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Numerical study on the influence of cross-sectional shape on strength and deformation behaviors of rocks under uniaxial compression
Computers and Geotechnics 84 (2017) 129–137
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
Numerical study on the influence of cross-sectional shape on strength and deformation behaviors of rocks under uniaxial compression Yuhang Xu, Ming Cai ⇑ Bharti School of Engineering, Laurentian University, Sudbury, Ont. P3E 2C6, Canada MIRARCO, Laurentian University, Sudbury, Ont. P3E 2C6, Canada
a r t i c l e
i n f o
Article history: Received 11 March 2016 Received in revised form 6 October 2016 Accepted 23 November 2016
Keywords: Cross-sectional shape Numerical experiment Uniaxial compression Hoop tension effect UCS Post-peak behavior
a b s t r a c t Cross-sectional shape effect, which has not been well studied, is one of the geometry effects that influence rock laboratory test results. In order to investigate the influence of cross-sectional shape on the strength and deformation behaviors of rocks, a comprehensive numerical experiment is carried out to simulate the deformation responses of circular, square, and rectangular cross-sectionally shaped specimens in uniaxial compression. The validity of the numerical model is first examined by comparing the uniaxial compressive strengths (UCS) of cylinder and square prism specimens obtained from the numerical modeling with these obtained in laboratory tests. Both the numerical modeling and laboratory test results show that the cross-sectional shape has a very small influence on the UCS of rocks. However, the numerical results show that the cross-sectional shape affects the post-peak behaviors of rocks considerably. It is also concluded that hoop tension contributes little to affecting rock strength. It is revealed through the numerical study that in the laboratory tests because the square prism specimens with a slenderness (defined by specimen height divided by specimen width) the same as that of a cylinder specimen have an equivalent diameter larger than that of the cylinder specimens, a slightly higher strength of the square prism specimens is thus observed. It is suggested to use the equivalent diameter of a non-circular cross-section to define the slenderness of a specimen to present laboratory test and numerical simulation results consistently. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Laboratory test results of rocks depend on many factors. Test results have shown that compared with smaller specimens larger specimens of the same slenderness (height to width or diameter ratio) tend to have a lower rock strength [1], and this is known as size effect. Aside from specimen size, the shape of specimens can affect laboratory test results. The shape effect includes the influence of the slenderness and the cross-sectional shape (circular or square) of a rock specimen on its uniaxial compressive strength (UCS) and post-peak stress–strain curve. It is documented that due to the end effect caused by friction at the specimen-platen contacts, the UCS of rocks increases with the decrease of the slenderness of the specimens [2,3]. However, the influence of crosssectional shape on UCS of rocks is not well studied because circular rather than square cross-sectionally shaped specimens are often used in rock mechanical property testing, as suggested by the ⇑ Corresponding author at: MIRARCO, Laurentian University, 935 Ramsey Lake Road, Sudbury P3E 2C6, Canada. E-mail addresses: [email protected] (Y. Xu), [email protected] (M. Cai). http://dx.doi.org/10.1016/j.compgeo.2016.11.017 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.
International Society for Rock Mechanics (ISRM) [4]. True triaxial testing has become popular in recent years and prismatic (square or rectangular cross-sectionally shaped) specimens are routinely employed in true triaxial testing [5–10]. Unfortunately, the influence of specimen’s cross-sectional shape on rock’s peak strength under true triaxial loading is again not well studied. The importance of obtaining not only the peak strength but also the complete stress–strain curve of rocks from laboratory tests has been recognized because the post-peak behavior of rocks affects rock stability [11–14]. For instance, a good knowledge of the post-peak behavior of rocks is needed to estimate the depth of failure accurately for rock support design [15] and to avoid violent pillar failure [16,17]. Thus, it is also important to study the influence of cross-sectional shape on the post-peak behavior of rocks. Few systematic studies focused on the cross-sectional shape effect. Hoop tension [18–21] can be induced by the geometry of a cylinder specimen in compression (Fig. 1b) and it may influence crack propagation and hence the strengths of rocks in laboratory and in-situ. Hoop tension also exists in square prism specimens but due to the shape difference, it would be less than that in cylinder specimens. For a rectangular prism specimen, hoop tension
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expansion under compression
hoop tension crack (a)
(b)
Fig. 1. Illustrations of hoop tension and its influence on crack propagation: (a) crack propagation on a non-circular shaped cross section (e.g., near an excavation boundary); (b) crack propagation on a circular shaped cross section (e.g., in a cylinder specimen).
would be even less. If hoop tension does influence rock strength, it can be hypothesized that the strength of a cylinder specimen should be higher than the strength of a square or a rectangular prism specimen. Uniaxial compression tests using cylinder and square prism specimens, sampled from a large block of Beishan granite, were conducted by Zhao et al. [22] and the test results indicate that there is no significant cross-sectional shape effect on the UCS of the rock. Due to the difficulty in preparing and conducting UCS tests using rectangular prism specimens, the conclusion on the hoop tension effect was inclusive. Furthermore, the post-peak deformation behavior of the rocks was not investigated in the laboratory tests. Inspired by the laboratory test conducted by Zhao et al. [22], we plan to use numerical experimental approach to study the crosssectional shape effect in the present study. First, the strength parameters of Beishan granite [22,23] are obtained for the numerical modeling through a detailed model calibration. Rock strengths of cylinder and square prism specimens are then obtained from the numerical modeling and compared with the laboratory results [22]. Subsequently, more numerical experiments are conducted to investigate the influence of cross-sectional shape on the UCS by adding results of rectangular prism specimens with different slenderness. Finally, the influence of cross-sectional shape on the post-peak behavior of rocks is studied. 2. Influence of cross-sectional shape on UCS of rocks 2.1. Numerical model and model parameters A numerical experiment using the ABAQUS/Explicit FEM tool is carried out to study the cross-sectional shape effect in uniaxial compression tests. ABAQUS/Explicit is a powerful tool in solving highly nonlinear structure system problems under transient loads by employing the explicit numerical scheme. It is also robust to solve problems involving complex boundary conditions with efficient contact convergence and oscillation control. The material properties of Beishan granite are calibrated first to simulate the uniaxial compression tests. The pre-peak behavior of the rock specimens in compression is simplified as linear elastic, and the elastic properties (Poisson’s ratio and Young’s modulus), obtained from the uniaxial compression test results, are summarized in Table 1. The peak strength and the post-peak behavior of the rock specimens are governed by strength parameters. MohrCoulomb failure criterion with a tension cut-off is employed. Based on the fitting equation for the triaxial compression test results of Beishan granite under low confinement (0–5 MPa) [23], the frictional strength parameter of the rock is obtained (Table 1). Then,
based on the UCS of Beishan granite [22], cohesive strength parameters of the rock (Table 2) are calibrated and used in this numerical experiment. Tensile strength was not provided in the test results of Beishan granite; thus the calibration for tension cut-off was based on the data compiled in [24,25], and the strength ratio of UCS to tensile strength was taken as 20. Steel loading platens used to apply a constant loading velocity onto the ends of the specimen are modeled to honor laboratory test conditions; steel property (E = 200 GPa, m = 0.3) is assigned to the platens. 3D simulation models of the specimens with standard slenderness (H/W = 2 or H/D = 2, where H is the height of the specimens, W is the width of the square prism specimen, and D is the diameter of the cylinder specimen) subjected to uniaxial compression by two steel platens are shown in Fig. 2. The geometry of the cylinder specimen is 50 mm in diameter and 100 mm in height (Fig. 2a), and that of the square prism specimen is 50 mm in width and 100 mm in height (Fig. 2b), which are same as the dimensions of the laboratory test specimens [22]. In the laboratory tests, the ends of the specimens were lubricated with a thin layer of Vaseline to reduce the end effect. A coefficient of friction (l) of 0.1, recommended from some researchers [26,27], is used for the specimen-platen contacts in the numerical modeling. In addition to the simulation of the cylinder and square prism specimens used in the laboratory tests, a rectangular prism specimen is considered in our numerical modeling to study the shape effect. The cross-section of the rectangular prism specimen is 70 mm in length and 35 mm in width, with a cross-sectional area of 2450 mm2, which is very close to the cross-sectional area of the square prism specimen (2500 mm2). The height of the rectangular prism specimen is 100 mm, which is the same as that of the cylinder and the square prism specimens. 2.2. Modeling results Fig. 3 presents the modeling results of the UCS of the cylinder, square prism, and rectangular prism specimens with different slenderness varying from 1.0 to 2.5, along with the UCS obtained from the laboratory tests for a slenderness of 2.0 [22]. Because there is no agreed definition of the slenderness of a rectangular prism specimen, the slenderness of the rectangular prism specimens presented in the figure is defined by the specimen height divided by the specimen’s equivalent width that results in the same crosssection area as the square prism specimen. Due to the end effect that can activate confined zones near the specimen ends (Fig. 4), the UCS of the specimens increases as the slenderness decreases. It is seen that there is no significant difference of UCS among different cross-sectionally shaped specimens with the same slenderness. The UCS of the square prism specimens is, in fact, slightly higher than that of the cylinder specimens of the same slenderness. The difference of the UCS between the cylinder and the square prism specimens increases with the decrease of the slenderness due to the increased end effect. The numerical simulation results are in good agreement with the laboratory results of Zhao et al. [22]. In their laboratory tests, eight specimens of each shape with a slenderness of 2.0 were tested and the mean UCS and the standard deviations of the cylinder and the square prism specimens were 132.1 MPa, 4.58 MPa, and 135.8 MPa, 7.20 MPa, respectively. The laboratory results showed that using carefully prepared specimens for testing, the average UCS of the cylinder and the square prism specimens were close to each other. In fact, the average UCS of the square prism specimens is 2.7% higher than that of the cylinder specimens, and our numerical modeling result is in good agreement with the test result. Therefore, it is proven both experimentally and numerically that the cross-sectional shape has a limited influence on the UCS of rocks.
