Damage Evolution and Deformation of Rock Salt Under Creep-Fatigue Loading

Abstract

During the operation of a compressed air energy storage (CAES) salt cavern, the surrounding rock experiences creep damage during the stages of constant internal pressure and undergoes fatigue damage due to the periodical injection-production. To describe the damage evolution of salt rock under creep-fatigue loading, a novel damage accumulation model based on the ductility exhaustion concept is proposed by applying a nonlinear summation method to represent the synergistic effect of creep and fatigue damage. Low-cycle fatigue (LCF) and creep-fatigue tests of rock salt were conducted under stress-control mode for various cycle stress amplitudes and hold times. Results show that the deformation of rock salt under creep-fatigue loading consists of initial, steady and accelerated phases. The proposed model matches well with the test data and can accurately describe the damage evolution as the applied stress amplitudes and dwell times change. The introduction of the hold times at the upper limit stress causes a strain increment and life reduction, which become more evident as the duration periods prolong and can be understood by the dislocation theory of crystals.

Introduction

Energy has become increasingly crucial in social, economic and scientific progresses. It is categorized into fossil fuels, nuclear resources and renewable energy resources (Panwar et al. 2011). Because nuclear energy can cause serious health problems for humans (Manzano-Agugliaro et al. 2013) and large-scale consumption of fossil fuels leads to environment pollution (Beccali et al. 2007), renewable energy is a significant concern in the long term. As a renewable energy, electricity can be generated by solar or wind. However, these technologies cannot directly provide either continuous base-load power or peak-load power due to their low power density and discontinuous nature (Bilgili et al. 2015). Compressed air energy storage (CAES), which compresses air into a suitable storage medium and reconverts it to electricity by a turbine generator (Budt et al. 2016), is characterized by large capacity and continuity in providing electrical power. Because of the unique properties of rock salt including rheology (Munson 1997), low-permeability (Chen et al. 2019) and self-repair after suffering damage (Yin et al. 2018), salt caverns are recognized as ideal storage facilities. They have been used for compressed air in Germany and the United States (Raju and Kumar Khaitan 2012). During the operation of a CAES system, the internal pressure fluctuation of the salt cavern consists of constant high pressure, production, constant low pressure and injection. The periodic injection-production leads to the cyclic loading and unloading of the surrounding rock, which damages the host rock salt and may cause roof fall and gas leakage (Liu et al. 2020; Wang et al. 2018). Therefore, it is essential to study the mechanical behaviour and damage evolution of rock salt under such loading conditions.

Many researchers have investigated the mechanical characteristics of rock salt. Because the cyclic loading conditions are critical for cavern safety and economic benefits (Han et al. 2020), the effects of stress level, frequency, temperature and confining pressure on the mechanical behaviour of rock salt have been systemically investigated by uniaxial and triaxial cyclic loading tests (He et al. 2019; Ma et al. 2013; Song et al. 2013). Similar to the typical creep behaviour of rock salt, the axial strain consists of initial, steady and accelerating deformation stages under the condition of cyclic loading (Guo et al. 2012). Compared with static loading, the salt presents strain-hardening and tends to be more ductile (Ma et al. 2013). The salt compressive strength and elasticity decrease with increasing number of loading cycles, which usually lead to fatigue failure (Fuenkajorn and Phueakphum 2010). This indicates that the parameters obtained from the static creep test to study the long-term stability of storage caverns may indicate detrimental behaviour. In addition to the behaviour analysis, several models have been proposed to predict the fatigue life and evaluate the damage accumulation of rock salt (He et al. 2018, 2019; Liu et al. 2013). These studies provide a guide to the stability design of salt caverns. Apart from continuous fatigue tests, Fan et al. (2016, 2019) conducted discontinuous cyclic loading tests and found that the presence of intervals between each cycle leads to the life reduction and more considerable plastic deformation compared with continuous fatigue tests. For a CAES system, the surrounding rock salt experiences fatigue damage during the air injection and withdraw stages and undergoes creep damage during the stages of constant air pressure. This loading pattern indicates creep-fatigue loading. However, the damage evolution and mechanical properties of salt rock under creep-fatigue loading remain unclear.

