# Creep constitutive model for salt rock considering loading and unloading path based on state variables

Jinyang Fan, Fan Yang, Zongze Li, Yang Zou, Jie Chen, Deyi Jiang, Daniel

Nelias

► To cite this version:

Jinyang Fan, Fan Yang, Zongze Li, Yang Zou, Jie Chen, et al. Creep constitutive model for salt rock considering loading and unloading path based on state variables. Geomechanics and Geophysics for Geo-Energy and Geo-Resources, 2025, 11 (1), 10.1007/s40948-025-00944-4. [hal-05015851](https://hal.science/hal-05015851v1)

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Submitted on 22 May 2025

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# Creep constitutive model for salt rock considering loading and unloading path based on state variables

Jinyang Fan · Fan Yang · Zongze Li · Yang Zou · Jie Chen · Deyi Jiang · Daniel Nelias

Received: 8 October 2024 / Accepted: 15 February 2025
© The Author(s) 2025

**Abstract** Creep behaviour in rocks is a typical mechanical property that is directly linked to the stability of underground engineering. The deformations and rate of rocks creep are not only influenced by time but also by the loading and unloading history. To more accurately predict creep mechanical behaviour of salt rocks, the rocks hardening is described by introducing a state variable. A new three-dimensional creep constitutive model of salt rocks was established to describe the loading and unloading history effect of the rheological properties. In this paper, salt rocks creep tests under various loading and unloading histories were conducted to investigate how different loading routes affect the creep behaviour of salt rocks. The effects of the model state variables were analysed through different indicators. An example verification was carried out with the results of plastic deformation

tests performed at different loading paths. The findings indicated the creep rate of stepped loading and the stepped unloading under the same stress level were significantly affected by the loading history. The proposed constitutive model can accurately fit the creep test curves of different loading paths, indicate that it can provided a prediction of the historical effect of the creep behaviour of salt rocks. Different parameters affect the different phases of the creep curve. The parameter k primarily affects the overall shape of the creep curves. Parameters m and c primarily influence the steady-state creep length and creep rates, excluding the initial cycle.

## Highlights

A new creep constitutive model of salt rocks is established that takes loading and unloading history into account.

State variables are introduced to describe the degree of strain hardening.

Model verification of salt rocks creep tests with different loading and unloading histories.

The variation characteristics of creep strain for salt rock with state variables are studied by different parameters.

<footnote>J. Fan · F. Yang · J. Chen · D. Jiang
State Key Laboratory of Coal Mine Disaster Dynamics and Control, School of Resources and Safety Engineering, Chongqing University, Chongqing 400044, China</footnote>

<footnote>Z. Li (✉) · Y. Zou (✉)
Division of Mining and Geotechnical Engineering, Luleå University of Technology, 97187 Luleå, Sweden
e-mail: zongze.li@associated.ltu.se</footnote>

<footnote>Y. Zou
e-mail: yang.zou@ltu.se</footnote>

<footnote>D. Nelias
INSA Lyon, CNRS, LaMCoS, UMR5259, 69621 Villeurbanne, France</footnote>
**Keywords** Salt rock · Creep model · Strain hardening · State variables · Loading and unloading

# 1 Introduction

Salt rock is a unique geological material known for its dense structure, low permeability, and self-healing properties, which has high stability and excellent sealing to salt caverns (Schulze et al. 2001; Lyu et al. 2021; Ślizowski et al. 2017; Grgic et al. 2021). These characteristics make salt caverns ideal for natural gas and oil storage, compressed air energy storage, helium and hydrogen storage, etc., and play a crucial role in energy system peaking (Wang et al. 2018; Tang et al. 2024). Salt rocks mineral resources are plentiful and distributed in major industrial countries worldwide. For instance, in Europe, numerous salt mines have been in operation since the 1920s, and salt caverns formed after water-soluble mining have been utilized as energy storage facilities (Blanco and Faaij 2018), as shown in Fig. 1.