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Y. Xu, M. Cai / Computers and Geotechnics 84 (2017) 129–137 Table 1 Laboratory test data for model parameter calibration. Elastic properties
Mohr - Coulomb strength parameters
Poisson’s ratio: Young’s modulus (GPa): UCS (MPa): Data source:
0.22 38
(r2 = r3, r1) (MPa):
132 [22]
Fitting equation: Data source:
Table 2 Strength and deformation parameters of Beishan granite used in the numerical experiment. Parameters
Value
Poisson’s ratio Young’s modulus (GPa) Cohesion (MPa) Residual cohesion (MPa) Tension cut-off (MPa) Residual tension (MPa) Friction angle (°)
0.22 38 19 1 7 0.01 58
3. Influence of cross-sectional shape on post-peak behavior of rocks 3.1. Strain-softening behavior of rocks In this section, we study the influence the cross-sectional shape of a specimen on its post-peak behavior. Comparing with the prepeak deformation behaviors and peak strengths for rock specimens with the same geometry in the laboratory results, the post-peak stress–strain curves are variable. This is because that the mechanical behavior of rocks is influenced by rock heterogeneity, and this is especially the case in the post-peak deformation stage, where localized failure normally takes place [29,30]. Obviously, it is difficult to investigate the problem using laboratory tests and the numerical experimental approach is used. Moreover, as justified later that instead of using the circumferential strain-controlled loading as that used in the laboratory testing, the axial strain controlled-loading is used in the simulation. Because different loading methods were used in the laboratory test and numerical
(0, 145.99) (2, 163.83) (5, 206.91) r1 = 12.129 r3 + 142.86 [23]
modeling, it is hence not possible to compare the post-peak stress–strain curves. The post-peak behaviors of rocks in uniaxial compression are usually classified into Class I and Class II failure types [31]. Class I failure type shows a strain-softening behavior. Class II failure type, on the other hand, shows that the post-peak strength decreases with the decrease of axial strain. Both Class I and Class II failure types can be observed in uniaxial compression test results of the same rock type, depending on the servo-control loading methods (axial strain control or lateral or circumferential strain control) used. Therefore, the post-peak behavior of rocks is loading condition dependent [2,32,33]. The risk of violent rock failure often forces tests to be conducted using lateral or circumferential strain controlled-loading. Thus, Class II type stress–strain curves are commonly seen in most laboratory test results. Class I failure type is considered for the Beishan granite specimens in this numerical experiment. This is because that we want to focus on investigating the influence of cross-sectional shape on rock deformation behaviors, and it is better to keep the loading condition simple. Class II failure type can only be obtained using the lateral or circumferential-strain-controlled loading, but it is simple to use the axial-strain-controlled loading in a numerical experiment. More importantly, using the axial strain-controlled loading might be more appropriate to reflect the actual loading condition in situ. For instance, the loading condition in a pillar is axial deformation controlled, and the in situ complete load–deformation curves of pillars obtained by field testing so far are all Class I [34,35]. Strain-softening behavior is assumed in the numerical experiment to capture the post-peak behaviors of different crosssectionally shaped specimens in uniaxial compression tests. This study focuses on studying the post-peak stress–strain curves of rocks, and the simulation of the explicit fracturing process is
ram steel platen
rock specimen
steel platen ram
(a)
(1, 154.63) (3.5, 182.44)
(b)
(c)
Fig. 2. 3D models of (a) cylinder, (b) square prism, and (c) rectangular prism specimens for uniaxial compression test simulation.
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3.2. Modeling results
160 cylinder (laboratory) square prism (laboratory) cylinder (simulation) square prism (simulation) rectangular prism (simulation)
UCS (MPa)
150
Fig. 5 presents the complete stress–strain curves of different cross-sectionally shaped specimens in uniaxial compression. It is seen that the peak strengths of different cross-sectionally shaped specimens are similar to each other. However, the post-peak behaviors of the specimens are influenced by their crosssectional shapes. The post-failure slope of the square prism specimen is the smallest, the rectangular prism specimen is the largest, and the cylinder specimen is between these two.
140
130 4. Discussions
120 1
2
3
4.1. End effect for peak load
Slenderness
c
c Fig. 4. Illustration of end effect in specimens with different slenderness in rock uniaxial compression tests (redraw based on [28]).
Cohesion (MPa)
Plastic strain
19 16 9.5 1
0 0.008 0.035 0.090
120 100
50 mm
80 60
70 mm
40
35 mm
Table 3 Strain-softening parameters of rock used in numerical modeling.
50 mm 140
50 mm
beyond the scope this research. Because the fracturing process during the post-peak deformation stage leads to a cohesion loss [36], the strain-softening behavior of the rock can be defined by degrading the rock’s cohesion strength as a function of plastic strain (defined in Table 3). A fundamental problem of incorporating strain-softening material models in a standard continuum model such as ABAQUS is the inherent mesh size sensitivity, which is an unsettled problem in numerical modeling in continuum mechanics [37,38]. The mesh sizes for all the specimens with different cross-sectional shapes are relatively fine and the mesh shapes (hexahedron) are the same in each model. In this fashion, numerical errors associated with mesh size are minimized and a relative comparison of the simulation results can be made because all the models are within the same order of numerical errors.
The modeling results demonstrate that there is no significant influence of cross-sectional shape on the peak strength of rocks, which is supported by the laboratory test results [22]. An advantage of numerical modeling is that it allows a detailed investigation of the mechanism that governs the observed phenomenon. Fig. 6 presents the distributions of confined elements (r3 < 0, compressive stress is negative in ABAQUS) in the specimens (the 1st row), the contours of the minimum principal stress (r3) on the side, top, and two vertical surfaces revealed from a quarter cut of the specimens (the 2nd row), and the distribution of confined elements with relatively high confinements (r3 < 1.5 MPa) in the specimens (the 3rd row) at peak load. The distributions of the confined elements and the r3 contours in different cross-sectionally shaped specimens are similar at their peak loads because their slenderness and contact friction are the same. Consequently, specimens with different cross-sectional shapes have similar peak strengths. The confined elements whose confinements (absolute values) are greater than 1.5 MPa at peak load are lumped near the specimen’s ends, forming cone-shaped confined zones, and the tops of which are approximately 15 mm from the specimen’s ends. The elements in these ‘‘highly” confined zones have higher peak strengths due to increased confinements. Therefore, hourglassing failure mode is normally observed in uniaxial compression tests [39,40]; it can also be observed in pillar failure in the field [41] because of strong end constraint to the pillars. Both the numerical simulation and the laboratory test results show that the UCS of a square prism specimen with a slenderness of 2.0 is higher than that of a cylinder specimen of the same slenderness. Table 4 presents the percentages of the confined elements (r3 < 0) to the total number of elements in each specimen, along with the average minimum principal stresses (r3) in the whole specimen, in the portion of 15 mm to the specimen’s end and in
Stress (MPa)
Fig. 3. UCS of circular, square, and rectangular cross-sectionally shaped specimens with different slenderness obtained from uniaxial compression test simulation. Laboratory test results for a slenderness of 2.0 are also shown [23].