In this paper, a novel damage accumulation model is proposed to describe the damage evolution of salt rock under creep-fatigue loading. To verify this model, LCF tests and creep-fatigue tests with various stress amplitudes and hold times were conducted under stress-control mode. The deformation behaviour of rock salt in the LCF and creep-fatigue tests was then analyzed. Finally, the effect of hold time is discussed based on the deformation discrepancy between the LCF tests and creep-fatigue tests. This study can enrich the comprehensiveness of the stability analysis of CAES salt caverns.

Damage Evolution Model and Damage Variable

Damage mechanics studies the initiation and propagation of internal microcracks under various loading conditions which deteriorate the mechanical properties of the material. When discussing damage laws, dilatancy is considered and the octahedral shearing overstress above the dilatancy boundary is responsible for the damage (Cristescu 1999; DeVries 2006). Dilatancy is also mentioned when investigating the permeation and gas transport properties of rock salt (Popp et al. 2002, 2012). However, the dilation point of rock salt under creep-fatigue loading varies with the testing samples. In this study, to investigate the damage evolution and deformation behaviour of the three stages, each salt sample needs to be cyclically loaded to failure. To simplify the damage model and make the damage variable easier to define and calculate, the initial loading point was selected as the start point of damage. The value of the damage variable at the initial loading point is zero, and it is defined as one when the samples fracture. This hypothesis comes from continuous damage mechanics (Lemaitre and Chaboche 1994) and has been used in some damage models (Eftekhari and Fatemi 2016; Guo et al. 2012; He et al. 2019).

The proposed damage model is based on the ductility exhaustion theory, which assumes that the ductility of the material exhausts continuously during the creep-fatigue loading, and the material fails once the accumulated strain reaches a critical ductility (Priest and Ellison 1981). Under creep-fatigue loading, the damage is divided into creep damage during the hold time and fatigue damage during the loading stage. In this model, creep damage and fatigue damage are calculated separately. The creep-fatigue damage evolution model is derived by a nonlinear summation method to include creep and fatigue damage. Finally, the damage variable suitable to rock salt is defined to represent the damage evolution during the creep-fatigue tests.

Damage Evolution Model

Creep deformation and void development are associated with the ductility exhaustion of the material. When the accumulated strain in the material reaches a critical value (Wen et al. 2016), the material fails. Thus, the creep damage variable can be defined as follows:

$$D_{c} = \mathop \smallint \limits_{0}^{{t_{H} }} \frac{{\dot{\varepsilon }}}{{\varepsilon_{C}^{*} }}dt$$
(1)

It is assumed that the damage process starts from the initial loading point and lasts until the specimen fails, so \(D_{c}\) values range from zero for the un-damaged state to one for the fracture condition. \(t_{H}\) is the hold time per cycle in creep-fatigue tests. \(\dot{\varepsilon }\) is the steady-state creep strain rate which can be calculated by the Norton powerlaw (Norton 1929):

$$\dot{\varepsilon } = A \cdot \left( {\overline{\sigma }} \right)^{n}$$
(2)

where \(A\) is a material constant and \(n\) denotes the creep exponent. The values of the two parameters can be obtained from uniaxial creep tests with different applied stress. \(\overline{\sigma }\) denotes the deviatoric stress, \(\overline{\sigma } = \sigma_{1} - \sigma_{3}\), \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses, respectively. The Norton powerlaw has been commonly used for describing the salt creep deformation (Bérest et al. 2001; Wang et al. 2016b). \(\varepsilon_{C}^{*}\) is the multi-axial creep ductility, which is defined on the basis of the uniaxial creep fracture strain, according to the Cocks and Ashby model (Oh et al. 2011). This model assumes that grain-boundary cavities nucleate at inclusions and subsequently grow by the power-law creep of the surrounding matrix until they link. Hence, the creep exponent \(n\) is used to determine the multi-axial fracture strain:

$$\varepsilon_{C}^{*} = \frac{{\varepsilon_{C} }}{{\sinh \left[ {2\left( {n - 0.5} \right)/3\left( {n + 0.5} \right)} \right]/\sinh \left[ {2\sigma_{m} \left( {n - 0.5} \right)/\sigma_{eq} \left( {n + 0.5} \right)} \right]}}$$
(3)

where \(\varepsilon_{C}\) is the uniaxial creep fracture strain; \(\sigma_{m}\) is the mean stress, and \(\sigma_{eq}\) represents the equivalent stress; \(h\) is the stress triaxiality which is defined by the ratio of \(\sigma_{m}\) and \(\sigma_{eq}\):

$$h = \frac{{\sigma_{m} }}{{\sigma_{eq} }} = \frac{{\sigma_{1} + \sigma_{2} + \sigma_{3} }}{{3\sigma_{eq} }}$$
(4)
$$\sigma_{eq} = \left[ {\frac{1}{2}\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{3} - \sigma_{1} } \right)^{2} } \right]^{1/2}$$
(5)

where \(\sigma_{2}\) is the intermediate principal stress.

Under creep-fatigue loading, the creep damage is accumulated mainly during the hold period and results in nucleation and growth of grain-boundary cavities (Nam 2002). Therefore, the creep damage is calculated only during the hold time while the effect of the loading–unloading stage is ignored.

Similar to creep rupture, fatigue is characterized by a damage accumulation with a continuous deterioration of the material properties. The material ductility reduces and energy dissipates gradually during the cyclic loading (Fuenkajorn and Phueakphum 2010). Based on the ductility exhaustion theory (Cheng and Plumtree 1998; Priest and Ellison 1981; Ye and Wang 2001), the fatigue damage of each cycle in the LCF tests is calculated as:

$$\frac{{dD_{F} }}{dN} = \frac{1}{{\varepsilon_{F} }}\frac{{d\varepsilon_{p} }}{dN}$$
(6)

where \(D_{F}\) is the fatigue damage variable; \(\varepsilon_{F}\) is the fatigue ductility;\(\varepsilon_{p}\) is the cumulative plastic strain at the end of \((N - 1)\) cycles, and \(N\) denotes the number of cycles. Under uniaxial stress, the plastic strain per cycle develops only during the loading cycle. It can be calculated using an evolution model of rock plastic strain proposed by Xu et al. (2012):

$$\frac{{d\varepsilon_{p} }}{dN} = \frac{{\left( {\sigma_{\max }^{f} - \sigma_{\min }^{f} } \right)}}{{S^{f} \left( {\sigma_{\max } - \sigma_{\min } } \right)^{a} N^{c} }}\left( {1 - \left( {\frac{N}{{N_{F} }}} \right)^{1 - c} } \right)^{{\frac{ - f}{{b + 1}}}}$$
(7)

where \(\sigma_{\max }\) and \(\sigma_{\min }\) are the upper and lower limit stresses of the cyclic loading, respectively; \(S\) is the load force; \(N_{F}\) is the number of cycles to failure; \(a, b, c\) and \(f\) are material constants. For the periodic cyclic loading, the boundary condition can be defined as: \(D_{F} = 0\) when \(N = 0\) and \(D_{F} = 1\) when \(N = N_{F}\). Thus, the fatigue damage variable is calculated by integrating Eq. 7 and applying the boundary condition as follows:

$$D_{F} = 1 - \left( {1 - \left( {\frac{N}{{N_{F} }}} \right)^{1 - c} } \right)^{m}$$
(8)

where \(m = 1 - f/\left( {b + 1} \right)\). Only the fatigue damage during the loading–unloading stage is included.

Under creep-fatigue, a strong interaction between creep and fatigue damage occurs, which accelerates the accumulation of creep and fatigue damage (Eftekhari and Fatemi 2016; Lemaitre and Plumtree 1979). Thus, a nonlinear summation method is used to indicate this synergistic effect (Lagneborg and Attermo 1971):

$$D = k_{1} D_{C}^{p} + k_{2} D_{F}^{1 - p}$$
(9)

where \(D\) represents the total damage variable. \(D_{C}\) and \(D_{F}\) denote the creep and fatigue damage variable, respectively. \(p\) is the creep-fatigue interaction damage exponent and reflects the relative magnitude of creep and fatigue damage. \(k_{1}\) and \(k_{2}\) are material constants.