Understanding the long-term rheological characteristics and stability of salt rocks is crucial, especially as it serves as an ideal surrounding rock for underground reservoirs (Li et al. 2023a; Azabou et al. 2021). Salt rocks have significant creep mechanical behaviour under the dual effects of geo-stress and time. Over extended periods of injection and production operations, salt rock creep can lead to accidents like cavity collapse, posing significant risks to the safety of gas storage reservoirs (He et al. 2023; AbuAisha et al. 2021; Blanco and Faaij 2018; Jin and Cristescu 1998). There are three phases to its usual creep curves: the decay phase, when the creep rate decreases, the steady-state phase, when the creep rate is relatively constant, and the accelerated phase, when the creep rate increases.

Over the past three decades, catastrophic accidents have been recurring in salt rock underground oil and gas reservoirs worldwide. These disasters, including oil and gas leakage, cavity failure, and surface subsidence, have had severe environmental and energy reserve security consequences (Peng et al. 2022). Particularly, when the depth of oil and gas storage in the salt cavern ranges between 1000 and 2000 m, the stability and the rate of convergence of the salt caverns can be affected by the formation of elastic-plastic damage zones. The Eminence salt

acupoint in Mississippi, USA, situated at a depth of 1700 m to 2000 m, experienced a remarkable volume convergence rate of up to 40% within two years after its completion in the 1970s (Desai and Varadarajan 1987). Similarly, the Tersanne salt cavern in France exhibited a high-volume convergence rate of around 30% over a period of 10 years (Liu et al. 2021), as illustrated in Fig. 2. To guarantee the stability of salt cavern, it is essential to comprehensively understand the long-term creep characteristics of salt rocks and build constitutive models that can predict deformations behaviours (He et al. 2022; Zhang et al. 2024; Ma et al. 2017; Arson et al. 2012).

Due to the complexity and diversity of practical engineering projects, the loading is constantly changing, so geomaterials undergoing rheological changes may be affected by the loading and unloading history (Liu et al. 2020b; Fan et al. 2020; Li et al. 2023b, 2024b; Jiang et al. 2016). For example, underground mine roadway engineering subject to long-term rheological action is affected by surrounding mining activities, underground salt rock gas storage is disturbed by frequent injection and mining activities, and a concrete dam body is subject to the combined action of long-term rheological stress and periodic stress. The long-term rheological effects and loading and unloading effects of these engineering projects are coupled and interact with each other, which increases the difficulty of recognizing and evaluating their stability. Roberts et al. (2015) studied the effect of cyclic loading and unloading on the capacity expansion characteristics and creep response of Avery Island salt rocks in the United States and found that cyclic loading and unloading had little effect on creep. Wu et al. (2020) established a viscoelastic-plastic rock model based on the cyclic triaxial loading and unloading creep test of rock. Hamami, (2006) performed triaxial multiphase creep tests to study the strain hardening properties of salt rocks during creep. Fan et al. (2019) researched the fatigue process of salt rocks during periodic loading and unloading and the changes in this process with different test time intervals. They showed that loading and unloading significantly impacted the deformations characteristics of salt rocks. Liu et al. (2020a) carried out cyclic axial stress loading and unloading tests on sandstone under different loading and unloading rate combinations, which showed that the loading and unloading history significantly impacted the axial stress of sandstone.
<figure><img src="image_2.png" /><figcaption>**Fig. 1** Distribution of underground salt deposits and caverns fields in Europe</figcaption></figure>

To quantitatively describe the creep deformation behaviour of rocks, various creep constitutive models have been proposed to depict the time-dependent deformation evolution of rocks, such as empirical models and component combination creep models.

(Wang et al. 2015) An empirical model is a functional expression established based on a large number of rock creep test results using regression fitting methods in mathematical statistics (Huang et al. 2020). Liu et al. (2010) pointed out, both the exponential
<figure><img src="image_3.png" /><figcaption>**Fig. 2** Creep contraction in Eminence (USA), Kiel (Germany) and Tersanne (France). The dashed surface represents the insoluble sediment at the bottom of the caverns. Volume losses for the Kiel cavern are not available</figcaption></figure>

function-type empirical model and the logarithmic function-type empirical model can better reflect the creep deformation behaviour of limestone. Cristescu (1993) established an empirical model that can describe the decay and steady creep stages based on the results of salt rock creep tests. The greatest advantage of empirical models is their simple form and fewer parameters. However, such models can only describe the creep behaviour of rocks under specific experimental conditions. Additionally, the physical significance of the model parameters is unclear, making them difficult to apply in numerical simulations and practical engineering.