20 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strain (%) Fig. 5. Stress–strain curves of different cross-sectionally shaped specimens in uniaxial compression.
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Confined elements with 3 < 0 in specimens (compressive stress is negative in ABAQUS)
Contour of (unit: Pa)
3
Confined elements with 3 -1.5 MPa in specimens (compressive stress is negative in ABAQUS)
(a)
(b)
(c)
Fig. 6. Distributions of confined elements at two r3 thresholds and r3 contours in different cross-sectionally shaped specimens at peak load: (a) cylinder, (b) square prism, and (c) rectangular prism specimens.
Table 4 Percentages of confined elements to the total number of elements and average minimum principal stresses (r3) in various portions of the specimens at peak load.
Total number of elements Number of confined elements Percentage of confined elements at peak load Average r3 (MPa) in the whole specimen at peak load Average r3 (MPa) in the portion of 15 mm to the specimen end at peak load Average r3 (MPa) in the 70 mm middle portion of the specimen at peak load
Cylinder specimen
Square prism specimen
Rectangular prism specimen
8844 6836 77%
9537 8531 89%
9108 7178 79%
1.22
1.40
1.15
3.95
4.00
3.70
0.04
0.27
0.04
the 70 mm middle portion of the specimen at peak load. The r3 value in each confined element due to end constraint controls the overall strength enhancement of the specimen. The square prism specimen has the highest percentage of confined elements in its volume and the highest confinement (absolute value, or the lowest r3 value) than the other two specimens. This is why it has a higher UCS than the other two specimens. For all specimens,
the average confinement in the specimen’s end zones is much higher than that in the middle portions of the specimens, showing that the end effect has a large influence on the stress condition near the specimen’s end which can influence the peak load [42]. 4.2. End effect for post-peak behavior As seen in Fig. 5, the post-peak behavior of a specimen depends on its cross-sectional shape. Fig. 7 presents the distributions of confined elements at two r3 thresholds and the r3 contours in the three cross-sectionally shaped specimens at axial strain e = 0.5% in the post-peak deformation stage. It is seen that due to the end effect, the square prism specimen has the highest local confinement in the post-peak deformation stage and the rectangular prism specimen has the lowest local confinement. Table 5 presents the percentages of confined elements and the average minimum principal stresses in the three cross-sectionally shaped specimens at e = 0.5%. The results show that locally confined elements in a specimen are reduced (comparing with the numbers shown in Table 4) due to the progressive failure of the rocks, and the degrees of decrease of the confined elements in the postpeak deformation stage is different for the three specimens. The square prism specimen has the highest percentage of confined elements and the smallest tensile stress r3; hence, its strength reduction in the post-peak deformation stage is the smallest. In contrast,
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Confined elements ( 3 < 0) in specimens (compressive stress is negative in ABAQUS)
Contour of (unit: Pa)
3
Confined elements ( 3 -1.5 MPa) in specimens (compressive stress is negative in ABAQUS)
(a)
(b)
(c)
Fig. 7. Distributions of confined elements at two r3 thresholds and r3 contours in different cross-sectionally shaped specimens at e = 0.5% in the post-peak deformation stage: (a) cylinder, (b) square prism, and (c) rectangular prism specimens.
Table 5 Percentages of confined elements to the total number of elements and average minimum principal stress (r3) in various portions of the specimens at e = 0.5% in the post-peak deformation stage.
Total number of elements Number of confined elements Percentage of confined elements at e = 0.5% Average r3 (MPa) in the whole specimen at e = 0.5% Average r3 (MPa) in the portion of 15 mm to the specimen end at e = 0.5% Average r3 (MPa) in the 70 mm middle portion of the specimen at e = 0.5%
Cylinder specimen
Square prism specimen
Rectangular prism specimen
8844 3978 45%
9537 5582 59%
9108 3670 40%
1.54
0.12
2.68
1.36
0.18
3.05
1.62
0.09
2.62
the rectangular prism specimen has the lowest percentage of confined elements and the largest tensile stress r3, and its post-peak curve is the most brittle. As can be noticed from Tables 4 and 5, the average r3 in each specimen changes from compression at peak load to tension in the post-peak loading stage at e = 0.5%. Fig. 8 presents the evolu-
tion of tensile stress (r3 > 0) at different loading stages in the three cross-sectionally shaped specimens. Tensile stress becomes more prominent with the increase of axial deformation in the postpeak deformation stage. At peak load, tensile stress appears only near the edge in the central areas of the specimens. In the postpeak deformation stage, the minimum principal stresses in the cone-shaped confined zones seen before at peak load gradually become tensile as deformation increases. As mentioned above, the lateral expansion rate of the rock is higher than that of the steel platens, and the elements under the platens are restricted by the steel platens and are subjected to compressive stresses if there is no rock failure. On the other hand, tensile stresses are generated in these elements in the post-peak deformation stage because their lateral contraction rate is higher than that of the steel platens as the load decreases. However, the evolutions of the minimum principal stress in the post-peak loading stage are different for different cross-sectionally shaped specimens. The rectangular prism specimen shows the largest tensile stress zones in the specimen, while the cylinder and the square prism specimens show relatively smaller tensile stress zones. The slenderness effect becomes more significant in uniaxial compression tests when the slenderness of a specimen decreases (Fig. 4). In essence, it is the change of the geometry of the specimen that leads to an increase or a decrease of the influence of the end constraint on the rock deformation behaviors. In other words, the
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tensile 3 (unit: Pa) at peak load
3 at = 0.4% in the post-peak deformation stage
3 at = 0.5% in the post-peak deformation stage
3 at = 0.6% in the post-peak deformation stage
(a)
(b)
(c)
Fig. 8. Evolutions of tensile minimum principal stresses (r3 > 0 contours) in different cross-sectionally shaped specimens at different deformation stages: (a) cylinder, (b) square prism, and (c) rectangular prism specimens.
geometry effect of a rock specimen is essentially manifested by the distribution of confined zones caused by the end effect. The crosssectional shape effect can be seen clearly in the post-peak deformation stage, when progressive failure of elements gradually decreases the effective bearing area of a specimen and consequently changes the effective specimen geometry. Failures occur first locally in elements whose confinement is small, and the locations of these elements depend on the crosssectional shape of the specimen. The degree of the effective specimen geometry change is also dependent on the specimen’s crosssectional shape. Therefore, the confined zones in a specimen are more dependent on the specimen’s cross-sectional shape in the post-peak deformation stage and so is the strain-softening curve. For instance, the elements of the rectangular prism are nonuniformly confined (Fig. 7c) and there are many of them subjected to tensile minimum principal stresses (Fig. 8c); hence, the overall post-peak strength of the rectangular prism specimen is lower than these of the other two specimens. The cylinder and square prism specimens are uniformly restricted (Fig. 7a and b) and there are less elements whose minimum principal stresses are tensile within the specimens (Fig. 8a and b) in the post-peak deformation stage. Therefore, the post-peak strengths of these two specimens are relatively high. Furthermore, the post-peak strengths of the cylinder and the square prism specimens are different because
the r3 contours in the two specimens are not the same. Thus, it is concluded that the cross-sectional shape affects the post-peak behavior of rocks. 4.3. Hoop tension effect Because a continuum numerical tool is used in this study, it is not possible to capture explicitly the crack initiation and propagation processes that lead to discontinuous failure of rocks [43]. Although the influence of hoop tension on a dilating crack cannot be simulated explicitly, it does not inhibit us from comparing the confined elements and the r3 values in the cylinder specimen with those in the prism specimens. Based on the hypothesis of the hoop tension theory, it can be reckoned that the number of confined elements and the average compressive r3 value in a cylinder specimen should be greater than that in a square or a rectangular prism specimen. Hence, the strength of the cylinder specimen should be the highest because the circular shape favors hoop tension development. The simulation results show that for the same slenderness (defined by H/D and H/W for the cylinder and square prism specimens, respectively) of different shaped specimens, it is the square prism specimen rather than the cylinder specimen that has the highest percentage of confined elements and the highest confine-
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ment (or the lowest tension), both at peak and in the post-peak loading stages (Tables 4 and 5). In addition, both the laboratory and the numerical results show that the strength of the square prism specimen is higher than that of the cylinder specimen (Fig. 3). The strength of the rectangular prism specimen is also very close to that of the cylinder and the square prism specimens. Thus, it is concluded that hoop tension contributes little to affecting rock strength. A major difference between the strengths obtained from laboratory tests (using specimens) and field data interpretation is that the interpreted field rock strength depends on the interpretation model used. When a numerical model reflects the field condition better, the interpreted rock strength of massive rocks is closer to that obtained from laboratory tests and this was demonstrated by Cai and Kaiser [44] using the Mine-by tunnel case history. Although the height (100 mm) and the slenderness (2.0) of the cylinder (H/D) and the square prism (H/W) specimens are the same in the laboratory tests, the cross-sectional area of the square prism specimen (50 50 = 2500 mm2) is larger than that of the cylinder specimen (p 252 = 1963 mm2). In such a case, the equivalent diameter of the square specimen is 56.4 mm, resulting in an equivalent slenderness of H/D = 1.77 that is squatter than the cylinder specimen with H/D = 2.0. Therefore, the end effect has a greater influence on the strength of the square prism specimen. Additional numerical simulation results shown in Fig. 9 reveal that if a square prism specimen with a width of 44 mm is used in uniaxial compression tests, it gives almost the same cross-sectional area (1936 mm2) as that of a cylinder specimen with a diameter of 50 mm. In such a case, the UCS of the square prism specimens is very close to that of the cylinder specimens.