Damage Variable

The damage variable is an internal thermodynamic variable used to describe the damage evolution inside the material. Because the damage cannot be measured directly, it is usually calculated by a descriptive method. The mechanical parameters, such as elastic modulus, ultrasonic waves and cycle stress amplitudes, were used in a previous study (Lemaitre and Dufailly 1987). But these methods are not strictly appropriate when describing the damage properties of rock salt, which is characterized by plastic deformation (Xiao et al. 2010).

Under creep-fatigue loading, cyclic loading causes fatigue damage to the salt sample, and static loading during the hold time causes creep damage. The stress–strain hysteresis curve per cycle is not closed. The energy dissipated per cycle in the material can be a measurement of damage (Payten et al. 2010). Therefore, the strain energy is used to define the damage variable, because it comprehensively reflects the creep and fatigue damage. Thus, the total damage variable at the end of \(N\) cycles can be expressed as (Ye and Wang 2001):

$$D_{N} = \frac{{W_{d} }}{{W_{0} }}$$
(10)

where \(W_{0}\) is the cumulative plastic strain energy at the end of the final cycle. \(W_{d}\) is the cumulative plastic strain energy at the end of the \(N\) cycles. The strain energy per cycle equals the work decrement of the dissipative system in cyclic loading, and is generally calculated as the area of the hysteresis cycle:

$$W_{dN} = \oint {\sigma_{N} d\varepsilon_{N} }$$
(11)

where \(W_{dN}\) is the plastic strain energy of cycle \(N\), \(\varepsilon_{N}\) is the strain amplitude of cycle \(N\), and \(\sigma_{N}\) is the stress amplitude. During creep-fatigue loading, the cumulative plastic strain energy means the aggregation of strain energy per cycle from the initial loading to the current cyclic number \(N\):

$$W_{d} = \mathop \sum \limits_{i = 1}^{N} W_{dN}$$
(12)

Mechanical Tests

Materials and Experimental Conditions

The salt samples were collected from the Khewra salt mine, Pakistan. They consist of NaCl (more than 96%), K2SO3 (around 3.1%), some mud and other undissolvable substances (less than 0.9%) (Fan et al. 2017; Zhao et al. 2020). Samples were processed into standard cylinders by wire cutting, with 50 mm diameter and 100 mm length. The upper and lower surfaces are dry-polished, and the parallelism is controlled within ± 0.2 mm. All the tests were conducted at constant temperature and relative humidity (\(25 \pm 3 \,^\circ {\text{C}}\) and \(60 \pm 10\%\), respectively), because temperature and humidity influence the mechanical properties of salt rock. To reduce the sample-to-sample variation of rock salt, specimens without obvious cracks were selected. The longitudinal wave velocities of these specimens were measured, and samples with velocities of \(3000 \pm 300 \,{\text{m/s}}\) were chosen for tests.

The testing machine is a digitally controlled electro-hydraulic servo machine. The capacities of axial load, frequency, and loading rate are 2000 kN, 5 Hz and 100 kN/s, respectively. The axial strain was obtained by monitoring the position of the loading piston by an LVDT (Linear Variable Differential Transformer) sensor. The lateral strain was measured by an extensometer set on the specimen circumferential surface by a chain (Fig. 1). To reduce the interface friction that strongly influences the stress–strain variation, the ends of the sample were separated from the steel loading platens using a pair of gaskets. In addition, samples were enclosed by a tight thin plastic wrapping to maintain a stable humidity and avoid any corrosion of the equipment from salt fragments (Fig. 1).

Fig. 1
figure1

The rock salt sample and test setup

Uniaxial Compression Tests

The upper and lower limit stresses in the LCF tests, and creep-fatigue tests are determined by a percentage of the uniaxial compression strength (UCS) of the rock salt. Therefore, uniaxial compression tests were conducted first. To improve the reliability of the results, three replicated tests were performed to obtain the average UCS. These tests were conducted with the constant strain rate (\(2.5 \times 10^{ - 5} {\text{ s}}^{ - 1}\)), and the UCS was defined as the maximum stress during the test. The UCS of the tested samples ranges from 21.95 to 22.15 MPa. Its average value is 22.06 MPa, which matches the uniaxial compression strength of pure salt (15–32 MPa) (Hansen and Mellegard 1984). The axial strains corresponding to the maximum stress of the tested samples are 4.578%, 4.584% and 4.670%, respectively.

