Introduction

To study the mechanical properties of jointed rock mass is challenging, as the strength and deformability of rock mass are significantly influenced by the distribution of joints. The failure modes of rock mass also vary markedly with the variation of joint orientation, spacing, and number of joints (Zhao et al. 2015). Joint spacing and joint connectivity rate are the two critical characteristics of joint geometry. At present, some researchers have carried out experimental and numerical studies on the influence of joint spacing and joint connectivity rate on the mechanical properties of jointed rock mass.

Chen et al. (2012) investigated the combined influence of joint inclination and joint continuity factor on deforming behavior of jointed rock mass for gypsum specimens with a set of non-persistent open flaws under uniaxial compression. Bahaaddini et al. (2013) conducted numerical investigation on the effect of joint geometrical parameters on the mechanical properties of a non-persistent jointed rock mass under uniaxial compression, considering joint orientation, joint spacing, joint persistency, and joint aperture. Chen et al. (2013) carried out a series of uniaxial compression tests on gypsum specimens with regularly arranged multiple parallel pre-existing joints, to study the dependence of cracking process of jointed rock mass on joint orientation and joint continuity factor. Bahaaddini et al. (2015) studied the effect of joint geometrical parameters of non-persistent rock mass on uniaxial compressive strength (UCS) and the deformation modulus using PFC3D. Fan et al. (2015a, b) studied the crack initiation stress and strain of jointed rock mass containing multiple cracks under uniaxial compressive loading using PFC3D, considering effects of joint inclination and joint continuity factor. Yang et al. (2016) conducted numerical simulation of rock blocks with non-persistent open joints under uniaxial compression using the particle flow modeling method. Liu et al. (2017) experimentally investigated the influences of joint dip angle, joint persistency, joint density, and joint spacing on the fatigue of synthetic jointed rock models. Yang et al. (2017) carried out a physical experiment to better understand the mechanical behavior of jointed rock mass, considering the effects of joint interaction, joint spacing, and joint dip angle. Chen et al. (2018) numerically studied the effect of joint strength mobilization on the mechanical behavior of jointed rock mass using PFC2D. Yang and Qiao (2018) investigated the shear behavior and failure mechanism of granite material bridges between discontinuous joints subjected to direct shearing using a flat-joint modeling approach.

The above scholars only studied the variation of mechanical properties of specimens containing non-persistent joints with the geometric parameters of the joints by conducting experiments or numerical simulations. Few scholars have proposed corresponding formulas to analyze the variation of compressive strength of jointed rock mass with joint spacing and joint connectivity rate.

In this study, various compression tests on preset intermittent fractured specimens were performed, in order to understand the effects of joint spacing and joint connectivity rate on the strength and deformation of jointed rock mass systematically. The fitting formulas for the variations of the UCS and elastic modulus of jointed rock mass with joint spacing and connectivity rate are proposed.

Experimental setup

Specimen size and arrangement of joints

The specimens used in this study have dimensions of 10 cm × 10 cm × 10 cm. Non-persistent joints were created by inserting planks with thickness of 1 mm into a mold after setting the material. The jointed rock mass specimen with joint connectivity rates of 0, 0.2, 0.4, 0.6, and 0.8 was prepared. The joint spacing is maintained at 30 mm, as shown in Fig. 1.

Fig. 1
figure 1

Jointed rock mass specimens with different joint connectivity rates (unit: mm). k = 0.2 (a), k = 0.4 (b), k = 0.6 (c), k = 0.8 (d)

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The jointed rock mass specimen with joint spacings of 10 mm, 20 mm, 30 mm, and 40 mm and connectivity rate of 0.4 is shown in Fig. 2.

Fig. 2
figure 2

Jointed rock mass specimens with different joint spacings (unit: mm). s = 10 mm (a), s = 20 mm (b), s = 30 mm (c), s = 40 mm (d)

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Preparation of model materials and specimens

Specimens were cast by pouring cement mortar with water/cement ratio of 0.65 into the mold. Cement No. 425 used in this study was produced by the China Building Materials Academy. The ISO standard sand was used which is conformed to Chinese National Standard GB/T17671-1999. Its particle size ranges from 0.5 to 1.0 mm.

