1 Introduction

The mechanical characteristics of intermittently jointed rocks play a dominant role in the overall mechanical behavior of many mining and civil engineering structures, such as underground tunnels, bridge abutments, and road foundations. Since these rock structures are likely to be subjected to cyclic loading resulting from earthquakes, quarrying, and rockbursts, it is thus crucial to accurately characterize the fatigue properties and failure mechanism of intermittently jointed rocks for the rational design and long-term stability analysis of rock structures under different cyclic loading conditions.

Existing attempts to understand the mechanical properties of jointed rocks were mainly concentrated on static loading. Researchers have performed numerous physical experiments on rock models containing various joints, including static uniaxial compression tests (Kulatilake et al. 1997, 2001; Singh et al. 2002; Bahaaddini et al. 2013; Fan et al. 2015; Feng et al. 2017), biaxial or triaxial compression tests (Einstein and Hirschfeld 1973; Tiwari and Rao 2006; Prudencio and Van Sint 2007; Sagong et al. 2011), and direct shear tests (Lajtai 1969; Gehle and Kutter 2003; Zhang et al. 2006; Park and Song 2009; Bahaaddinia et al. 2014). They concluded that the static strength and the deformation behavior of jointed rocks were significantly affected by the geometrical parameters of the joints, involving joint length, orientation, spacing and density, etc. In addition, the fracture coalescence behavior of jointed rock specimens was systematically assessed under uniaxial or biaxial compression (Bobet and Einstein 1998; Wong and Chau 1998; Wong et al. 2001; Sagong and Bobet 2002; Wong and Einstein 2009a, b, c); they first identified the nature of the coalescence cracks and proposed associated coalescence classification schemes. Wong and Einstein (2009a, b, c) further improved the classification scheme to describe the newly identified crack types and features. Recently, combining the numerical approaches, the progressive failure process of rocks containing single, two, or multiple joints were revealed under static loading conditions (Wong et al. 2006; Zhang and Wong 2012, 2013; Bi et al. 2016; Cao et al. 2016).

Compared with efforts on studying static mechanical properties of rock materials, investigations of cyclic mechanism were mostly limited to intact rocks. Some early experimental studies revealed the hysteresis of the stress–strain curve of intact rocks under cyclic loading and reported that the plastic deformation gradually accumulates with increasing number of cycles (Burdine 1963; Attewell and Farmer 1973; Tao and Mo 1990; Ray et al. 1999). Subsequently, Bagde and Petroš (2005a) defined the term “fatigue” as the tendency of materials to fail or the process of damage accumulation under cyclic loading conditions. Researchers further reported that both the fatigue strength and the deformation modulus of intact rocks exponentially decreased with increasing cycles, and the fatigue mechanical properties were significantly affected by cyclic loading parameters, including cyclic frequency, maximum stress and amplitude (Bagde and Petroš 2005b; Fuenkajorn and Phueakphum 2010; Ma et al. 2013; Liu et al. 2017a). To describe the damage evolution of rock materials under cyclic loading, Xiao et al. (2009, 2010) derived an inverted-S fatigue damage model for intact rocks (Fig. 1a). They pointed out that the fatigue damage variable shows a three-phase development with increasing relative cycle (i.e., ratio of the current cycle number n to the total cycle number N), and it increased with the increase in maximum cyclic stress (Fig. 1b). In addition, the energy characteristics of intact rocks under cyclic loading were investigated (Hua and You 2001; Bagde and Petroš 2009); they reported that the sustained energy of intact rocks could be regarded as a rock characteristic, which affected the rock failure mechanism. Momeni et al. (2015) further reported that a dramatic decrease in the energy density (i.e., areas under the loading portion of the stress–strain curve ranging between minimum and maximum stress, the region ABC in Fig. 1c) occurred for the initial cycles, and the energy density increased with increasing amplitude or decreasing frequency (Fig. 1c, d).

Fig. 1
figure 1

Schematic of the fatigue damage variable and energy density of the intact rocks under cyclic loading reproduced from previous studies: a Xiao et al. (2009); b Xiao et al. (2010); c and d Momeni et al. (2015)

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Since the mechanical behavior of jointed rocks significantly differs from that of intact rocks, the fatigue properties of jointed rocks under cyclic uniaxial compression was also investigated (Brown and Hudson 1974; Prost 1988; Li et al. 2001; Liu et al. 2017b). Earlier research (Brown and Hudson 1974; Prost 1988) pointed out that the intermittently jointed rocks were extremely sensitive to cyclic loading and that the fatigue life of jointed rocks increased with decreasing maximum cyclic stress and amplitude. Recently, Li et al. (2001) reported that the cyclic loading frequency significantly affected the fatigue strength and residual strength of jointed rock specimens. Liu et al. (2017b) systematically revealed the influence of joint geometry configurations on the mechanical properties of intermittently jointed rocks under given cyclic loads.