Uniaxial Creep Tests

To obtain the parameters in the proposed model (\(A\) and \(n\)), uniaxial creep tests were performed under different axial stresses. The axial stress was first loaded to the applied stress using displacement control, with constant strain rate (\(2.5 \times 10^{ - 5} {\text{ s}}^{ - 1}\)). The control was then switched to force control, and the axial load was held constant until the sample fails. Experimental conditions and the steady-state strain rate are listed in Table. 1. The steady-state creep rate grows with increasing deviatoric stress. Based on Eq. 2 and the test data in Table 1, the Norton creep parameters \(A\) and \(n\) were \(1.2524 \times 10^{ - 10}\, {\text{MPa}}^{ - 3.7542} /{\text{s}}^{ - 1}\) and \(3.7542\), respectively.

Table 1 Uniaxial creep test condition and tests results

Low-Cycle Fatigue Tests and Creep-Fatigue Tests

To verify the proposed damage model, creep-fatigue tests were performed at various stress levels and hold times. Factors that affect fatigue behaviour are stress level, cyclic stress amplitude, frequency, loading rate and loading waveform (He et al. 2019). Hold times also affect creep behaviour. Therefore, the upper limit stress \(\sigma_{\max }\), the lower limit stress \(\sigma_{min}\) and the hold time per cycle \(t_{H}\) were selected as variables in the creep-fatigue tests (Wang et al. 2017a; Wei and Yang 2009). A trapezoidal wave was used for the creep-fatigue tests, and the samples were loaded and unloaded at 1.4 kN/s (\(0.71 \,{\text{MPa/s}}\)) in the loading–unloading stage. To further investigate the effect of hold time, LCF tests were conducted for comparison with the same loading velocity (Fig. 2).

Fig. 2
figure2

The loading waveforms used in the tests: a low-cycle fatigue test, b creep-fatigue test

Due to the limitations of experimental equipment, there are some deviations between the test conditions and engineering practice, including cycle period and loading stress level. Despite these deviations, this study can provide a new understanding of mechanical properties of rock salt and provide a reference for analyzing the stability of salt caverns for a CAES system.

Results and Discussion

Mechanical Behaviour

The loading conditions of each sample are presented in Table 2. The F1–F6 samples and the CF1–CF10 samples belong to the LCF tests and the creep-fatigue tests, respectively. Figures 3 and 4, respectively, show the strain–time curves for the F1–F3 and CF1–CF5 samples. Similar curves were obtained for samples F4–F6 and CF6–CF10, so the figures for these samples are omitted. Similar to the typical creep behavior of rock salt (Yang et al. 1999), the strain–time curves of the LCF and creep-fatigue tests both present three deformation stages (initial, steady and accelerated). In the initial stage, the axial strain develops rapidly, and the deformation accumulates quickly. In the second stage, the axial deformation grows steadily, and this phase accounts for most of the lifetime. In the accelerated stage, the strain increases rapidly until the tested sample fails.

Table 2 Results of the low-cycle fatigue tests and creep-fatigue tests
Fig. 3
figure3

Axial strain–time curves for samples F1-F3

Fig. 4
figure4

Axial strain–time curves for samples CF1-CF5

Figures 5, 6 and 7, respectively, show the stress–strain curves for the F1–F3 and CF1–CF5 samples. Similar curves were obtained for samples F4-F6 and CF6-CF10, so the figures for these samples are omitted. The plastic deformation increases with the number of cycles. Figures 5 and 6 show that the spacing of the hysteresis loop becomes tighter as the cyclic stress amplitude decreases. Figure 7 shows that the spacing of the hysteresis loop becomes looser with increasing the hold time per cycle. The area between the loading curve and unloading curve per cycle denotes the amount of energy dissipated (Liu et al. 2013), which corresponds to the damage level per cycle due to the external loading. The salt samples in the LCF and creep-fatigue tests fail at the accelerating deformation stage, although the upper limit stress is lower than the UCS. This is because of the accumulated plastic deformation and damage per cycle during the cyclic loading. The behaviour of the hysteresis loops is consistent with that under different cyclic loading scenarios (Guo et al. 2012; Han et al. 2020; Song et al. 2013).