5. Conclusions The influence of the cross-sectional shape of a rock specimen on its strength in uniaxial compression tests was studied using the numerical experimental approach. It is found that the crosssectional shape has a very small influence on the peak strength of rocks, but it affects the post-peak behavior of rocks significantly. The study revealed that it is the relation between the distribution of confined and tensile zones during the post-peak deformation process that actually contributes to the cross-sectional shape effect. It is shown in our modeling results, which are supported by laboratory test results, that the strength of a square prism specimen is actually higher than that of a cylinder specimen for the
160
cylinder (D = 50 mm) square prism (W = 44 mm)
UCS (MPa)
150
140
130
120 1
2
3
Slenderness Fig. 9. UCS of circular (diameter D = 50 mm) and square (width W = 44 mm) crosssectionally shaped specimens with different slenderness obtained from uniaxial compression test simulation.
same slenderness calculated by the specimen height divided by the width (square specimen) or the diameter (cylinder specimen). The strength of a cylinder specimen is not significantly higher than that of a rectangular prism specimen, implying that hoop tension contributes little to affecting rock strength. The measurable strength difference between the cylinder specimens and the square prism specimens observed in the laboratory test and the numerical simulation results is attributed to the relatively large cross-section areas of the square prism specimens used. If the slenderness of a square prism specimen is calculated by dividing the specimen height by an equivalent diameter showing the same cross-section area as a cylinder specimen, then the square prism specimen will have the same strength as the cylinder specimen. Insight gained from this study can assist in interpreting laboratory test results more objectively. One suggestion is to use the equivalent diameter of a non-circular cross-section to define the slenderness of a specimen. In this fashion, consistent presentation of the test results can be achieved. For the same cross-section area and slenderness, a rectangular pillar has the most brittle post-peak behavior. Hence, it is beneficial not creating rectangular crosssectionally shaped pillars in underground mines to avoid violent pillar failure. Acknowledgement The work presented in this paper was financially supported by NSERC (Natural Science and Engineering Research Council of Canada). References [1] Hoek E, Brown ET. Underground excavations in rock; 1980. [2] Hudson JA, Brown ET, Fairhurst C. Shape of the complete stress-strain curve for rock. Stability of Rock Slopes: ASCE; 1972. p. 773–95. [3] Das MN. Influence of width/height ratio on post-failure behaviour of coal. Int J Min Geol Eng 1986;4:79–87. [4] Fairhurst C, Hudson J. Draft ISRM suggested method for the complete stressstrain curve for intact rock in uniaxial compression. Int J Rock Mech Min Sci 1999;36:279–89. [5] Bobet A. Influence of the loading apparatus on the stresses within biaxial specimens. ASTM Geotech Test J 2001;24:256–72. [6] Mogi K. Fracture and flow of rocks under high triaxial compression. J Geophys Res 1971;76:1255–69. [7] Takahashi M, Koide H. Effect of the intermediate principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than 2000 m. In: ISRM international symposium. International Society for Rock Mechanics; 1989. [8] Haimson B, Chang C. A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. Int J Rock Mech Min Sci 2000;37:285–96. [9] He M, Miao J, Feng J. Rock burst process of limestone and its acoustic emission characteristics under true-triaxial unloading conditions. Int J Rock Mech Min Sci 2010;47:286–98. [10] Sheng D, Westerberg B, Mattsson H, Axelsson K. Effects of end restraint and strain rate in triaxial tests. Comput Geotech 1997;21:163–82. [11] Lockner D, Byerlee J, Kuksenko V, Ponomarev A, Sidorin A. Quasi-static fault growth and shear fracture energy in granite. Nature 1991;350:39–42. [12] Zipf Jr RK. Analysis of stable and unstable pillar failure using a local mine stiffness method. In: Proceedings of the workshop on coal pillar mechanics and design. Washington (DC): U.S. Bureau of Mines; 1992. p. 128–43. [13] Hoek E, Brown E. Practical estimates of rock mass strength. Int J Rock Mech Min Sci 1997;34:1165–86. [14] Joseph T, Barron K. The post-failure characteristics of rock. CIM Bull 2003;96:66–74. [15] Cai M, Kaiser P, Tasaka Y, Minami M. Determination of residual strength parameters of jointed rock masses using the GSI system. Int J Rock Mech Min Sci 2007;44:247–65. [16] Salamon MDG. Stability, instability and design of pillar workings. Int J Rock Mech Min Sci Geomech Abstr 1970;7:613–31. [17] Wiles T. Loading system stiffness-a parameter to evaluate rockburst potential. In: First international seminar on deep and high stress mining, Australian Centre for Geomechanics. Perth; 2002. [18] Henkel D, Gilbert G. The effect measured of the rubber membrane on the triaxial compression strength of clay samples. Geotechnique 1952;3:20–9. [19] Meyerhof G. The bearing capacity of concrete and rock⁄. Mag Concr Res 1953;4:107–16.
Y. Xu, M. Cai / Computers and Geotechnics 84 (2017) 129–137 [20] Geertsma J. Some rock-mechanical aspects of oil and gas well completions. Soc Petrol Eng J 1985;25:848–56. [21] Vernik L, Zoback MD. Estimation of maximum horizontal principal stress magnitude from stress-induced well bore breakouts in the Cajon Pass Scientific Research borehole. J Geophys Res: Solid Earth (1978–2012) 1992;97:5109–19. [22] Zhao X, Cai M, Wang J, Li P. Strength comparison between cylindrical and prism specimens of Beishan granite under uniaxial compression. Int J Rock Mech Min Sci 2015:10–7. [23] Zhao X, Cai M, Wang J, Ma L. Damage stress and acoustic emission characteristics of the Beishan granite. Int J Rock Mech Min Sci 2013;64:258–69. [24] Brook N. The measurement and estimation of basic rock strength. Compr Rock Eng 1993;3:41–66. [25] Cai M. Practical estimates of tensile strength and Hoek-Brown strength parameter mi of brittle rocks. Rock Mech Rock Eng 2010;43:167–84. [26] Grote K-H, Antonsson EK. Springer handbook of mechanical engineering. Springer Science & Business Media; 2009. [27] Rashed G, Peng SS. Change of the mode of failure by interface friction and width-to-height ratio of coal specimens. J Rock Mech Geotech Eng 2015;7 (3):256–65. [28] Van Vliet M, Van Mier J. Experimental investigation of concrete fracture under uniaxial compression. Mech Cohesive-Frict Mater 1996;1:115–27. [29] Rudnicki JW, Rice J. Conditions for the localization of deformation in pressuresensitive dilatant materials. J Mech Phys Solids 1975;23:371–94. [30] Bobet A, Einstein H. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int J Rock Mech Min Sci 1998;35:863–88. [31] Wawersik WR, Fairhurst Ch. A study of brittle rock fracture in laboratory compression experiments. Int J Rock Mech Min Sci Geomech Abstr 1970;7:561–75.