However, according to the available literature, the fatigue responses of intermittently jointed rocks to cyclic loading, including the fatigue deformation characteristic, the fatigue energy and damage evolution, and the fatigue failure and progressive behavior, remain far from being thoroughly understood. The influence of cyclic loading parameters, namely different cyclic frequencies, maximum stresses, and amplitudes, on these fatigue properties have never been systematically investigated in the laboratory. It is thus the intention of this study to experimentally investigate the fatigue mechanical properties of intermittently jointed rock models under cyclic uniaxial compression with different loading parameters.

The remainder of this text is organized as follows. Section 2 introduces the experimental setup, including the specimen preparation, the test equipment, and the test schemes, followed by the determination of fatigue mechanical properties and experimental results of monotonic loading tests in Sect. 3. Section 4 comprehensively analyzes and discusses the experimental results of cyclic uniaxial compression tests, regarding the influence of the three cyclic loading parameters on the fatigue deformation characteristic, the fatigue energy and damage evolution, and the fatigue failure and progressive behavior. Section 5 summarizes the study.

2 Experimental Setup

2.1 Specimen Preparation and Test Equipment

Considering that directly fabricating joints in real rocks is difficult, artificial rock-like materials are thus widely used to prepare jointed rock models in the laboratory investigations ever since the late 1960s (Einstein et al. 1969; Einstein and Hirschfeld 1973; Shen et al. 1995; Bobet and Einstein 1998; Singha and Rao 2005; Wong and Einstein 2009a, b, c). In this study, the synthetic rock-like materials are prepared with fine sand, high-strength cement, water, silicon powder, and water-reducing agent at mass ratio of 1.2:1:0.35:0.15:0.015, respectively, as shown in Fig. 2. The preparation process of the intermittently jointed rock models is as follows: (i) assembling the specified jointed mold (length × thickness × height = 200 mm × 100 mm × 100 mm) and inserting the steel sheets (length × thickness × height = 15 mm × 0.4 mm × 150 mm) into corresponding slots pre-fabricated in jointed mold (Fig. 2); (ii) pouring the mixed synthetic materials into the assembled mold, and vibrating the mold with mixed materials for 2 min on a shaking table to minimize air bubbles inside the specimens; (iii) removing these steel sheets after 10 h of curing and disassembling the jointed mold after 24 h of curing; (iv) keeping the cast intermittently jointed rock models for 28 days in a curing room (at 20 °C, 95% relative humidity and an atmospheric pressure). To keep the mechanical properties of the jointed specimens as consistent as possible, the mix proportion of synthetic materials remains the same, and the preparation processes are carefully controlled.

Fig. 2
figure 2

Sketch of the components of synthetic materials and the mold for fabricating jointed rock models

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Figure 3a presents the joint geometry configuration selected in this study, which is determined by the number and position of steel sheets inserted in the jointed molds. All jointed rock models have the same geometry configuration in our tests, including the joint length a = 15 mm, the rock bridge length b = 20 mm, the spacing d = 24 mm, the dip angle θ = 45°, the persistence k = 0.318, and the intensity ρ = 6.75 × 10−3 mm−1. Note that the persistence is defined as the ratio of the sum of individual joint surface areas to the surface of a coplanar reference plane, and the intensity is defined as the total joint area per unit volume (Dershowitz and Einstein 1988). Typical prepared intermittently jointed rock specimens depicted in Fig. 3b, c show the MTS-793 Rock and Concrete Test System employed in our tests. This test system is composed of software and hardware components, which provides a closed-loop control of servo-hydraulic equipment. Consisting of a compression loading frame, an axial dynamic loading system, and a data acquisition system, this equipment is capable of performing uniaxial static and dynamic compression tests. The loading frame has a 2500 KN and a 2750 KN compression load capacity for static and dynamic tests, respectively. The loading mode can be automatically varied via the dynamic control transducers, and the axial deformation and load can be simultaneously recorded by the data acquisition system. Specifically, for cyclic tests, the loading waveform can be selected from ramp, square, and sinusoidal waves, and the loading frequency can be varied from 0.01 to 20 Hz. In addition, in our tests, high vacuum grease is utilized to lubricate the contact surfaces between the tested specimens and the loading plates, and a JAI SP-5000 M high-resolution industrial camera is used to monitor the cracking process of jointed specimens with images taken at a frame rate of 134 fps.