Fig. 5
figure5

Stress–strain curves for samples F1-F3

Fig. 6
figure6

Stress–strain curves of rock salt under creep-fatigue tests: Group 1 (various stress amplitudes)

Fig. 7
figure7

Stress–strain curves of rock salt under creep-fatigue tests: Group 2 (various hold times per cycle)

As shown in Table 2, in Group 1 (including CF1, CF2 and CF3), the samples are maintained at the same hold time and upper limit stress, and the lower limit stress of samples CF1, CF2 and CF3 is 10%, 20% and 30% of the UCS, respectively. Their cycle numbers to failure are 8, 40 and 85, and their steady strain rates are \(1.43 \times 10^{ - 7}\, {\text{s}}^{ - 1}\), \(3.50 \times 10^{ - 8} \,{\text{s}}^{ - 1}\), and \(2.31 \times 10^{ - 8}\, {\text{s}}^{ - 1}\), respectively. This indicates that the creep-fatigue life and the steady strain rate increase as the lower limit stress increases under the same upper limit stress in the experimental range. This is mainly because the damage of each cycle is reduced as the stress amplitude decreases, which slows the crack propagation (Song et al. 2013).

In Group 2 (including CF3, CF4 and CF5), the samples are maintained at the same stress levels, and the hold times per cycle of samples CF3, CF4 and CF5 are 90 s, 120 s and 150 s, respectively. Their cycle numbers to failure are 85, 35 and 15, and their steady strain rates are \(2.31 \times 10^{ - 8} \,{\text{s}}^{ - 1}\), \(3.39 \times 10^{ - 8} \,{\text{s}}^{ - 1}\), and \(6.06 \times 10^{ - 8} \,{\text{s}}^{ - 1}\), respectively. The creep-fatigue life reduces with increasing hold time per cycle, while the steady strain rate shows the opposite trend. These results show that the magnitudes of damage per cycle, specifically the creep damage, increase with the increase of the hold time in the experimental range. The behaviour of the CF6-CF10 samples with the \(\sigma_{\max }\) of 75% of the UCS shows a similar trend in creep-fatigue life and steady-state strain rate (Table 2), so the detailed descriptions are omitted.

To improve the reliability and make the experimental condition conform to the engineering practice of a CAES salt cavern, the creep-fatigue test with changing the upper limit stress on the same sample was conducted (Fig. 8). To quantitatively study the deformation behavior under various stress levels, the steady strain rates at each step were calculated (Table 3). The steady strain rate at each step grows with increasing the upper limit stress under the same hold time per cycle and lower limit stress.

Fig. 8
figure8

Stress-time and strain–time curves of rock salt under various upper limit stress per cycle

Table 3 The loading conditions and results of the creep-fatigue tests with changing the stress level and hold time on one sample

The Prediction Ability of the Proposed Model

Samples CF1-CF10 are used to verify the proposed damage model. All the parameters and coefficients are summarized in Table 4. The predicted results match well with the test data despite some minor fluctuations at certain points (Fig. 9). This indicates that the proposed model can be used to describe the damage evolution of rock salt under creep-fatigue loading.

Table 4 Parameters of the proposed damage model (\(R^{2}\) is the coefficient of determination.)
Fig. 9
figure9

Comparison of the propose model and test data: a CF1, b CF2, c CF3, d CF4, e CF5, f CF6, g CF7, h CF8, i CF9, j CF10

Similar to creep damage (Yang et al. 1999) and fatigue damage (He et al. 2019), the damage evolution of rock salt under creep-fatigue loading consists of three stages. During the first phase, the compaction and closure of microcracks and the growth of new cracks lead to a rapid increase of the damage variable. The damage variable grows steadily during the second stage because the microcracks are closed, and the newly-grown cracks gradually become stable. This stage accounts for most of the lifetime. When the damage accumulates to a certain degree, a large number of new cracks appear and gradually connect and penetrate. At this time, the damage variable increases rapidly. When a large number of cracks are intensified and penetrated into macroscopic cracks, the salt rock fails.