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[32] Labuz J, Biolzi L. Class I vs Class II stability: a demonstration of size effect. Int J Rock Mech Min Sci Geomech Abstr 1991:199–205. [33] Bieniawski ZT, Denkhaus H, Vogler U. Failure of fractured rock. Int J Rock Mech Min Sci Geomech Abstr 1969:323IN29331–30IN32341. [34] Van Heerden W. In situ complete stress-strain characteristics of large coal specimens. J South Afr Inst Min Metall 1975;75:207–17. [35] Cook NGW, Hodgson K, Hojem JPM. A 100 MN jacking system for testing coal pillars underground. J South Afr Inst Min Metall 1971;71:215–24. [36] Martin CD. Seventeenth Canadian geotechnical colloquium: the effect of cohesion loss and stress path on brittle rock strength. Can Geotech J 1997;34:698–725. [37] Needleman A. Material rate dependence and mesh sensitivity in localization problems. Comput Meth Appl Mech Eng 1988;67:69–85. [38] Cook RD. Concepts and applications of finite element analysis. John Wiley & Sons; 2007. [39] You M. True-triaxial strength criteria for rock. Int J Rock Mech Min Sci 2009;46:115–27. [40] Xu Y, Cai M. Numerical simulation of end constraint effect on post-peak behaviors of rocks in uniaxial compression. In: 49th US rock mechanics/geomechanics symposium. San Francisco, USA. [41] Martin C, Maybee W. The strength of hard-rock pillars. Int J Rock Mech Min Sci 2000;37:1239–46. [42] Brady B. Effects of inserts on the elastic behavior of cylindrical materials loaded between rough end-plates. Int J Rock Mech Min Sci Geomech Abstr 1971;8:357–69. [43] Senent S, Jimenez R, Reyes A. Numerical simulation of the influence of smallscale defects on the true-triaxial strength of rock samples. Comput Geotech 2013;53:142–56. [44] Cai M, Kaiser P. In-situ rock spalling strength near excavation boundaries. Rock Mech Rock Eng 2014;47:659–75.
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
Numerical study on the influence of cross-sectional shape on strength and deformation behaviors of rocks under uniaxial compression Yuhang Xu, Ming Cai ⇑ Bharti School of Engineering, Laurentian University, Sudbury, Ont. P3E 2C6, Canada MIRARCO, Laurentian University, Sudbury, Ont. P3E 2C6, Canada
a r t i c l e
i n f o
Article history: Received 11 March 2016 Received in revised form 6 October 2016 Accepted 23 November 2016
Keywords: Cross-sectional shape Numerical experiment Uniaxial compression Hoop tension effect UCS Post-peak behavior
a b s t r a c t Cross-sectional shape effect, which has not been well studied, is one of the geometry effects that influence rock laboratory test results. In order to investigate the influence of cross-sectional shape on the strength and deformation behaviors of rocks, a comprehensive numerical experiment is carried out to simulate the deformation responses of circular, square, and rectangular cross-sectionally shaped specimens in uniaxial compression. The validity of the numerical model is first examined by comparing the uniaxial compressive strengths (UCS) of cylinder and square prism specimens obtained from the numerical modeling with these obtained in laboratory tests. Both the numerical modeling and laboratory test results show that the cross-sectional shape has a very small influence on the UCS of rocks. However, the numerical results show that the cross-sectional shape affects the post-peak behaviors of rocks considerably. It is also concluded that hoop tension contributes little to affecting rock strength. It is revealed through the numerical study that in the laboratory tests because the square prism specimens with a slenderness (defined by specimen height divided by specimen width) the same as that of a cylinder specimen have an equivalent diameter larger than that of the cylinder specimens, a slightly higher strength of the square prism specimens is thus observed. It is suggested to use the equivalent diameter of a non-circular cross-section to define the slenderness of a specimen to present laboratory test and numerical simulation results consistently. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Laboratory test results of rocks depend on many factors. Test results have shown that compared with smaller specimens larger specimens of the same slenderness (height to width or diameter ratio) tend to have a lower rock strength [1], and this is known as size effect. Aside from specimen size, the shape of specimens can affect laboratory test results. The shape effect includes the influence of the slenderness and the cross-sectional shape (circular or square) of a rock specimen on its uniaxial compressive strength (UCS) and post-peak stress–strain curve. It is documented that due to the end effect caused by friction at the specimen-platen contacts, the UCS of rocks increases with the decrease of the slenderness of the specimens [2,3]. However, the influence of crosssectional shape on UCS of rocks is not well studied because circular rather than square cross-sectionally shaped specimens are often used in rock mechanical property testing, as suggested by the ⇑ Corresponding author at: MIRARCO, Laurentian University, 935 Ramsey Lake Road, Sudbury P3E 2C6, Canada. E-mail addresses: [email protected] (Y. Xu), [email protected] (M. Cai). http://dx.doi.org/10.1016/j.compgeo.2016.11.017 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.
International Society for Rock Mechanics (ISRM) [4]. True triaxial testing has become popular in recent years and prismatic (square or rectangular cross-sectionally shaped) specimens are routinely employed in true triaxial testing [5–10]. Unfortunately, the influence of specimen’s cross-sectional shape on rock’s peak strength under true triaxial loading is again not well studied. The importance of obtaining not only the peak strength but also the complete stress–strain curve of rocks from laboratory tests has been recognized because the post-peak behavior of rocks affects rock stability [11–14]. For instance, a good knowledge of the post-peak behavior of rocks is needed to estimate the depth of failure accurately for rock support design [15] and to avoid violent pillar failure [16,17]. Thus, it is also important to study the influence of cross-sectional shape on the post-peak behavior of rocks. Few systematic studies focused on the cross-sectional shape effect. Hoop tension [18–21] can be induced by the geometry of a cylinder specimen in compression (Fig. 1b) and it may influence crack propagation and hence the strengths of rocks in laboratory and in-situ. Hoop tension also exists in square prism specimens but due to the shape difference, it would be less than that in cylinder specimens. For a rectangular prism specimen, hoop tension
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expansion under compression
hoop tension crack (a)
(b)
Fig. 1. Illustrations of hoop tension and its influence on crack propagation: (a) crack propagation on a non-circular shaped cross section (e.g., near an excavation boundary); (b) crack propagation on a circular shaped cross section (e.g., in a cylinder specimen).