The component combination model is based on the assumptions of rock viscosity, elasticity, and plasticity, combining basic components through series, parallel, or similar configurations (Yan et al. 2020; Li et al. 2022). Due to the relatively clear physical significance of the parameters in the component combination model, it can intuitively characterize the complex mechanical properties of rocks. Mansouri and Ajalloeian, (2018) investigated the creep behavior of salt rock under uniaxial compression (constant axial stress and cyclic confining pressure) using the Burgers model. Liu et al. (2020a, b) obtained a viscoelastic-plastic model describing the creep behavior of salt rock by first connecting the plastic element and the Kelvin model in parallel, and then connecting

them in series with the Maxwell model. However, in traditional component combination models, mechanical parameters such as elastic modulus and viscosity coefficient are typically treated as constants, leading to poor fitting performance and an inability to capture the accelerated creep stage.

In order to clarify the deformation progression of rocks, several creep constitutive models have been put forth. The classical component combination model is capable of effectively representing the complex mechanical characteristics of rocks, the parameters are usually regarded as constants (Mahmoudi et al. 2019; Singh et al. 2018). There are certain differences between a test data curve and the corresponding model-predicted curve, particularly during the accelerated creep phase (Shao et al. 2006; Gao et al. 2022; Wang et al. 2023; Firme et al. 2016). To describe the accelerated creep behaviour, a number of investigations have taken into account parameters of the rocks by incorporating a variable that changes with time in new creep models. Xue et al. (2021) considered the combined effect of temperature and volumetric stress in deep salt rocks formations. A damage variable was constructed based on the Weibull distribution of the voxel strength. The Hooke model, which took into account plastic damage, was developed to characterise damage at the scale of specimens. Thereby, a static constitutive model was established under
triaxial compression. Zhou et al (2011) studied the concepts of continuum damage mechanic and fractional derivative. They introduced a modification to the Nishihara model by substituting the viscous element with a variable-order Abel damper. Therefore, they put forth a fractional-order creep model of salt rocks.

According to rock damage evolution formulars proposed by Xie et al (2011). Eslami et al (2019) proposed nonlinear creep constitutive formulars reflecting the process of salt rocks creep. These models have established an excellent foundation for clarifying the creep behaviour of rocks, but few considered the impact of the loading and unloading history on the rheological property. Since there was no reliable way to differentiate among the plastic strain, viscoelastic creep strain, and viscoplastic creep strain, these model parameters were not determined with sufficient accuracy (Liu et al. 2023).

The creep characteristics of salt rock are not only closely related to the loading process under constant stress but are also significantly influenced by the history of loading and unloading. The aforementioned scholars have all conducted highly effective research work, laying a solid foundation for establishing a reasonable rock creep model. However, the description of the loading and unloading process is relatively simple, making it difficult to accurately simulate the impact of stress history on creep behavior. This study investigates the rheological properties of salt rock using different loading paths. Based on state variables that characterize the hardening degree of the rock, a novel salt rock creep constitutive model is established, taking into account the effects of loading and unloading history. This enables a more accurate representation and prediction of the deformation behavior of salt rock.

## 2 Creep model

### 2.1 Basic features of creep

Usually, there are three phases to the rocks creep process (Shkuratnik et al. 2019). Namely, the decay (variable), creep, the steady-state creep, and the accelerated creep. The Norton model is usually used to characterise the correlation between the deformation

rate and stress during the steady-state creep phase, such as (Yoshioka et al. 2022; Pan and Zhou 2023):

$$ \dot{\epsilon} = A\sigma^n \quad (1) $$

where $\dot{\epsilon}$ is the deformations rate, representing the first derivative of strain to time. $\sigma$ is stress. A and n are the material coefficients. The model does not adequately consider the temporal correlation of the creep rate, particularly in regard to the varied creep rate observed during the first and third phases. Some researchers have developed nonlinear creep constitutive models based on the creep behaviour observed in the first two stages. For instance, the Abel component model based on the nonlinear change in creep rate is (Deng et al. 2022):

$$ \epsilon = \frac{\sigma^{n+1}}{\eta_1} (t + m\sqrt{t}) - \frac{\sigma}{\eta_2} \exp \left[ -\frac{E_2}{\eta_2} \sigma^n (t + m\sqrt{t}) \right] \quad (2) $$

where $\epsilon$ is strain. $t$ is time. $\eta_1$, $\eta_2$, $m$ are all material parameters. $E_2$ is elasticity modulus.

