen from the figure that the strength for specimen containing a flaw increases as the infill strength increases. And the specimen strength keeps almost constant as the infill strength exceeds its critical value. For β = 0°, the specimen strength increases from 31 MPa to 59.5 MPa as the infill strength approaches its critical value (s = 0.3). The specimen strength increases from 44 MPa to 60 MPa for case of β = 90° as increasing the infill
The primary aim of this study is to investigate the influences of filled flaws on cracking behaviour of rock-like materials. Therefore, influences of the infill heterogeneity are discussed in detail. Two typical cases are discussed: β = 30° and β = 60°. According to the discussed mentioned above, the infill has little influence on the stress-strain response if the infill strength exceeds its critical value. Therefore, only infill strength lower than its critical value are included in this section. Fig. 18 gives the influence of heterogeneity index r for infill on stress-strain response at the inclination angle of β = 30°, where four values for r are adopted: r = 1.0, r = 3.0, r = 5.0, r = 7.0. It can be seen from Fig. 18 that the peak stress and the corresponding strain increase by increasing the value of r. The peak stress increases from 53 MPa to 60 MPa as r increases from 1.0
Fig. 12. Influences of flaw inclination angle on wing crack initiation angle.
Fig. 13. Influence of infill strength factor on wing crack initiation angles.
6. Influences of heterogeneity index r for infill
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Fig. 14. Critical infill strength factors for different flaw inclination angles. Fig. 16. Influences of infill strength factors on wing crack initiation (β = 30°).
to 7.0 for case of s = 0.1. However, for case of s = 0.4, the peak stress increases from 54 MPa to 67 MPa. For β = 60° (Fig. 19), the peak stress increases from 47 MPa to 52 MPa as r increases from 1.0 to 7.0 for case of s = 0.1. For case of s = 0.5, the peak stress increases from 50 MPa to 64 MPa. One can conclude that the mechanical response of the specimen containing a filled flaw is more sensitive to heterogeneity of the infill for case of higher infill factor.
7. Conclusions and discussion This study focuses on the influences of infill strength on the crack behaviour and mechanical performances of the rock specimen containing one filled flaw. The numerical code, RFPA (Tang, 1997) was used to model behaviour of rock specimen containing one flaw. The important
Fig. 15. Typical stress-strain response for specimen with various infill strength factors.
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Fig. 17. Evolution of specimen strength with various infill strength factors.
contribution in the current work is a thorough analysis of the effect of infill strength on the cracking behaviours of rocks. An extensive study based on the numerical model leads to the following conclusions: (1) There exists a critical value for the infill strength which controls the crack propagation and the crack pattern. If the infill strength is higher than critical value, the filled flaw has little influence on the crack propagation from the flaw. The critical infill strength is
Fig. 19. Influences of heterogeneity index for infill on the stress-strain response (β = 60°).
related to the flaw inclination angle and reaches its peak value at β = 50°. Further analysis indicates that the crack inclination angle decreases as the infill strength increases to the critical value. (2) The stress-strain response shows that the specimen with a filled flaw has higher stiffness than that with an unfilled flaw before peak stress. The peak stress and the corresponding strain increase as the infill strength increases. If the infill strength reaches its critical value, the infill strength has little influence on the stress-strain response. The infill can delay the initiation of wing crack and a higher infill strength can lead to a higher stress level at which wing crack initiates. (3) The flaw infill strength controls not only the crack pattern of a specimen but also its strength. The specimen strength increases as the infill strength increases. And the specimen strength keeps almost constant as the infill strength exceeds its critical value.
Fig. 18. Influences of heterogeneity index for infill on the stress-strain response (β = 30°).
In RFPA code, the heterogeneity index is introduced to reflect the random nature of rock-like materials by the method that mechanical properties of the elements are assumed to follow the statistical Weibull distribution. However, it is a very difficult work to select the heterogeneity index since there is no standard to determine it. In this paper, the adopted heterogeneity index is determined by limited test specimens and should be further validated by more specimens.
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Acknowledgements The work described in this paper was fully supported by National Natural Science Foundation of China (No. 51674100), Science and Technology Project of Henan Province (No. 162102210064), the Program for Science &Technology Innovation Talents Universities of Henan Province (Project No. 18HASTIT017) and the Fundamental Research Funds for the Universities of Henan Province (Project No. NSFRF140206). The authors also appreciate the comments from the reviewers. References Ali, A., Jakobsen, M., 2011. Seismic characterization of reservoirs with multiple fracture sets using velocity and attenuation anisotropy data. J. Appl. Geophys. 75, 590–620. Behraftar, S., Torres, S.A.G., Scheuermann, A., Williams, D.J., Marques, E.A.G., Avarzaman, H.J.A., 2017. Calibration methodology to obtain material parameters for the representation of fracture mechanics based on discrete element simulations. Comput. Geotech. 81, 31–38. Belytschko, T., Black, T., 1999. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45 (5), 601–620. Bobet, A., 2000. The initiation of secondary cracks in compression. Eng. Fract. Mech. 66, 187–219. Bobet, A., Einstein, H.H., 1998. Numerical Modeling of Fracture Coalescence in a Model Rock Mat. Damjanac, B., Chundall, P., 2016. Application of distinct element methods to simulation of hydraulic fracturing in naturally fractured reservoirs. Comput. Geotech. 71, 283–294 (Int. J. Fracture 92, 221-252). Duriez, J., Scholtès, L., Donzé, F.-V., 2016. Micromechanics of wing crack propagation for different flaw properties. Eng. Fract. Mech. 153, 378–398. Erdogan, F., Sih, G., 1963. On the crack extension in plates under plane loading and transverse shear. J. Basic Eng. 85 (4), 519–527. Fan, T., 2003. Fracture Theory Foundation. Science Press, Beijing. Hoek, E., Bieniawski, Z.T., 1965. Brittle fracture propagation in rock under compression. Int. J. Fract. 1, 137–155. Horii, H., Nemat-Nasser, S., 1985. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J. Geophys. Res. 90 (B4), 3105–3125. Indraratna, B., Premadasa, W., Brown, E.T., Gens, A., Heitor, A., 2014. Shear strength of rock joints influenced by compacted infill. Int. J. Rock Mech. Min. Sci. 70, 296–307. Jiefan, H., Ganglin, C., Yonghong, Z., Ren, W., 1990. An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics 175, 184–269. Lehner, F., Kachanov, M., 1996. On modeling of “winged” cracks forming under compression. Int. J. Fract. 77, 65–75. Li, Y.P., Chen, L.Z., Wang, Y.H., 2005. Experimental research on pre-cracked marble under compression. Int. J. Solids Struct. 42, 2505–2516. Mahabadi, O.K., Lisjak, A., Munjiza, A., Grasselli, G., 2012. Y-Geo: new combined finite-discrete element numerical code for geomechanical applications. Int. J. Geomech. 12, 676e88.
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