Hold Time Analysis

To quantitatively study the effect of hold time on the life reduction, the life reduction ratio \(L_{r}\) is defined as \(1 - N_{cf} /N_{f0}\) (Wang et al. 2017a) where \(N_{f0}\) and \(N_{cf}\) are the cycle numbers to failure in the LCF and creep-fatigue tests, respectively. The values of \(L_{r}\) of the CF3-CF5 samples are 0.721, 0.885 and 0.951, and the value of \(L_{r}\) of the CF8-CF10 samples are 0.650, 0.762 and 0.937. These results mean that \(L_{r}\) increases with increasing the hold time per cycle. This is similar to the hold time sensitivity of metals under tension-hold-only and compression-hold-only condition (Wang et al. 2016a, 2017b).

Figure 10 shows the axial and lateral strain per cycle in the LCF tests and creep-fatigue tests. Under the same stress levels, the strain per cycle in the creep-fatigue tests is larger than that in the LCF tests. Because the volumetric strain correlates well with the damage (Chen et al. 2016; Roberts et al. 2015), we further compared the volumetric strain per cycle between the F1 sample and the CF1 sample (Fig. 11). Similar to the axial and lateral strain, the CF1 sample had more significant volumetric strain per cycle than the F1 sample. These results show that the presence of hold time leads to more damage per cycle and accelerates the failure of the salt sample.

Fig. 10
figure10

Axial strain and lateral strain per cycle of F1 and CF1. The values for the first cycle and the last cycle, far larger than the others, are not shown in the plot (F1, axial strain, 1.917% for the first cycle and 0.141% for the last cycle, lateral strain, −2.405% for the first cycle and −3.399% for the last cycle; CF1: axial strain, 2.783% for the first cycle and 0.452% for the last cycle, lateral strain, −5.542% for the first cycle and −4.279% for the last cycle)

Fig. 11
figure11

Volumetric strain per cycle of F1 and CF1. The values for the first cycle and the last cycle, far larger than the others, are not shown in the plot (F1, −2.893% for the first cycle and −6.659% for the last cycle, CF1, 8.301% for the first cycle and 8.106% for the last cycle)

The tested specimens contain natural cleavage and different components. These components have different mechanical properties (e.g. elastic modulus, Poisson’s ratio, etc.), leading to a mechanically heterogeneous material (Toribio et al. 2016). Thus, loading and unloading of such material results in plastic deformation (residual deformation) and residual stress. In the LCF tests, the crystals inside the rock salt undergo a time-dependent dislocation slip during the loading stage (Carter and Hansen 1983). Some dislocations are blocked by obstacles while in other microcracks can evolve directly (Le and Tran 2016; Senseny et al. 1992). During the unloading stage, some crystal dislocations slip back driven by residual stress. This reverse dislocation slip can overcome some obstacles and move toward the crystal boundary, which may induce micro-fractures and increase the material damage. This could explain the plastic deformation and the cumulative damage in the LCF tests.

The introduction of dwell times can accelerate the accumulation of plastic deformation and damage mainly for two reasons. First, the number of dislocations and the degree of slip increase with time because this process is time-dependent. The presence of hold time increases the duration per cycle, and thus increases the damage per cycle. Second, the plastic deformation of crystals is the result of dislocation on the slip surface. During the slip stage, the dislocations accumulate and cause the blocking effect of the dislocation groups when they face obstacles. This effect results in stress concentration, which can overcome obstacles and rupture different types of crystals. The introduction of hold times increases the intensity of the stress concentration and thus reduces the number of blocks. Therefore, the dislocation slips back more easily and more rapidly during unloading, which accelerates the formation of micro-fractures and material damage compared with the LCF tests (Chen et al. 2015; Mamun et al. 2017).