Fractional order models include (Wang et al. 2021)

$$ \epsilon = \frac{\sigma}{\eta_1 \Gamma(\beta + 1)} + \frac{\sigma - \sigma_s}{\eta_2 \Gamma(\gamma + 1)} t^{\gamma} \quad (3) $$

where $\sigma_s$ is the yield limit, $\gamma$ is material parameter. The model described in formular (3) can be simply summarized as follows.

$$ \epsilon = a\sigma^n f(t) \quad (4) $$

Here, a is a constant. $\sigma^n$ is the stress correlation term. $f(t)$ is the time correlation term. As time approaches infinite, the first derivative of the $f(t)$ part of the model with respect to time can be simplified to

$$ f_1(t) = t \quad (5) $$

$$ f_2(t) = t + m\sqrt{t} \quad (6) $$

$$ f_3(t) = t^{\beta} + bt^{\gamma} \quad (7) $$

where $\beta$ is the material parameter. $b$ is a constant.

Except for the result of formular (5), the results of above formulars increase nonlinearly when the time approaches infinity. This model cannot accurately represent the nonlinear characteristics of rocks, resulting in significant errors in predicting the long-term creep behaviour of rocks.
Lyu et al (2021) researched creep behaviour of salt rocks at different low-stress levels through long-term uniaxial creep experiments. At a pressure of 15 MPa, the test data obtained from the creep test of salt rocks were processed. The researches indicated that the creep rates of decay and accelerated creep are more significant than that of steady-state creep. whether axial or radial creep. The duration of steady-state creep is approximately 3/4 of the total test time, which was much longer than the duration of decay creep and accelerated creep. Therefore, steady-state creep is an essential and nonnegligible part of long-term salt rock creep tests.

## 2.2 New creep model development

Given this, according to the stage characteristics of creep deformations, the following component creep model is proposed, which consists of a damping component and friction component, as shown in Fig. 3.

The friction component characterizes the starting conditions for plastic deformation. Damping component 1 represents the steady-state creep process of salt rocks with a linear relationship between the strain rate and the power of the stress.

$$ \dot{\epsilon} = a\sigma^n \qquad (8) $$

where a is the material parameter.

Damping component 2 characterizes the salt rock behaviour during a variable creep rate, as in the first deformation phase. The relationship between strain and stress power and time follows a hyperbolic relationship.

$$ \epsilon = \frac{b\sigma^n}{1 + c\sigma^m t^k} \qquad (9) $$

where c and k are the material parameters. $\sigma^m$ is the stress correlation term.

Combining formulars (8) and (9), the following relationship is obtained:

$$ \epsilon = a\sigma^n \left( t + \frac{b}{1 + c\sigma^m t^k} \right) \qquad (10) $$

Salt rock is NaCl polycrystalline. The microcosmic mechanism of plastic-creep interaction is the coupling between the slip motion of the dislocation within the grain and the climb motion of the dislocations within the grain boundary and its interference surface (Ha et al. 2017; He et al. 2018). At the initial stage of deformation, the density of slip dislocation increases rapidly, and the grain boundary is formed by climbing (Gleason and Tullis 1995). When the applied stress is greater than the internal stress, more dislocation is generated and the internal stress is increased, and the deformation rate is gradually slowed down. When the applied stress reaches the internal stress, it begins to enter the yield state. The proliferation and slip of the dislocation tend to be stable, and the deformation rate reaches a stable state. In this paper, the stress-strains relationship of component 2 was used to define the state variables, representing the hardening degree $\sigma^*$ associated with the internal structure and dislocation density of the salt rocks. The state variable can be expressed as

$$ \frac{\sigma^*}{\sigma} = \frac{1}{1 + c\sigma^m t^k} \qquad (11) $$

<figure><img src="image_4.png" /><figcaption>Fig. 3 Schematic of the creep model components E: Elastic element; $\eta_1$: damping component 1, characterizing the salt rock variable speed creep process; $\eta_2$: damping component 2,</figcaption></figure>

characterizing the salt rock steady creep process; $\sigma_{s0}$: friction component 1, characterizing plastic starting conditions; $\sigma_{s1}$: friction component 2, characterizing loading plasticity
$$\sigma^* = \frac{\sigma}{1 + c\sigma^m t^k} \qquad (12)$$