Fig. 3
figure 3

a Schematic of the joint geometrical parameters, b the prepared intermittently jointed rock models, and c the MTS-793 Rock and Concrete Test System

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2.2 Test Procedure and Schemes

To measure the static mechanical properties of the jointed rock models, monotonic uniaxial compression tests are first conducted on four jointed specimens in an axial displacement control mode with a strain rate of 5 × 10−5 s−1. The obtained static uniaxial compression strength provides the reference values of cyclic loading parameters. For the cyclic compression tests, a periodic sinusoidal waveform is specified, and the loading path is shown in Fig. 4, in which the following cyclic loading parameters are defined: T is the cycle period, the frequency F is defined as F = 1/T, σ max and σ min are the maximum and minimum cyclic stress, respectively, and σ amp = σ max − σ min and σ avr = (σ max + σ min)/2 denote the cyclic amplitude and the average stress, respectively. Initially, the jointed specimens are loaded from 0 to σ avr at the strain rate of 5 × 10−5 s−1, and then the cyclic loading and unloading are conducted with specified loading parameters.

Fig. 4
figure 4

Schematic of the loading path in cyclic uniaxial compression tests and the characteristic parameters of cyclic loading

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In this study, 30 cyclic uniaxial compression tests with different loading parameters are performed to investigate the influence of cyclic loading parameters, including different frequencies F of 1, 3, 5, and 10 Hz, different maximum stress levels M of 0.80, 0.85, 0.90, and 0.95 and different amplitude levels A of 0.40, 0.50, 0.60, and 0.70; thereinto, the maximum stress level M and amplitude level A are defined as the ratio of the maximum cyclic stress and amplitude to the static uniaxial compressive strength of jointed rock specimens, respectively. The different cyclic loading conditions are summarized in Table 1, in which specimens M85, A5, and F1 are subjected to the same cyclic loading. For each cyclic loading condition, three jointed specimens are tested.

Table 1 Summary of the cyclic loading parameters and the representative cyclic testing results
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3 Determination of Fatigue Properties and Results of Monotonic Loading Tests

3.1 Determination of Fatigue Energy Density and Damage Variable

Based on the first law of thermodynamics, the variation in work of the external forces δW is equal to the variation of internal energy δU for each volume element under adiabatic conditions (static equilibrium and no net heat flow). According to Solecki and Conant (2003), δW can be calculated in terms of the stress components σ and corresponding strain components ε as follows:

$$\delta W = \int\limits_{V} {\left( {\sigma_{xx} \delta \varepsilon_{xx} + \sigma_{yy} \delta \varepsilon_{yy} + \sigma_{zz} \delta \varepsilon_{zz} + 2\sigma_{xy} \delta \varepsilon_{xy} + 2\sigma_{yz} \delta \varepsilon_{yz} + 2\sigma_{zx} \delta \varepsilon_{zx} } \right){\text{d}}V}$$
(1)

Expressing the internal energy U for volume V in terms of the internal energy per unit volume, i.e., the energy density u, there is,

$$\begin{aligned} \delta U = \int\limits_{V} {\delta u{\text{d}}V} \hfill \\ \, = \int\limits_{V} {\left( {\sigma_{xx} \delta \varepsilon_{xx} + \sigma_{yy} \delta \varepsilon_{yy} + \sigma_{zz} \delta \varepsilon_{zz} + 2\sigma_{xy} \delta \varepsilon_{xy} + 2\sigma_{yz} \delta \varepsilon_{yz} + 2\sigma_{zx} \delta \varepsilon_{zx} } \right){\text{d}}V} \hfill \\ \end{aligned}$$
(2)

Thus, the variation of energy density u can be expressed as follows:

$$\delta u = \left( {\sigma_{xx} \delta \varepsilon_{xx} + \sigma_{yy} \delta \varepsilon_{yy} + \sigma_{zz} \delta \varepsilon_{zz} + 2\sigma_{xy} \delta \varepsilon_{xy} + 2\sigma_{yz} \delta \varepsilon_{yz} + 2\sigma_{zx} \delta \varepsilon_{zx} } \right)$$
(3)