would be even less. If hoop tension does influence rock strength, it can be hypothesized that the strength of a cylinder specimen should be higher than the strength of a square or a rectangular prism specimen. Uniaxial compression tests using cylinder and square prism specimens, sampled from a large block of Beishan granite, were conducted by Zhao et al. [22] and the test results indicate that there is no significant cross-sectional shape effect on the UCS of the rock. Due to the difficulty in preparing and conducting UCS tests using rectangular prism specimens, the conclusion on the hoop tension effect was inclusive. Furthermore, the post-peak deformation behavior of the rocks was not investigated in the laboratory tests. Inspired by the laboratory test conducted by Zhao et al. [22], we plan to use numerical experimental approach to study the crosssectional shape effect in the present study. First, the strength parameters of Beishan granite [22,23] are obtained for the numerical modeling through a detailed model calibration. Rock strengths of cylinder and square prism specimens are then obtained from the numerical modeling and compared with the laboratory results [22]. Subsequently, more numerical experiments are conducted to investigate the influence of cross-sectional shape on the UCS by adding results of rectangular prism specimens with different slenderness. Finally, the influence of cross-sectional shape on the post-peak behavior of rocks is studied. 2. Influence of cross-sectional shape on UCS of rocks 2.1. Numerical model and model parameters A numerical experiment using the ABAQUS/Explicit FEM tool is carried out to study the cross-sectional shape effect in uniaxial compression tests. ABAQUS/Explicit is a powerful tool in solving highly nonlinear structure system problems under transient loads by employing the explicit numerical scheme. It is also robust to solve problems involving complex boundary conditions with efficient contact convergence and oscillation control. The material properties of Beishan granite are calibrated first to simulate the uniaxial compression tests. The pre-peak behavior of the rock specimens in compression is simplified as linear elastic, and the elastic properties (Poisson’s ratio and Young’s modulus), obtained from the uniaxial compression test results, are summarized in Table 1. The peak strength and the post-peak behavior of the rock specimens are governed by strength parameters. MohrCoulomb failure criterion with a tension cut-off is employed. Based on the fitting equation for the triaxial compression test results of Beishan granite under low confinement (0–5 MPa) [23], the frictional strength parameter of the rock is obtained (Table 1). Then,
based on the UCS of Beishan granite [22], cohesive strength parameters of the rock (Table 2) are calibrated and used in this numerical experiment. Tensile strength was not provided in the test results of Beishan granite; thus the calibration for tension cut-off was based on the data compiled in [24,25], and the strength ratio of UCS to tensile strength was taken as 20. Steel loading platens used to apply a constant loading velocity onto the ends of the specimen are modeled to honor laboratory test conditions; steel property (E = 200 GPa, m = 0.3) is assigned to the platens. 3D simulation models of the specimens with standard slenderness (H/W = 2 or H/D = 2, where H is the height of the specimens, W is the width of the square prism specimen, and D is the diameter of the cylinder specimen) subjected to uniaxial compression by two steel platens are shown in Fig. 2. The geometry of the cylinder specimen is 50 mm in diameter and 100 mm in height (Fig. 2a), and that of the square prism specimen is 50 mm in width and 100 mm in height (Fig. 2b), which are same as the dimensions of the laboratory test specimens [22]. In the laboratory tests, the ends of the specimens were lubricated with a thin layer of Vaseline to reduce the end effect. A coefficient of friction (l) of 0.1, recommended from some researchers [26,27], is used for the specimen-platen contacts in the numerical modeling. In addition to the simulation of the cylinder and square prism specimens used in the laboratory tests, a rectangular prism specimen is considered in our numerical modeling to study the shape effect. The cross-section of the rectangular prism specimen is 70 mm in length and 35 mm in width, with a cross-sectional area of 2450 mm2, which is very close to the cross-sectional area of the square prism specimen (2500 mm2). The height of the rectangular prism specimen is 100 mm, which is the same as that of the cylinder and the square prism specimens. 2.2. Modeling results Fig. 3 presents the modeling results of the UCS of the cylinder, square prism, and rectangular prism specimens with different slenderness varying from 1.0 to 2.5, along with the UCS obtained from the laboratory tests for a slenderness of 2.0 [22]. Because there is no agreed definition of the slenderness of a rectangular prism specimen, the slenderness of the rectangular prism specimens presented in the figure is defined by the specimen height divided by the specimen’s equivalent width that results in the same crosssection area as the square prism specimen. Due to the end effect that can activate confined zones near the specimen ends (Fig. 4), the UCS of the specimens increases as the slenderness decreases. It is seen that there is no significant difference of UCS among different cross-sectionally shaped specimens with the same slenderness. The UCS of the square prism specimens is, in fact, slightly higher than that of the cylinder specimens of the same slenderness. The difference of the UCS between the cylinder and the square prism specimens increases with the decrease of the slenderness due to the increased end effect. The numerical simulation results are in good agreement with the laboratory results of Zhao et al. [22]. In their laboratory tests, eight specimens of each shape with a slenderness of 2.0 were tested and the mean UCS and the standard deviations of the cylinder and the square prism specimens were 132.1 MPa, 4.58 MPa, and 135.8 MPa, 7.20 MPa, respectively. The laboratory results showed that using carefully prepared specimens for testing, the average UCS of the cylinder and the square prism specimens were close to each other. In fact, the average UCS of the square prism specimens is 2.7% higher than that of the cylinder specimens, and our numerical modeling result is in good agreement with the test result. Therefore, it is proven both experimentally and numerically that the cross-sectional shape has a limited influence on the UCS of rocks.
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Y. Xu, M. Cai / Computers and Geotechnics 84 (2017) 129–137 Table 1 Laboratory test data for model parameter calibration. Elastic properties
Mohr - Coulomb strength parameters
Poisson’s ratio: Young’s modulus (GPa): UCS (MPa): Data source:
0.22 38
(r2 = r3, r1) (MPa):
132 [22]
Fitting equation: Data source:
Table 2 Strength and deformation parameters of Beishan granite used in the numerical experiment. Parameters
Value
Poisson’s ratio Young’s modulus (GPa) Cohesion (MPa) Residual cohesion (MPa) Tension cut-off (MPa) Residual tension (MPa) Friction angle (°)
0.22 38 19 1 7 0.01 58
3. Influence of cross-sectional shape on post-peak behavior of rocks 3.1. Strain-softening behavior of rocks In this section, we study the influence the cross-sectional shape of a specimen on its post-peak behavior. Comparing with the prepeak deformation behaviors and peak strengths for rock specimens with the same geometry in the laboratory results, the post-peak stress–strain curves are variable. This is because that the mechanical behavior of rocks is influenced by rock heterogeneity, and this is especially the case in the post-peak deformation stage, where localized failure normally takes place [29,30]. Obviously, it is difficult to investigate the problem using laboratory tests and the numerical experimental approach is used. Moreover, as justified later that instead of using the circumferential strain-controlled loading as that used in the laboratory testing, the axial strain controlled-loading is used in the simulation. Because different loading methods were used in the laboratory test and numerical
(0, 145.99) (2, 163.83) (5, 206.91) r1 = 12.129 r3 + 142.86 [23]
modeling, it is hence not possible to compare the post-peak stress–strain curves. The post-peak behaviors of rocks in uniaxial compression are usually classified into Class I and Class II failure types [31]. Class I failure type shows a strain-softening behavior. Class II failure type, on the other hand, shows that the post-peak strength decreases with the decrease of axial strain. Both Class I and Class II failure types can be observed in uniaxial compression test results of the same rock type, depending on the servo-control loading methods (axial strain control or lateral or circumferential strain control) used. Therefore, the post-peak behavior of rocks is loading condition dependent [2,32,33]. The risk of violent rock failure often forces tests to be conducted using lateral or circumferential strain controlled-loading. Thus, Class II type stress–strain curves are commonly seen in most laboratory test results. Class I failure type is considered for the Beishan granite specimens in this numerical experiment. This is because that we want to focus on investigating the influence of cross-sectional shape on rock deformation behaviors, and it is better to keep the loading condition simple. Class II failure type can only be obtained using the lateral or circumferential-strain-controlled loading, but it is simple to use the axial-strain-controlled loading in a numerical experiment. More importantly, using the axial strain-controlled loading might be more appropriate to reflect the actual loading condition in situ. For instance, the loading condition in a pillar is axial deformation controlled, and the in situ complete load–deformation curves of pillars obtained by field testing so far are all Class I [34,35]. Strain-softening behavior is assumed in the numerical experiment to capture the post-peak behaviors of different crosssectionally shaped specimens in uniaxial compression tests. This study focuses on studying the post-peak stress–strain curves of rocks, and the simulation of the explicit fracturing process is
ram steel platen
rock specimen
steel platen ram
(a)
(1, 154.63) (3.5, 182.44)
(b)
(c)
Fig. 2. 3D models of (a) cylinder, (b) square prism, and (c) rectangular prism specimens for uniaxial compression test simulation.
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3.2. Modeling results
160 cylinder (laboratory) square prism (laboratory) cylinder (simulation) square prism (simulation) rectangular prism (simulation)
UCS (MPa)
150
Fig. 5 presents the complete stress–strain curves of different cross-sectionally shaped specimens in uniaxial compression. It is seen that the peak strengths of different cross-sectionally shaped specimens are similar to each other. However, the post-peak behaviors of the specimens are influenced by their crosssectional shapes. The post-failure slope of the square prism specimen is the smallest, the rectangular prism specimen is the largest, and the cylinder specimen is between these two.
140
130 4. Discussions
120 1
2
3
4.1. End effect for peak load
Slenderness
c
c Fig. 4. Illustration of end effect in specimens with different slenderness in rock uniaxial compression tests (redraw based on [28]).
Cohesion (MPa)
Plastic strain
19 16 9.5 1
0 0.008 0.035 0.090
120 100
50 mm
80 60
70 mm
40
35 mm
Table 3 Strain-softening parameters of rock used in numerical modeling.
50 mm 140
50 mm
beyond the scope this research. Because the fracturing process during the post-peak deformation stage leads to a cohesion loss [36], the strain-softening behavior of the rock can be defined by degrading the rock’s cohesion strength as a function of plastic strain (defined in Table 3). A fundamental problem of incorporating strain-softening material models in a standard continuum model such as ABAQUS is the inherent mesh size sensitivity, which is an unsettled problem in numerical modeling in continuum mechanics [37,38]. The mesh sizes for all the specimens with different cross-sectional shapes are relatively fine and the mesh shapes (hexahedron) are the same in each model. In this fashion, numerical errors associated with mesh size are minimized and a relative comparison of the simulation results can be made because all the models are within the same order of numerical errors.