After the initial setting of the cement mortar, wood plank was inserted into position as marked by the short-line shown in Figs. 1 and 2. The length of the wood planks is 10 cm, and the wood planks remained permanently in the specimens.

Test machine

Uniaxial compression tests were conducted using WAW-600C universal testing machine. The maximum displacement loading rate of the testing machine is 60 mm/min, and the maximum compression load is 600 kN. A picture of the testing machine is shown in Fig. 3.

Fig. 3
figure 3

WAW-600C universal testing machine

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In this study, the uniaxial compression tests were performed under displacement loading rate of 0.6 mm/min, approximately to a strain rate of 10−4 s−1.

Test groups

When conducting compression test on each jointed rock mass specimen, the loading was applied in a direction that was either parallel or perpendicular the joint plane, as shown in Figs. 1 and 2. As for the jointed rock mass specimen with the joint connectivity rate of 0.4 and the joint spacing of 30 mm, the loading direction was set either perpendicular or parallel to the joint plane as shown in Fig. 4. Angle β of joint plane is considered as 0° when the loading direction is perpendicular to the joint plane and 90° when the loading direction is parallel to the joint plane.

Fig. 4
figure 4

The relationship between the loading direction and the joint plane. β = 0° (a), β = 90° (b)

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The uniaxial compression tests were divided into several groups as shown in Table 1. The uniaxial compression tests were classified according to three factors, i.e., the joint connectivity rate, the joint spacing, and the geometric relationship between the loading orientation and the joint plane.

Table 1 Experimental groups of uniaxial compression tests
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Test results of jointed rock mass with varied joint connectivity rates

Peak UCS of jointed rock mass

The comparison of UCS of jointed rock mass specimen with various joint connectivity rates, k, when the loading direction is perpendicular or parallel to the joint plane, is shown in Fig. 5.

Fig. 5
figure 5

The peak uniaxial compressive strengths of jointed rock mass specimen with different joint connectivity rates, k

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Test results show that the peak uniaxial compressive strengths of jointed rock mass specimen with joint connectivity rates of 0, 0.2, 0.4, 0.6, and 0.8, when the loading direction is parallel to the joint plane, are greater than that when the loading direction is perpendicular to the joint plane. This coincides with the conclusions obtained by Bahaaddini et al. (2013), Fan et al. (2015a, b), Bahaaddini et al. (2016), Yang et al. (2016), Yang et al. (2017), and Chen et al. (2018).

When the joint dip angle is low (e.g., β = 0° and 15°), with the axial compression increasing, wing cracks developed from the pre-existing joint tips. The wing cracks are tensile cracks that initiated at an angle from the joint tips and propagated in the direction of maximum compression, and macro failure planes going through the entire specimens fail the jointed rock mass with the development of wing cracks (Yang et al. 2017). When the joint dip angle is 90°, although cracks occurred to the rock bridges between pre-existing joints and the joint plane, the joint plane has negligible effect on the failure behavior of the jointed rock mass, and this failure model can be defined as intact material failure mode (Yang et al. 2017). Therefore, the peak uniaxial compressive strength of jointed rock mass specimen with preset non-persistent joints when the loading direction is parallel to the joint plane is higher than that when the loading direction is perpendicular to the joint plane.

The difference of peak UCS of jointed rock mass specimen between when β is 90° and 0° increases with joint connectivity rate. Therefore, the anisotropy of UCS of jointed rock mass tends to increase with joint connectivity rate. The equivalent peak uniaxial compressive strengths of jointed rock mass specimen with different joint connectivity rates are shown in Fig. 6. In this figure, σJR and σR are the peak uniaxial compressive strengths of jointed rock mass specimen with and without joints, respectively.

Fig. 6
figure 6

Variation curves of equivalent peak UCS of jointed rock mass with joint connectivity rate when β are 90° and 0°

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In Fig. 6, it can be noted that the peak UCS of jointed rock mass model decreases rapidly with joint connectivity rate when the loading direction is perpendicular to the joint plane. The relationship curve between the equivalent peak UCS and the joint connectivity rate is approximately linear when the loading direction is parallel to the joint plane. Therefore, the effect of joint connectivity rate on the peak UCS, when the loading direction is parallel to the joint plane, is relatively less than that on when the loading direction is perpendicular to the joint plane.