For uniaxial compression tests, there is only an axial stress component and a corresponding strain component, and thus the energy density u can be calculated as follows:

$$u = \int\limits_{\varepsilon } {\sigma {\text{d}}\varepsilon }$$
(4)

For cyclic uniaxial compression tests, the fatigue energy density can be calculated by integrating the fatigue stress–strain curve. As shown in Fig. 5a, the area under the loading curve represents the total energy density u (i.e., the region ABCD), the area under the unloading portion indicates the elastic energy density u e (i.e., the region CDEF), and the area difference between u and u e represents the hysteresis energy density u d (i.e., the region ABCFE); the elastic energy is released in the unloading process, and the hysteresis energy is released inducing the internal damage and irreversible deformation. The following equations are generally employed to determine the fatigue energy density parameters (Meng et al. 2016):

$$\left\{ {\begin{array}{*{20}l} {u = \int\limits_{{\varepsilon_{1} }}^{{\varepsilon_{\text{m}} }} {\sigma {\text{d}}\varepsilon } = \sum\limits_{i = 1}^{n} {\frac{1}{2}\left( {\sigma_{i2} + \sigma_{i1} } \right)\left( {\varepsilon_{i2} - \varepsilon_{i1} } \right)} } \hfill \\ {u^{\text{e}} = \int\limits_{{\varepsilon_{2} }}^{{\varepsilon_{\text{m}} }} {\sigma {\text{d}}\varepsilon } = \sum\limits_{i = 1}^{n} {\frac{1}{2}\left( {\sigma_{i2} + \sigma_{i1} } \right)\left( {\varepsilon_{i2}^{\text{e}} - \varepsilon_{i1}^{\text{e}} } \right)} } \hfill \\ {u^{d} = u - u^{\text{e}} = \int\limits_{{\varepsilon_{1} }}^{{\varepsilon_{\text{m}} }} {\sigma {\text{d}}\varepsilon } - \int\limits_{{\varepsilon_{2} }}^{{\varepsilon_{\text{m}} }} {\sigma {\text{d}}\varepsilon } } \hfill \\ \end{array} } \right.$$
(5)

where ε 1 and ε 2 are the strain corresponding to the minimum stress σ min in the loading and unloading curve, respectively; ε m is the strain corresponding to the maximum stress σ max in the loading curve; ε i1, ε i2, \(\varepsilon_{i1}^{\text{e}}\), and \(\varepsilon_{i2}^{\text{e}}\) are the strain at an integral step; σ i1 and σ i2 are the cyclic stress at an integral step, respectively.

Fig. 5
figure 5

a Calculation of the fatigue energy density parameters, and b schematic of determination of the fatigue damage variable

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The fatigue damage variable, which reflects the accumulated damage evolution of rock materials under cyclic loads, is one of the most critical parameters in fatigue analysis. Xiao et al. (2010) pointed out that the damage variable can be defined based on the hysteresis energy. In this study, this definition method is used to calculate the fatigue damage variable D of the jointed rock models. A schematic of the determination of the fatigue damage variable is shown in Fig. 5b, and the equation can be expressed as follows:

$$D = {{\sum\limits_{i = 1}^{{N_{i} }} {\left| {U_{{_{i} }}^{{^{d} }} } \right|} \, } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^{{N_{i} }} {\left| {U_{{_{i} }}^{{^{d} }} } \right|} \, } { \, \sum\limits_{i = 1}^{N} {\left| {U_{{_{i} }}^{d} } \right|} }}} \right. \kern-0pt} { \, \sum\limits_{i = 1}^{N} {\left| {U_{{_{i} }}^{d} } \right|} }}$$
(6)

where N and N i are the cycle number at final and current time step, respectively, and \(U_{i}^{d}\) is the hysteresis energy in the ith cycle.