The modeling results demonstrate that there is no significant influence of cross-sectional shape on the peak strength of rocks, which is supported by the laboratory test results [22]. An advantage of numerical modeling is that it allows a detailed investigation of the mechanism that governs the observed phenomenon. Fig. 6 presents the distributions of confined elements (r3 < 0, compressive stress is negative in ABAQUS) in the specimens (the 1st row), the contours of the minimum principal stress (r3) on the side, top, and two vertical surfaces revealed from a quarter cut of the specimens (the 2nd row), and the distribution of confined elements with relatively high confinements (r3 < 1.5 MPa) in the specimens (the 3rd row) at peak load. The distributions of the confined elements and the r3 contours in different cross-sectionally shaped specimens are similar at their peak loads because their slenderness and contact friction are the same. Consequently, specimens with different cross-sectional shapes have similar peak strengths. The confined elements whose confinements (absolute values) are greater than 1.5 MPa at peak load are lumped near the specimen’s ends, forming cone-shaped confined zones, and the tops of which are approximately 15 mm from the specimen’s ends. The elements in these ‘‘highly” confined zones have higher peak strengths due to increased confinements. Therefore, hourglassing failure mode is normally observed in uniaxial compression tests [39,40]; it can also be observed in pillar failure in the field [41] because of strong end constraint to the pillars. Both the numerical simulation and the laboratory test results show that the UCS of a square prism specimen with a slenderness of 2.0 is higher than that of a cylinder specimen of the same slenderness. Table 4 presents the percentages of the confined elements (r3 < 0) to the total number of elements in each specimen, along with the average minimum principal stresses (r3) in the whole specimen, in the portion of 15 mm to the specimen’s end and in
Stress (MPa)
Fig. 3. UCS of circular, square, and rectangular cross-sectionally shaped specimens with different slenderness obtained from uniaxial compression test simulation. Laboratory test results for a slenderness of 2.0 are also shown [23].
20 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strain (%) Fig. 5. Stress–strain curves of different cross-sectionally shaped specimens in uniaxial compression.
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Confined elements with 3 < 0 in specimens (compressive stress is negative in ABAQUS)
Contour of (unit: Pa)
3
Confined elements with 3 -1.5 MPa in specimens (compressive stress is negative in ABAQUS)
(a)
(b)
(c)
Fig. 6. Distributions of confined elements at two r3 thresholds and r3 contours in different cross-sectionally shaped specimens at peak load: (a) cylinder, (b) square prism, and (c) rectangular prism specimens.
Table 4 Percentages of confined elements to the total number of elements and average minimum principal stresses (r3) in various portions of the specimens at peak load.
Total number of elements Number of confined elements Percentage of confined elements at peak load Average r3 (MPa) in the whole specimen at peak load Average r3 (MPa) in the portion of 15 mm to the specimen end at peak load Average r3 (MPa) in the 70 mm middle portion of the specimen at peak load
Cylinder specimen
Square prism specimen
Rectangular prism specimen
8844 6836 77%
9537 8531 89%
9108 7178 79%
1.22
1.40
1.15
3.95
4.00
3.70
0.04
0.27
0.04
the 70 mm middle portion of the specimen at peak load. The r3 value in each confined element due to end constraint controls the overall strength enhancement of the specimen. The square prism specimen has the highest percentage of confined elements in its volume and the highest confinement (absolute value, or the lowest r3 value) than the other two specimens. This is why it has a higher UCS than the other two specimens. For all specimens,
the average confinement in the specimen’s end zones is much higher than that in the middle portions of the specimens, showing that the end effect has a large influence on the stress condition near the specimen’s end which can influence the peak load [42]. 4.2. End effect for post-peak behavior As seen in Fig. 5, the post-peak behavior of a specimen depends on its cross-sectional shape. Fig. 7 presents the distributions of confined elements at two r3 thresholds and the r3 contours in the three cross-sectionally shaped specimens at axial strain e = 0.5% in the post-peak deformation stage. It is seen that due to the end effect, the square prism specimen has the highest local confinement in the post-peak deformation stage and the rectangular prism specimen has the lowest local confinement. Table 5 presents the percentages of confined elements and the average minimum principal stresses in the three cross-sectionally shaped specimens at e = 0.5%. The results show that locally confined elements in a specimen are reduced (comparing with the numbers shown in Table 4) due to the progressive failure of the rocks, and the degrees of decrease of the confined elements in the postpeak deformation stage is different for the three specimens. The square prism specimen has the highest percentage of confined elements and the smallest tensile stress r3; hence, its strength reduction in the post-peak deformation stage is the smallest. In contrast,
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Confined elements ( 3 < 0) in specimens (compressive stress is negative in ABAQUS)
Contour of (unit: Pa)
3
Confined elements ( 3 -1.5 MPa) in specimens (compressive stress is negative in ABAQUS)
(a)
(b)
(c)
Fig. 7. Distributions of confined elements at two r3 thresholds and r3 contours in different cross-sectionally shaped specimens at e = 0.5% in the post-peak deformation stage: (a) cylinder, (b) square prism, and (c) rectangular prism specimens.
Table 5 Percentages of confined elements to the total number of elements and average minimum principal stress (r3) in various portions of the specimens at e = 0.5% in the post-peak deformation stage.
Total number of elements Number of confined elements Percentage of confined elements at e = 0.5% Average r3 (MPa) in the whole specimen at e = 0.5% Average r3 (MPa) in the portion of 15 mm to the specimen end at e = 0.5% Average r3 (MPa) in the 70 mm middle portion of the specimen at e = 0.5%
Cylinder specimen
Square prism specimen
Rectangular prism specimen
8844 3978 45%
9537 5582 59%
9108 3670 40%
1.54
0.12
2.68
1.36
0.18
3.05
1.62
0.09
2.62
the rectangular prism specimen has the lowest percentage of confined elements and the largest tensile stress r3, and its post-peak curve is the most brittle. As can be noticed from Tables 4 and 5, the average r3 in each specimen changes from compression at peak load to tension in the post-peak loading stage at e = 0.5%. Fig. 8 presents the evolu-
tion of tensile stress (r3 > 0) at different loading stages in the three cross-sectionally shaped specimens. Tensile stress becomes more prominent with the increase of axial deformation in the postpeak deformation stage. At peak load, tensile stress appears only near the edge in the central areas of the specimens. In the postpeak deformation stage, the minimum principal stresses in the cone-shaped confined zones seen before at peak load gradually become tensile as deformation increases. As mentioned above, the lateral expansion rate of the rock is higher than that of the steel platens, and the elements under the platens are restricted by the steel platens and are subjected to compressive stresses if there is no rock failure. On the other hand, tensile stresses are generated in these elements in the post-peak deformation stage because their lateral contraction rate is higher than that of the steel platens as the load decreases. However, the evolutions of the minimum principal stress in the post-peak loading stage are different for different cross-sectionally shaped specimens. The rectangular prism specimen shows the largest tensile stress zones in the specimen, while the cylinder and the square prism specimens show relatively smaller tensile stress zones. The slenderness effect becomes more significant in uniaxial compression tests when the slenderness of a specimen decreases (Fig. 4). In essence, it is the change of the geometry of the specimen that leads to an increase or a decrease of the influence of the end constraint on the rock deformation behaviors. In other words, the
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tensile 3 (unit: Pa) at peak load
3 at = 0.4% in the post-peak deformation stage
3 at = 0.5% in the post-peak deformation stage
3 at = 0.6% in the post-peak deformation stage
(a)
(b)
(c)
Fig. 8. Evolutions of tensile minimum principal stresses (r3 > 0 contours) in different cross-sectionally shaped specimens at different deformation stages: (a) cylinder, (b) square prism, and (c) rectangular prism specimens.