Conclusion

For underground salt caverns in a compressed air energy storage (CAES) system, the surrounding rock salt is under creep-fatigue loading, and its deformation behaviour needs to be investigated. This study presents low-cycle fatigue and creep-fatigue tests in a stress-controlled mode under different cycle stress amplitudes and hold times. Several conclusions can be outlined:

  1. (1)

    The deformation of rock salt under creep-fatigue loading consists of initial, steady and accelerated stages, similar to typical creep and fatigue behaviour. The steady strain rate increases with increasing the duration periods under the same stress levels, while it decreases with the increase of the lower limit stress under the same hold time and upper limit stress.

  2. (2)

    A novel damage model is derived based on the ductility exhaustion concept, which uses a nonlinear summation method to consider the synergistic effect of the creep and fatigue damage. The proposed model matches well with the test data and reflects the damage evolution of the three deformation phases in rock salt. The strain energy density is suitable to define the damage variable under creep-fatigue loading.

  3. (3)

    The presence of hold times at the upper limit stress causes deformation enhancement and life reduction, which can be interpreted by a dislocation mechanism.

Abbreviations

D :

Total damage

\(D_{C}\) :

Creep damage

\(D_{F}\) :

Fatigue damage

\(D_{N}\) :

Total damage at the end of \(N\) cycles

\(\overline{\sigma }\) :

Deviatoric stress

\(t_{H}\) :

Hold time per cycle

\(L_{r}\) :

Life reduction ratio

\(\dot{\varepsilon }\) :

Steady-state creep strain rate

\(\varepsilon_{C}^{*}\) :

Multi-axial creep ductility

\(\varepsilon_{C}\) :

Axial creep fracture strain

\(\varepsilon_{F}\) :

Fatigue ductility

\(\varepsilon_{p}\) :

Cumulative plastic strain at the end of \(N - 1\) cycles

A, n :

Material- and temperature-dependent constants in Norton powerlaw

S :

Load force

a, b, c, f, m :

Material-dependent constants in Xu et al.’s plastic strain evolution formula

\(W_{dN}\) :

Plastic strain energy of cycle \(N\)

\(\sigma_{N}\) :

Stress amplitude of cycle \(N\)

\(N_{F}\) :

Fatigue life in the proposed damage model

\(N_{f0}\) :

Cycle numbers to failure in the LCF tests

\(N_{cf}\) :

Cycle numbers to failure in the creep-fatigue tests

\(N\) :

Cycle number

\(p\) :

Creep-fatigue interaction damage exponent

\(\sigma_{\max }\) :

The upper limit stress

\(\sigma_{\min }\) :

The lower limit stress

\(\sigma_{m}\) :

Mean stress

\(\sigma_{eq}\) :

Equivalent stress

\(R^{2}\) :

The coefficient of determination

h :

Stress triaxiality in Cocks and Ashby model

\(W_{d}\) :

Cumulative strain energy at the end of \(N\) cycles

\(W_{0}\) :

Cumulative strain energy at the end of the final cycle

\(k_{1}\), \(k_{2}\) :

Material-dependent constants

\(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) :

Maximum, intermediate and minimum principal stresses

\(\varepsilon_{N}\) :

Strain amplitude of cycle \(N\)

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Acknowledgements

The authors are sincerely grateful to Professor J. J. K. Daemen (Mackay School of Earth Sciences and Engineering, University of Nevada, USA) for his linguistic assistance during the preparation of the manuscript. The authors would gratefully like to acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51874274, 51774266) and Youth Innovation Promotion Association CAS (Grant No. 2019324). The authors would like to thank the Science and Technology Research Project of Jiangxi Provincial Department of Education (Grant No. GJJ200634).

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Zhao, K., Ma, H., Yang, C. et al. Damage Evolution and Deformation of Rock Salt Under Creep-Fatigue Loading. Rock Mech Rock Eng (2021). https://doi.org/10.1007/s00603-020-02342-6

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Keywords

  • Creep-fatigue
  • Rock salt
  • Damage evolution
  • Hold time
  • CAES