3.2 Results of Static Monotonic Loading Tests

Monotonic loading tests are performed on four jointed rock specimens to obtain the static uniaxial compressive strength, which provides the reference values for subsequent cyclic loading parameters. The monotonic stress–strain curves of the tested jointed specimens are shown in Fig. 6, featuring a slowly increasing part up to the peak strength and then a dramatic decreasing post-failure part. The details of experimental results are tabulated in Table 2, in which the measured average static uniaxial compressive strength is 32.76 MPa, and the average Young’s modulus is 8.13 GPa. Figure 7 depicts the failure modes at the peak strength of the four tested specimens under monotonic loading, in which the tensile splitting failure through intact materials is prominent in all tested specimens. Moreover, the four jointed specimens exhibit similar progressive failure behavior. Taking the specimen U4 as an example, Fig. 8 presents the progressive failure at six typical stress levels: i.e., initial state (i.e., 0 MPa), 20% peak strength (i.e., 6.72 MPa), 40% peak strength (i.e., 13.44 MPa), 60% peak strength (i.e., 20.15 MPa), 80% peak strength (i.e., 26.87 MPa), and 100% peak strength (i.e., 33.59 MPa). As the axial stress increases almost linearly up to 40% peak strength, the front surface of the tested specimen barely changes (Fig. 8a–c). At the stress level of 60% peak strength, the surface cracks initiate from the joint tips (Fig. 8d). Afterward, the axial stress continues to increase up to 80% peak strength, corresponding to the further propagation of cracks and the formation of the observable tensile wing cracks on the front surface of the tested specimen (Fig. 8e). As the stress reaches its maximum value, i.e., 100% peak strength, the wing cracks completely coalesce, triggering the inevitable tensile failure through the entire specimen (Fig. 8f).

Fig. 6
figure 6

Static uniaxial stress–strain curves of the four tested jointed rock specimens under monotonic loading

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Table 2 Summary of monotonic uniaxial compression tests
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Fig. 7
figure 7

Static uniaxial failure modes of the four tested jointed rock specimens under monotonic loading

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Fig. 8
figure 8

Progressive failure behavior of the jointed rock specimen U4 under monotonic loading (red arrows indicate the surface cracks of the tested jointed specimen) (color figure online)

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4 Experimental Results and Discussion of Cyclic Uniaxial Compression Tests

To investigate the influence of cyclic loading parameters (i.e., loading frequency, maximum stress and amplitude) on the fatigue mechanical properties of intermittently jointed rock models, 10 different cyclic loading conditions are applied on the synthetic jointed rock models with the same geometrical parameters (i.e., a = 15 mm, b = 20 mm, d = 24 mm, θ = 45°, k = 0.318 and ρ = 3 rows). For each loading condition, three jointed rock models are tested. The specifics of cyclic uniaxial compression tests are listed in Table 1, and the corresponding results are reported and discussed in this section.

4.1 Influence of Cyclic Loads on the Fatigue Deformation Characteristics

Figure 9a–c shows the representative stress–strain curves of intermittently jointed specimens under monotonic loading and different cyclic loading conditions. During an entire cyclic test, the number of hysteresis loops in the fatigue curve ranges from sparse to intensive and then to sparse. There are two parts of fatigue strain in each hysteresis loop, i.e., the elastic strain and the plastic strain; the elastic part is restored during the unloading progress, while the plastic part is irreversible and gradually accumulates with increasing cycles until the fatigue failure occurs. In addition, as shown in Fig. 10, the terminal fatigue strain (i.e., the axial strain at the fatigue failure point in the cyclic stress–strain curve) of the tested specimens is found to be approximately equal to the post-failure monotonic strain (i.e., the axial strain at the point with the maximum cyclic stress in the post-failure portion of the monotonic stress–strain curve). As compared in Table 1, the deviations between them are <8%, which indicates that the fatigue deformations of the jointed rocks are dominated by their static deformations.

Fig. 9
figure 9figure 9

Representative stress–strain curves of the jointed rock specimens obtained from monotonic and cyclic loading tests: a different loading frequencies, b different maximum stress levels, and c different amplitude levels (red line and black line denote the monotonic stress–strain curve and the cyclic stress–strain curve, respectively) (color figure online)

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Fig. 10
figure 10

Terminal fatigue strain of the tested jointed rock specimens under different cyclic loading conditions and the post-failure monotonic strain corresponding to maximum cyclic stress. Square, post-failure monotonic strain. Square with bar, terminal fatigue strain. a Different loading frequencies. b Different maximum stress levels. c Different cyclic amplitude levels

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Similar to the three-phase development of irreversible deformations for intact rocks (Xiao et al. 2009), the irreversible plastic strains of the tested jointed rock specimens under different cyclic loading conditions also develop in a three-stage manner, i.e., initial, steady, and accelerated stage, as marked in Fig. 11. For a given loading condition, the irreversible strain increases rapidly in stage I and remains steady in stage II until it accelerates again in stage III due to the sudden fatigue failure. In the accelerated stage, the irreversible strain increases more rapidly under cyclic loading conditions with higher maximum stress, higher amplitude, or lower cyclic frequency.