geometry effect of a rock specimen is essentially manifested by the distribution of confined zones caused by the end effect. The crosssectional shape effect can be seen clearly in the post-peak deformation stage, when progressive failure of elements gradually decreases the effective bearing area of a specimen and consequently changes the effective specimen geometry. Failures occur first locally in elements whose confinement is small, and the locations of these elements depend on the crosssectional shape of the specimen. The degree of the effective specimen geometry change is also dependent on the specimen’s crosssectional shape. Therefore, the confined zones in a specimen are more dependent on the specimen’s cross-sectional shape in the post-peak deformation stage and so is the strain-softening curve. For instance, the elements of the rectangular prism are nonuniformly confined (Fig. 7c) and there are many of them subjected to tensile minimum principal stresses (Fig. 8c); hence, the overall post-peak strength of the rectangular prism specimen is lower than these of the other two specimens. The cylinder and square prism specimens are uniformly restricted (Fig. 7a and b) and there are less elements whose minimum principal stresses are tensile within the specimens (Fig. 8a and b) in the post-peak deformation stage. Therefore, the post-peak strengths of these two specimens are relatively high. Furthermore, the post-peak strengths of the cylinder and the square prism specimens are different because
the r3 contours in the two specimens are not the same. Thus, it is concluded that the cross-sectional shape affects the post-peak behavior of rocks. 4.3. Hoop tension effect Because a continuum numerical tool is used in this study, it is not possible to capture explicitly the crack initiation and propagation processes that lead to discontinuous failure of rocks [43]. Although the influence of hoop tension on a dilating crack cannot be simulated explicitly, it does not inhibit us from comparing the confined elements and the r3 values in the cylinder specimen with those in the prism specimens. Based on the hypothesis of the hoop tension theory, it can be reckoned that the number of confined elements and the average compressive r3 value in a cylinder specimen should be greater than that in a square or a rectangular prism specimen. Hence, the strength of the cylinder specimen should be the highest because the circular shape favors hoop tension development. The simulation results show that for the same slenderness (defined by H/D and H/W for the cylinder and square prism specimens, respectively) of different shaped specimens, it is the square prism specimen rather than the cylinder specimen that has the highest percentage of confined elements and the highest confine-
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ment (or the lowest tension), both at peak and in the post-peak loading stages (Tables 4 and 5). In addition, both the laboratory and the numerical results show that the strength of the square prism specimen is higher than that of the cylinder specimen (Fig. 3). The strength of the rectangular prism specimen is also very close to that of the cylinder and the square prism specimens. Thus, it is concluded that hoop tension contributes little to affecting rock strength. A major difference between the strengths obtained from laboratory tests (using specimens) and field data interpretation is that the interpreted field rock strength depends on the interpretation model used. When a numerical model reflects the field condition better, the interpreted rock strength of massive rocks is closer to that obtained from laboratory tests and this was demonstrated by Cai and Kaiser [44] using the Mine-by tunnel case history. Although the height (100 mm) and the slenderness (2.0) of the cylinder (H/D) and the square prism (H/W) specimens are the same in the laboratory tests, the cross-sectional area of the square prism specimen (50 50 = 2500 mm2) is larger than that of the cylinder specimen (p 252 = 1963 mm2). In such a case, the equivalent diameter of the square specimen is 56.4 mm, resulting in an equivalent slenderness of H/D = 1.77 that is squatter than the cylinder specimen with H/D = 2.0. Therefore, the end effect has a greater influence on the strength of the square prism specimen. Additional numerical simulation results shown in Fig. 9 reveal that if a square prism specimen with a width of 44 mm is used in uniaxial compression tests, it gives almost the same cross-sectional area (1936 mm2) as that of a cylinder specimen with a diameter of 50 mm. In such a case, the UCS of the square prism specimens is very close to that of the cylinder specimens.
5. Conclusions The influence of the cross-sectional shape of a rock specimen on its strength in uniaxial compression tests was studied using the numerical experimental approach. It is found that the crosssectional shape has a very small influence on the peak strength of rocks, but it affects the post-peak behavior of rocks significantly. The study revealed that it is the relation between the distribution of confined and tensile zones during the post-peak deformation process that actually contributes to the cross-sectional shape effect. It is shown in our modeling results, which are supported by laboratory test results, that the strength of a square prism specimen is actually higher than that of a cylinder specimen for the
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cylinder (D = 50 mm) square prism (W = 44 mm)
UCS (MPa)
150
140
130
120 1
2
3
Slenderness Fig. 9. UCS of circular (diameter D = 50 mm) and square (width W = 44 mm) crosssectionally shaped specimens with different slenderness obtained from uniaxial compression test simulation.
same slenderness calculated by the specimen height divided by the width (square specimen) or the diameter (cylinder specimen). The strength of a cylinder specimen is not significantly higher than that of a rectangular prism specimen, implying that hoop tension contributes little to affecting rock strength. The measurable strength difference between the cylinder specimens and the square prism specimens observed in the laboratory test and the numerical simulation results is attributed to the relatively large cross-section areas of the square prism specimens used. If the slenderness of a square prism specimen is calculated by dividing the specimen height by an equivalent diameter showing the same cross-section area as a cylinder specimen, then the square prism specimen will have the same strength as the cylinder specimen. Insight gained from this study can assist in interpreting laboratory test results more objectively. One suggestion is to use the equivalent diameter of a non-circular cross-section to define the slenderness of a specimen. In this fashion, consistent presentation of the test results can be achieved. For the same cross-section area and slenderness, a rectangular pillar has the most brittle post-peak behavior. Hence, it is beneficial not creating rectangular crosssectionally shaped pillars in underground mines to avoid violent pillar failure. Acknowledgement The work presented in this paper was financially supported by NSERC (Natural Science and Engineering Research Council of Canada). References [1] Hoek E, Brown ET. Underground excavations in rock; 1980. [2] Hudson JA, Brown ET, Fairhurst C. Shape of the complete stress-strain curve for rock. Stability of Rock Slopes: ASCE; 1972. p. 773–95. [3] Das MN. Influence of width/height ratio on post-failure behaviour of coal. Int J Min Geol Eng 1986;4:79–87. [4] Fairhurst C, Hudson J. Draft ISRM suggested method for the complete stressstrain curve for intact rock in uniaxial compression. Int J Rock Mech Min Sci 1999;36:279–89. [5] Bobet A. Influence of the loading apparatus on the stresses within biaxial specimens. ASTM Geotech Test J 2001;24:256–72. [6] Mogi K. Fracture and flow of rocks under high triaxial compression. J Geophys Res 1971;76:1255–69. [7] Takahashi M, Koide H. Effect of the intermediate principal stress on strength and deformation behavior of sedimentary rocks at the depth shallower than 2000 m. In: ISRM international symposium. International Society for Rock Mechanics; 1989. [8] Haimson B, Chang C. A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. Int J Rock Mech Min Sci 2000;37:285–96. [9] He M, Miao J, Feng J. Rock burst process of limestone and its acoustic emission characteristics under true-triaxial unloading conditions. Int J Rock Mech Min Sci 2010;47:286–98. [10] Sheng D, Westerberg B, Mattsson H, Axelsson K. Effects of end restraint and strain rate in triaxial tests. Comput Geotech 1997;21:163–82. [11] Lockner D, Byerlee J, Kuksenko V, Ponomarev A, Sidorin A. Quasi-static fault growth and shear fracture energy in granite. Nature 1991;350:39–42. [12] Zipf Jr RK. Analysis of stable and unstable pillar failure using a local mine stiffness method. In: Proceedings of the workshop on coal pillar mechanics and design. Washington (DC): U.S. Bureau of Mines; 1992. p. 128–43. [13] Hoek E, Brown E. Practical estimates of rock mass strength. Int J Rock Mech Min Sci 1997;34:1165–86. [14] Joseph T, Barron K. The post-failure characteristics of rock. CIM Bull 2003;96:66–74. [15] Cai M, Kaiser P, Tasaka Y, Minami M. Determination of residual strength parameters of jointed rock masses using the GSI system. Int J Rock Mech Min Sci 2007;44:247–65. [16] Salamon MDG. Stability, instability and design of pillar workings. Int J Rock Mech Min Sci Geomech Abstr 1970;7:613–31. [17] Wiles T. Loading system stiffness-a parameter to evaluate rockburst potential. In: First international seminar on deep and high stress mining, Australian Centre for Geomechanics. Perth; 2002. [18] Henkel D, Gilbert G. The effect measured of the rubber membrane on the triaxial compression strength of clay samples. Geotechnique 1952;3:20–9. [19] Meyerhof G. The bearing capacity of concrete and rock⁄. Mag Concr Res 1953;4:107–16.
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