The strain formular can be rewritten as

$$\epsilon = a\sigma^n \left(t + \frac{\sigma^*}{\sigma}b\right) \qquad (13)$$

State variables can characterize the stress magnitude and its effect on salt rock creep. When applying stress to the rocks, the new state variables can be represented as

$$\sigma^* = \frac{\sigma}{1 + c\sigma^m (t + \sigma_0^{-*})^k} + \sigma_0^* \qquad (14a)$$

$$\sigma_0^{-*} = \left( \frac{\sigma - \sigma_0^*}{c\sigma^m \sigma_0^*} \right)^{1/k} \qquad (14b)$$

where $\sigma_0^*$ represents the original state. $\sigma_0^{-*}$ is the time variable corresponding to $\sigma^*$.

The strain is rewritten as

$$\epsilon = a\sigma^n \left[ t + \frac{b}{1 + c\sigma^m(t + \sigma_0^{-*})^k} \right] \qquad (15)$$

By calculating the first-order derivative of time for formular (14) and (15), we obtain the state rate as:

$$\frac{\partial \sigma^*}{\partial t} = \dot{\sigma}^* = - \frac{c k \sigma^{m+1} (t + \sigma_0^{-*})^{k-1}}{\left[ 1 + c \sigma^m (t + \sigma_0^{-*})^k \right]^2} \qquad (16)$$

The strain rate is obtained as

$$\frac{\partial \epsilon}{\partial t} = \dot{\epsilon} = a\sigma^n \left\{ 1 - \frac{b c k \sigma^m (t + \sigma_0^{-*})^{k-1}}{\left[ 1 + c \sigma^m (t + \sigma_0^{-*})^k \right]^2} \right\} \qquad (17)$$

From the above formulars, it can be observed that when stress changes, the strain rate correspondingly changes. When in a loading state, if the stress loading rate is $v$, the state variables can be obtained by integrating the following formular:

$$\sigma^* = c k \int \frac{(\sigma_0 + vt)^{m+1} (t + \sigma_0^{-*})^{k-1}}{\left[ 1 + c (\sigma_0 + vt)^m (t + \sigma_0^{-*})^k \right]^2} dt + \sigma_0^* \qquad (18)$$

Therefore, the strain during the loading process can be expressed as

$$\epsilon = a \int (\sigma_0 + vt)^n \left\{ 1 - \frac{b c k (\sigma_0 + vt)^m (t + \sigma_0^{-*})^{k-1}}{\left[ 1 + c (\sigma_0 + vt)^m (t + \sigma_0^{-*})^k \right]^2} \right\} dt + \epsilon_0 \qquad (19)$$

where $\epsilon_0$ is the initial strain. $v$ is the speed of stress.

## 2.3 Modification during unloading

When in an unloading state, it's possible that the externally applied stress is less than the state variables. In this case, the state variables can be expressed as:

$$\sigma^* = \frac{\sigma}{1 - c\sigma^m(t + \sigma_0^{-*})^k} \qquad (20)$$

When in an unloading state, the internal adjustments rely entirely on internal forces, resulting in a relatively slow adjustment rate. Introducing the decay creep unloading factor $U_1$, calculating the first-order derivative of time, we obtain the state rate as:

$$\dot{\sigma}^* = U_1 \frac{c k \sigma^{m+1} (t + \sigma_0^{-*})^{k-1}}{\left[ 1 - c \sigma^m (t + \sigma_0^{-*})^k \right]^2} \qquad (21a)$$

$$\sigma_0^{-*} = \left( \frac{\sigma_0^* - \sigma}{c\sigma^m\sigma_0^*} \right)^{1/k} \qquad (21b)$$