Chen et al. (2011) suggested that the nonlinear variation law of equivalent peak UCS of jointed rock mass with joint connectivity rate could be analyzed by the reciprocal expression of power function:

$$ {\sigma}_{JR}/{\sigma}_R=1/\left(1+{pk}^q\right) $$
(1)

where p and q are the fitting parameters.

Wang et al. (2011) suggested that the nonlinear variation law of equivalent peak UCS of jointed rock mass with joint connectivity rate could be analyzed by quadratic polynomial function:

$$ {\sigma}_{JR}/{\sigma}_R={ak}^2+ bk+1 $$
(2)

where a and b are the fitting parameters.

Equations (1) and (2) are firstly implemented into Origin software. Then, the values of the parameters of Eqs. (1) and (2) are obtained by using the nonlinear least square method. The same procedures were utilized in this paper for all subsequent formulas.

The test results shown in Fig. 6 were analyzed by using Eqs. (1) and (2), respectively. Figure 7 shows the results of Eq. (1) proposed by Chen et al. (2011) in comparison with Eq. (2) proposed by Wang et al. (2011). The square of the correlation coefficient (R2) and the sum of least error square (Q) are chosen as discrimination criteria (Yang and Cheng 2011). The back-analyzed parameters of fitting curves between the equivalent peak UCS and joint connectivity rate, and the coefficients of R2 and the coefficients of Q are listed in Tables 2 and 3.

Fig. 7
figure 7

The contrast result between Eq. (1) proposed by Chen et al. (2011) and Eq. (2) proposed by Wang et al. (2011). β = 0° (a), β = 90° (b)

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Table 2 Fitting parameters of Eq. (1) proposed by Chen et al. (2011)
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Table 3 Fitting parameters of Eq. (2) proposed by Wang et al. (2011)
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The model agrees well with the experimental result when R2 tends to 1 and Q approaches zero. Therefore, the fitting accuracy of Eq. (1) proposed by Chen et al. (2011) is higher than that of Eq. (2) proposed by Wang et al. (2011) in fitting the test data when the loading direction is perpendicular to the joint plane. The fitting accuracy of Eq. (1) proposed by Chen et al. (2011) is close to that of Eq. (2) proposed by Wang et al. (2011) in fitting the test data when the loading direction is parallel to the joint plane.

Equation (1) is the reciprocal expression of power function of joint connectivity rate, and the fitting accuracy of Eq. (1) is higher than that of Eq. (2). Therefore, the fitting accuracy of the reciprocal expression of Eq. (2) may be higher than that of Eq. (2). And the fitting accuracy of cubic polynomial function may also be higher than that of Eq. (2). These need further investigation to confirm.

Because of this, we used various functions to analyze the nonlinear variation law of equivalent peak UCS of jointed rock mass with joint connectivity rate. In this study, the quadratic polynomial function, the cubic polynomial function, and the fourth-degree polynomial function, in association with the reciprocal expression of the quadratic polynomial function, the reciprocal expression of the cubic polynomial function, and the reciprocal expression of the fourth-degree polynomial function were employed subsequently.

The reciprocal expression of the quadratic polynomial function is

$$ {\sigma}_{JR}/{\sigma}_R=1/\left({ak}^2+ bk+1\right) $$
(3)

The cubic polynomial function is

$$ {\sigma}_{JR}/{\sigma}_R={ak}^3+{bk}^2+ ck+1 $$
(4)

The reciprocal expression of the cubic polynomial function is

$$ {\sigma}_{JR}/{\sigma}_R=1/\left({ak}^3+{bk}^2+ ck+1\right) $$
(5)

The fourth-degree polynomial function is

$$ {\sigma}_{JR}/{\sigma}_R={ak}^4+{bk}^3+{ck}^2+ dk+1 $$
(6)

The reciprocal expression of the fourth-degree polynomial function is

$$ {\sigma}_{JR}/{\sigma}_R=1/\left({ak}^4+{bk}^3+{ck}^2+ dk+1\right) $$
(7)

The R2 and Q of Eqs. (2)–(7) when fitting the test data shown in Fig. 6 are given in Tables 4 and 5.