Fig. 11
figure 11

Development curves of the irreversible plastic strains of jointed rock specimens under various cyclic loading conditions. I Initial stage. II Steady stage. III Accelerated stage. a Different loading frequencies. b Different maximum stress levels. c Different cyclic amplitude levels

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Figure 12 shows the influence of the three cyclic loading parameters on the two representative strains, i.e., initial and terminal irreversible strains; the two strains are defined as the irreversible strains accumulated in the initial stage and in all three stages, respectively. Specifically, as the loading frequency increases from 1 to 10 Hz, the initial and terminal irreversible strains increase from 0.27 to 0.34% and from 0.32 to 0.37%, respectively. In contrast, the two strains feature a linear decrease with increasing maximum cyclic stress; as the maximum stress level increases from 0.80 to 0.95, the initial and terminal irreversible strains decrease from 0.29 to 0.22% and from 0.35 to 0.27%, respectively. Similar to the influence of maximum stress, as the amplitude level increases from 0.40 to 0.70, the initial and terminal irreversible plastic strains decrease from 0.32 to 0.15% and from 0.36 to 0.18%, respectively.

Fig. 12
figure 12

Influence of cyclic loading parameters on the initial and terminal irreversible strains: a, b loading frequency, c, d maximum stress level, and e, f amplitude level

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Two representative fatigue deformation moduli of the jointed rock specimens are also analyzed, namely, Young’s modulus and secant modulus. In this study, the Young’s modulus is defined as the slope of straight-line portion of the stress–strain curve, and the secant modulus is the slope of secant line at a stress level at 50% of the static uniaxial compressive strength. Figure 13 presents the relationships between the two fatigue moduli and the relative cycle (i.e., ratio of the current cycle number n to the total cycle number N) of the jointed rock specimens under different cyclic loading conditions. All the curves of fatigue moduli show a nonlinear decrease with increasing relative cycle, and both the Young’s modulus and the secant modulus decrease with increasing frequency or decreasing maximum stress and amplitude. In addition, compared with the variation of secant modulus, the Young’s modulus is more susceptible to cyclic loading parameters.

Fig. 13
figure 13

Influence of cyclic loading parameters on the Young’s modulus and secant modulus: a different loading frequencies, b different maximum stress levels, and c different amplitude levels

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4.2 Influence of Cyclic Loads on the Fatigue Energy Parameters

In this section, the fatigue energy parameters of jointed rock specimens under cyclic loading are determined employing Eq. (5), including the total energy density, the elastic energy density and the hysteresis energy density. Representative curves of energy density parameters of the tested jointed specimens under different cyclic loading conditions are depicted in Figs. 14 and 15, and the corresponding results are listed in Tables 3, 4, and 5. As shown in Fig. 14, as the relative cycle increases, the total energy density in a cycle decreases in the initial cycles, remains almost constant for a while, and then increases. In sharp contrast, the variation of the elastic energy density is opposite to that of the total energy density. Compared to that of the total energy density and the elastic energy density, the variation of the hysteresis energy density with respect to the relative cycle is more drastic. It features a dramatic increasing and decreasing transition in the initial cycles, and then it increases again (Fig. 15).

Fig. 14
figure 14

Influence of cyclic loading parameters on the total energy density and the elastic energy density: a, b different loading frequencies, c, d different maximum stress levels, and e, f different amplitude levels

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Fig. 15
figure 15

Influence of cyclic loading parameters on the hysteresis energy density: a different loading frequencies, b different maximum stress levels, and c different amplitude levels