Table 4 The coefficients of R2 using Eqs. (2)–(7) fitting
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Table 5 The coefficients of Q using Eqs. (2)–(7) fitting
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The fitting accuracy of multiple polynomial function can be significantly improved by increasing the number of times and the number of parameters of multiple polynomial functions when fitting the test data shown in Fig. 6. The R2 obtained by the fourth-degree polynomial function is 1.0. For the quadratic polynomial function or the cubic polynomial function, R2 will be improved and Q will be reduced when using its reciprocal expression fitting. The fitting accuracy of Eq. (6) or (7) seems to be high. The fitting curves of the test results using Eqs. (2)–(7) are shown in Fig. 8.

Fig. 8
figure 8

The fitting curves of Eqs. (2)–(7). β = 0° (a), β = 90° (b)

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Although the fitting accuracy of Eq. (6) or (7) is almost identical, the fitting curve of Eq. (6) is closer to the test data, and the fitting curve of Eq. (7) is a little deviated from the test data when k is in the range of 0 to 0.2, when β is 0°. Therefore, it is preferable to use Eq. (6) for analysis. When using Eq. (6), we have:

  1. 1.

    When β is 0°, the relation can be expressed by

$$ {\sigma}_{JR}/{\sigma}_R=-19.84365{k}^4+34.13722{k}^3-17.99275{k}^2+1.6806k+1 $$
(8)
  1. 2.

    When β is 90°, the relation can be expressed by

$$ {\sigma}_{JR}/{\sigma}_R=-6.5546{k}^4+10.59291{k}^3-5.21746{k}^2+0.56555k+1 $$
(9)

The fitting accuracy of Eq. (1) is higher than that of Eq. (2) when fitting the nonlinear variation law of equivalent peak UCS of jointed rock mass with joint connectivity rates in this study, whilst the fitting accuracy of the fourth-degree polynomial function is higher than that of Eq. (1).

Therefore, the multiple polynomial function is appropriate to analyze the nonlinear variation law of equivalent peak UCS of jointed rock mass with joint connectivity rate. The fitting accuracy of multiple polynomial function can be improved by increasing the number of times and the number of parameters, whilst the number of times and the number of parameters of multiple polynomial function need to be determined according to the lab and field test results of practical project.

Elastic modulus

The elastic modulus is calculated as a secant modulus from the origin to a defined point on the uniaxial compressive stress-strain curve, within 30% of the specimen’s peak UCS. The comparison of elastic modulus of jointed rock mass model with different joint connectivity rates, when the loading direction is perpendicular and parallel to the joint plane, is shown in Fig. 9. Variation curves of equivalent elastic modulus with joint connectivity rate of jointed rock mass specimen are shown in Fig. 10. In Fig. 10, EJR and ER are elastic moduli of jointed rock mass models with and without joint, respectively.

Fig. 9
figure 9

The elastic modulus of jointed rock mass specimen with different joint connectivity rates

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Fig. 10
figure 10

Variation curves of equivalent elastic modulus with joint connectivity rate of jointed rock mass when β are 90° and 0°

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The elastic modulus of jointed rock mass specimen decreases substantially with joint connectivity rate when the loading direction is perpendicular to the joint plane. The decreasing rate of equivalent elastic modulus of jointed rock mass specimen with joint connectivity rate when the loading direction is perpendicular to the joint plane is higher than that when the loading direction is parallel to the joint plane.

Test results of jointed rock mass with varied joint spacing

Peak UCS of jointed rock mass model

The uniaxial compressive strength of jointed rock mass specimen with different joint spacings, when the loading direction is perpendicular or parallel to the joint plane, is shown in Fig. 11.

Fig. 11
figure 11

The peak uniaxial compressive strengths of jointed rock mass specimen with different joint spacings

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In Fig. 11, it is clear that the peak uniaxial compressive strength of jointed rock mass model with joint spacings of 10 mm, 20 mm, 30 mm, and 40 mm, when the loading direction is parallel to the joint plane, is greater than those when the loading direction is perpendicular to the joint plane. The peak uniaxial compressive strengths of jointed rock mass models when β is 90° and 0° both increase with joint spacing.

The jointing index m is defined as the ratio of the length of jointed rock mass specimen to the joint spacing (Chen et al. 2014):

$$ m=L/s $$
(10)

where L is the length of jointed rock mass model, and s is the joint spacing.

In Eq. (10), it is known that the jointing index m is inversely proportional to the joint spacing. The denser the joint, the smaller the joint spacing and the greater the jointing index m is. The length of jointed rock mass specimen of 100 mm is adopted in this study; thus, the jointing indexes of jointed rock mass specimen with joint spacings of 100 mm, 40 mm, 30 mm, 20 mm, and 10 mm are calculated as 1.0, 2.5, 3.33, 5, and 10, respectively. Variation curves of equivalent peak uniaxial compressive strengths of jointed rock mass with jointing index, m, are shown in Fig. 12.

Fig. 12
figure 12

Variation curves of equivalent peak uniaxial compressive strengths of jointed rock mass with jointing index

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The peak UCS of jointed rock mass specimen decreases drastically with jointing index when the loading direction is perpendicular to the joint plane. The decreasing rate of the peak UCS of jointed rock mass specimen when the loading direction is parallel to the joint plane is relatively smaller than that when the loading direction is perpendicular to the joint plane with jointing index.

Chen et al. (2014) suggested that the nonlinear variation law of equivalent peak UCS of jointed rock mass with jointing index can be analyzed using the power function proposed by Goldstein et al. (1966):

$$ {\sigma}_{JR}/{\sigma}_R=a+\left(1-a\right){m}^{-b} $$
(11)

where a and b are the fitting parameters; m is the jointing index.

Although the fitting curve of Eq. (11) agrees well with the test data when β is 0° as shown in Fig. 10, it cannot converge when β is 90° as shown in Fig. 10. For this, we proposed a power function to analyze the test data:

$$ {\sigma}_{JR}/{\sigma}_R=a+{m}^{-b}+{m}^{-c} $$
(12)

where a, b and c are the fitting parameters; m is the jointing index.

The fitting results are shown in Fig. 13, and the fitting parameters are listed in Table 6.

Fig. 13
figure 13

Fitting curves of power function proposed in this study when analyzing the equivalent peak UCS with jointing index

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Table 6 Parameters of power function proposed in this study when analyzing the equivalent peak UCS with jointing index
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Both the fitting curves of power function proposed in this study agree well with the test data when β are 0° and 90° as shown in Fig. 10. Thus, Eq. (12) is more suitable to analyze the variation curve of equivalent peak UCS of jointed rock mass with jointing index.

Elastic modulus

The elastic moduli of jointed rock mass specimen with different joint spacings, when the loading direction is perpendicular and parallel to the joint plane, are shown in Fig. 14.

Fig. 14
figure 14

The elastic moduli of jointed rock mass specimen with different joint spacings

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The elastic moduli of jointed rock mass specimen with joint spacings of 10 mm, 20 mm, 30 mm, and 40 mm when the loading direction is parallel to joint plane are greater than those when the loading direction is perpendicular to joint plane. The elastic moduli of jointed rock mass models when β are 90° and 0° increase with joint spacing. Variation curves of equivalent elastic modulus with jointing index are shown in Fig. 15.

Fig. 15
figure 15

Variation curves of equivalent elastic modulus with jointing index

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It can be noted that the elastic modulus of jointed rock mass specimen decreases rapidly with increasing jointing index m when the loading direction is perpendicular to the joint plane. The decreasing rate of elastic modulus of jointed rock mass specimen when the loading direction is parallel to the joint plane is relatively smaller than that when the loading direction is perpendicular to the joint plane with jointing index. In this figure, it is evident that the variations of equivalent elastic modulus of jointed rock mass models are nonlinear with jointing index m when β are 90° and 0°.

Chen et al. (2014) proposed the same power function as the form of Eq. (11) to analyze the variation curve of equivalent elastic modulus with jointing index:

$$ {E}_{JR}/{E}_R=c+\left(1-c\right){m}^{-d} $$
(13)

where c and d are the fitting parameters; m is the jointing index.

When using Eq. (12) to analyze the variation curve of equivalent elastic modulus with jointing index, the left side of Eq. (12) needs to be replaced by EJR/ER.

The results of Eq. (12) in contrast to Eq. (13) are shown in Fig. 16 for analyzing the variation curve of equivalent elastic modulus with jointing index. The parameters of Eqs. (12)–(13) with respect to equivalent elastic modulus and jointing index m of jointed rock mass model and the R2 and Q are listed in Tables 7 and 8.