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Table 3 Total energy density (MJ/mm3) of the representative jointed rock specimens at different relative cycle
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Table 4 Elastic energy density (MJ/mm3) of the representative jointed rock specimens at different relative cycle
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Table 5 Hysteresis energy density (MJ/mm3) of the representative jointed rock specimens at different relative cycle
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Moreover, the energy density magnitudes are highly dependent on cyclic loading parameters. As the loading frequency increases, the duration of cyclic loads acting on tested specimen becomes shorter, leading to the decrease in the total energy. Taking the fatigue energy density parameters in the initial stage to analyze, as the loading frequency increases from 1 to 10 Hz, the initial total energy density, elastic energy density, and hysteresis energy density decrease from 0.042 to 0.028 MJ/mm3, from 0.037 to 0.027 MJ/mm3, and from 0.005 to 0.001 MJ/mm3, respectively. In addition, as the maximum stress level increases from 0.80 to 0.95, the initial total energy density, elastic energy density, and hysteresis energy density increase from 0.037 to 0.059 MJ/mm3, from 0.034 to 0.043 MJ/mm3, and from 0.003 to 0.016 MJ/mm3, respectively. With increasing cyclic amplitude, more energy is absorbed, and the area of hysteresis loop increases, indicating that more energy is released for fatigue failure of the tested specimen. Thus, as the amplitude level increases from 0.40 to 0.70, the initial total energy density, elastic energy density, and hysteresis energy density increase from 0.039 to 0.050 MJ/mm3, from 0.036 to 0.039 MJ/mm3, and from 0.003 to 0.011 MJ/mm3, respectively.

4.3 Influence of Cyclic Loads on the Fatigue Damage Evolution

Employing Eq. (6) in Sect. 3.2, the fatigue damage variables of jointed rock specimens can be calculated based on the hysteresis energy. Figure 16 depicts the corresponding cumulative damage evolution curves of the tested jointed specimens under different cyclic uniaxial compression; these curves show an inverted-S shape. Similar to the development of irreversible plastic strain of jointed rock specimens, these inverted-S-shaped curves can also be divided into three stages: initial, steady, and accelerated stages. The fatigue damage variable accumulates quickly in the first stage, and then it increases steadily. In the third stage, the damage variable increases rapidly again as the final fatigue failure of the specimen occurs.

Fig. 16
figure 16

Evolution of the fatigue damage variable of the jointed rock specimen under various cyclic loading conditions, and the influence of cyclic loading parameters on the initial damage variable and fatigue life: a, b different loading frequencies, c, d different maximum stress levels, and e, f different amplitude levels

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Considering the influence of the three cyclic loading parameters on the fatigue damage, Fig. 16 shows that the cumulative damage variable increases with decreasing frequency and increasing maximum stress or amplitude. The variation of fatigue damage with cyclic loading parameters can be explained as follows. Under a higher loading frequency, less energy is dissipated, leading to the decrease in damage variable. Specifically, as the loading frequency increases from 1 to 10 Hz, the damage variable in the initial stage decreases from 0.17 to 0.10. In sharp contrast, at a higher maximum stress or amplitude, more energy is released, resulting in the increase in the damage variable. Hence, as the maximum stress level and the amplitude level increase from 0.80 to 0.95 and from 0.40 to 0.70, respectively, the initial damage variable increases from 0.14 to 0.27 and from 0.13 to 0.21, respectively. Figure 16 also illustrates the relations between the fatigue lives (i.e., the number of cycles to fatigue failure) of jointed rock specimens and the three cyclic loading parameters. With increasing loading frequency or decreasing maximum stress and amplitude, the fatigue life of the tested jointed specimen shows an exponential increase; the specific values of fatigue lives under different cyclic loading parameters are listed in Table 1.

4.4 Influence of Cyclic Loads on the Fatigue Progressive Behavior

Figure 17a–c depicts the representative fatigue failure scenarios of jointed specimens under cyclic uniaxial compression with different loading parameters. In all cyclically failed jointed specimens, the tensile wing cracks are the primary cracks, similar to the tensile splitting failure of jointed specimens under monotonic loading (Fig. 7). These wing cracks propagate at a certain angle with the preexisting joints toward the direction of maximum compression (Sagong and Bobet 2002), leading to an inevitable tensile splitting failure through the intact materials. Based on the existing standard definition of crack coalescence categories proposed by Wong and Einstein (2009a), three different coalescence patterns are observed in this study. The classification schematics are shown in Fig. 18, and the representative coalescence patterns in our tests are depicted in Fig. 19. Category I denotes that the two preexisting joints are linked up by a T-I crack; this tensile wing crack probably initiates from the tip of joint 1 and then propagates toward the face of joint 2 at a distance from the joint tip (Fig. 18a), or it perhaps initiates from the face of joint 1 at a distance from the joint tip and then propagates toward the tip of joint 2 (Fig. 18b). In Category II, the joint tips are linked up at the same side by a T-II crack dominantly of tensile nature, forming almost a straight line along the axial loading direction (Fig. 18c); since this tensile crack occurs in the opposite direction to the conventional wing crack, it cannot be regarded as a T-I wing crack. In contrast, for Category III, the joint tips are linked up at the opposite side, i.e., from the right tip of joint 1 to the left tip of joint 2 (Fig. 18d), forming an oblique T-III crack; the angle between the oblique T-III crack and the loading direction is larger than that between the straight T-II crack and the loading direction. Note that there may be occasional short shear cracks along the coalescence direction in Categories II and III.