Fig. 16
figure 16

The fitting results using Eqs. (12) and (13) when analyzing the variation curve of equivalent elastic modulus with jointing index. Eq. (13) proposed by Chen et al. (2014) (a), Eq. (12) proposed in this study (b)

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Table 7 Parameters of Eq. (13) when analyzing the variation curve of equivalent elastic modulus with jointing index
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Table 8 Parameters of Eq. (12) when analyzing the variation curve of equivalent elastic modulus with jointing index
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Both the fitting curves of Eqs. (12) and (13) agree well with the test data, and the R2 and Q of Eqs. (12) and (13) match well. Therefore, Eqs. (12) and (13) are applicable to analyze the variation curve of equivalent elastic modulus of jointed rock mass with jointing index.

Discussion

Equation (12) is mainly composed of two power functions and a constant. The fitting curve of Eq. (12) proposed in this study agrees well with the test data when analyzing the variation curve of equivalent peak UCS or equivalent elastic modulus of jointed rock mass with jointing index. However, the fitting accuracy of Eq. (12) may be improved by increasing the number of power function. Therefore, we expand the number of power function of Eq. (12), i.e., one, two, and three, respectively.

  1. 1.

    When the number of power function is 1, Eq. (12) is changed to be

$$ {\sigma}_{JR}/{\sigma}_R=a+{m}^{-b} $$
(14)
  1. 2.

    When the number of power function is 3, Eq. (12) is expanded as

$$ {\sigma}_{JR}/{\sigma}_R=a+{m}^{-b}+{m}^{-c}+{m}^{-d} $$
(15)

When using Eqs. (14)–(15) to analyze the variation curve of equivalent elastic modulus with jointing index, the left side of Eqs. (14)–(15) also need to be replaced by EJR/ER.

The test data shown in Figs. 11 and 14 are analyzed using Eqs. (12), (14), and (15), and the fitting results are shown in Fig. 17. The values of R2 of Eqs. (12), (14), and (15) are listed in Tables 9 and 10.

Fig. 17
figure 17

The fitting curves of Eqs. (12), (14), and (15). Equivalent peak UCS (a), equivalent elastic modulus (b)

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Table 9 R2 of Eqs. (12), (14), and (15) when analyzing the variation curve of equivalent peak UCS with jointing index
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Table 10 R2 of Eqs. (12), (14), and (15) when analyzing the variation curve of equivalent elastic modulus with jointing index
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The R2 of Eq. (12) is obviously higher than that of Eq. (14), but the R2 of Eq. (15) is little higher than that of Eq. (12) when analyzing the variation curves of equivalent peak UCS and equivalent elastic modulus with jointing index, when β are 90° and 0°. Therefore, Eq. (12) is appropriate to analyze the variation curves of equivalent peak UCS and equivalent elastic modulus with jointing index. The fitting accuracy of Eq. (12) can be improved by increasing the number of power function. However, the number of power function needs to be determined according to field test results of practical project.

Conclusions

Using the tests on fractured rock mass models in this context, the following conclusions can be drawn:

  1. 1.

    Both the peak UCS and the elastic modulus of jointed rock mass specimen when the loading direction is parallel to the joint plane are greater than those when the loading direction is perpendicular to the joint plane with the same joint connectivity rate and/or the same joint spacing.

  2. 2.

    The difference of peak UCS of jointed rock mass specimen, when β is 90° and 0°, increases with joint connectivity rate. It can be concluded that the anisotropy of UCS of jointed rock mass tends to increase with joint connectivity rate.

  3. 3.

    The decreasing rates of peak UCS and elastic modulus of jointed rock mass when the loading direction is perpendicular to the joint plane are higher than those when the loading direction is parallel to the joint plane with joint connectivity rate and/or jointing index.

  4. 4.

    The fourth-degree polynomial function proposed in this study is appropriate to analyze the nonlinear variation law of equivalent peak UCS of jointed rock mass with joint connectivity rate. However, the number of times and the number of parameters of multiple polynomial function need to be determined according to lab and field test results of practical projects.

  5. 5.

    The power function proposed in this study is suitable to describe the nonlinear variation laws of equivalent peak UCS and equivalent elastic modulus with jointing index, whilst the number of power function need to be determined according to site-specific test results of practical project.