Fig. 17
figure 17figure 17

Representative fatigue failure modes of the jointed rock specimens under various cyclic compression tests: a different loading frequencies, b different maximum stress levels, and c different amplitude levels

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Fig. 18
figure 18

Characteristic graphs of the crack coalescence categories observed in the present study: a, b Category I, c Category II, and d Category III

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Fig. 19
figure 19

Representative crack coalescence patterns observed in the present study

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Compared with the progressive failure behavior under the monotonic loading (Fig. 8), the fatigue failure of jointed specimens under cyclic loading usually occurs more abruptly without obvious preceding signs. Taking the jointed specimen F1 as an example, Fig. 20 depicts the progressive fatigue failure during six stages: i.e., initial stage, 20% fatigue life, 40% fatigue life, 60% fatigue life, 80% fatigue life, and 100% fatigue life. From stage I (initial stage) to stage III (40% fatigue life), no obvious variation can be observed on the front surface of tested specimen, similar to the process that the axial stress increases from 0 to 40% peak strength in monotonic loading test (Fig. 8a–c). In stage IV (60% fatigue life), the surface cracks initiate from the joint tips due to the highly concentrated stress, while these initiation cracks are less obvious than those under monotonic loading (Fig. 8d); after that, these cracks further propagate forming the observable tensile wing cracks through stage IV to stage V (80% fatigue life). Rather than the gradual coalescence of surface cracks in monotonic loading tests, these surface tensile cracks of cyclically tested jointed specimens close and open at the same pace as the loading frequency, eventually triggering the sudden fatigue failure at stage VI (100% fatigue life).

Fig. 20
figure 20

Progressive failure behavior of the jointed rock specimen F1 under cyclic uniaxial compression (red arrows indicate the surface cracks of the tested jointed specimen) (color figure online)

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5 Summary and Conclusions

Intermittently jointed rocks are quite sensitive to cyclic loading conditions in mining and civil engineering structures. However, the systematic investigations on the fatigue mechanism of jointed rocks under different cyclic loading are rather limited. In this study, synthetic jointed rock models are employed to perform cyclic uniaxial compression tests with different loading parameters, including four frequencies, four maximum stresses and four amplitudes. Our experimental results systematically depict the influence of the three cyclic loading parameters on the fatigue properties of intermittently jointed rock models, regarding the fatigue deformation characteristic, the fatigue energy and damage evolution, and the fatigue failure and progressive behavior. The following conclusions can be drawn:

  1. 1.

    The terminal fatigue strain (i.e., the axial strain at the fatigue failure point in the cyclic stress–strain curve) of the tested specimens is approximately equal to the post-failure monotonic strain (i.e., the axial strain at the point with the maximum cyclic stress in the post-failure portion of the monotonic stress–strain curve). Both the Young’s modulus and the secant modulus feature a nonlinear decrease with increasing cycles, and the development of irreversible plastic strain can be divided into three stages, i.e., initial, steady, and accelerated stages (Figs. 9, 10, 11, 12, 13, 14).

  2. 2.

    The total energy density in each cycle decreases in the initial cycles, remains almost constant for a while, and then increases, while the variation of the elastic energy density is the opposite. The hysteresis energy density features a dramatic increasing and decreasing transition and then increases again. The three energy density parameters increase as the frequency decreases or the maximum cyclic stress and amplitude increase (Fig. 15).

  3. 3.

    Cumulative fatigue damage of jointed rock models exhibits an inverted-S shape with a three-stage evolution. This damage variable increases quickly in the initial cycles and then increases slowly until it increases rapidly again as the fatigue failure of the specimen occurs. With decreasing frequency or increasing maximum stress and amplitude, the cumulative fatigue damage variable increases, leading to the decrease in fatigue life (Fig. 16).

  4. 4.

    Tensile splitting failure through intact materials is prominent in all cyclically failed jointed models, and three crack coalescence patterns are observed based on the interaction between two adjacent joints. The tensile cracks on the front surface of jointed specimens always initiate from joint tips and then propagate at a certain angle with the joints toward the direction of maximum compression (Figs. 17, 18, 19, 20).