At this time, the deformations rate of steady-state creep will also be influenced. Introducing the steady-state creep unloading factor $U_2$, the strain formular can be expressed as:

$$\epsilon = a \int (\sigma_0 + vt)^n \left\{ U_2 + \frac{U_1 b c k (\sigma_0 + vt)^m (t + \sigma_0^{-*})^{k-1}}{\left[ 1 - c (\sigma_0 + vt)^m (t + \sigma_0^{-*})^k \right]^2} \right\} dt + \epsilon_0 \qquad (22)$$

The above formular is simplified to obtain the following formular:

$$\epsilon = a \int (\sigma_0 + vt)^n \left( U_2 + \frac{U_1 b \dot{\sigma}^*}{\sigma_0 + vt} \right) dt + \epsilon_0 \qquad (23)$$

Formular (23) represents the integrated form of the salt rock creep constitutive model that comprehensively considers the effects of loading and unloading.
## 2.4 Three-dimension form

The potential function is usually used to characterize the direction of plastic flow in plastic constitutive models (Khaledi et al. 2016). If the potential function is similar to the yield function, it is called the correlation criterion. If it is different, it is called the noncorrelation criterion. In most cases, the flow direction measured in the experiment does not have a strong relationship with the yield function. Therefore, the non-associated criterion is used here (Ren et al. 2021).

$$ \epsilon_{ij} = \gamma \langle F \rangle \frac{\partial Q_i}{\partial \sigma_{ij}} \qquad (24) $$

F is the yield function, which serves as the initiation condition for creep deformation. Q is the plastic potential function, where its partial derivative with respect to time characterizes the direction of plastic flow. γ represents the relationship between creep deformation and stress dynamics, which can be calculated using formular (23). Theoretically, this model has a particular proportional relationship between the plastic deformations in each direction. However, most test results do not reflect this relationship. The accumulation of NaCl polycrystalline particles forms salt rock, and its deformation is mainly dominated by dislocation generation and slip (Taheri et al. 2020; Zill et al. 2021; Thiemeyer et al. 2015; Yubero et al. 2021). Therefore, the direction of the slip is related to that of plastic flow.

When the salt rock is in steady-state creep, the internal dislocation and external forces reach a dynamic equilibrium. The generation, slip and removal of dislocations occur within an orderly and stable state. Dislocations primarily move along the direction of maximum shear stress, known as the easiest slip system direction, resulting in plastic deformation proportional to each direction (Li et al. 2024a; Lu et al. 2021). During the first stage, the external force was significantly larger than the internal force, leading to rapid proliferation, slip of dislocations. In the rapid generation and slip process, the "super-saturation" state of dislocations may occur, resulting in the emergence of new slip systems, such as climbing and other behaviours, thus causing changes in the proportions of plastic deformation in all directions. Therefore, a "double potential function" constitutive

model of the plastic potentials of the two phases is established in this paper.

$$ \epsilon_{ij} = \gamma_1 \langle F \rangle \frac{\partial Q_1}{\partial \sigma_{ij}} + \gamma_2 \langle F \rangle \frac{\partial Q_2}{\partial \sigma_{ij}} \qquad (25) $$

The viscoelastic-plastic constitutive formular is

$$ \epsilon_{ij} = \gamma_1 \langle F \rangle \frac{\partial Q_1}{\partial \sigma_{ij}} + \frac{\Delta \sigma}{E} + \gamma_2 \langle F \rangle \frac{\partial Q_2}{\partial \sigma_{ij}} + \frac{\Delta \sigma}{E} \qquad (26) $$

$\gamma_1$ and $\gamma_2$ respectively represent the kinetic functions for the first and second phases. $Q_1$ and $Q_2$ respectively represent the plastic potential functions for the first and second phases. F is the initiation condition shared by the first and second stages. E is the elastic modulus. Here, the plastic potential functions adopt a form similar to the D-P criterion.

$$ Q = \sqrt{J_2} - \alpha I_1 \qquad (27) $$

where $J_2$ is the second invariant of the stress deviator, $I_1$ is the first invariant of the stress tensor, and $\alpha$ represents the material parameter.

# 3 Model verification and improvement

## 3.1 Creep deformation under constant stresses

Curve fitting is a common method to determine the parameters of the creep constitutive model. This method can use all test data to ensure the minimum overall error of the theoretical and test results. Using model formular (23), the stress levels of various stages in the constant stress creep test (CTCT) on salt rock were fitted, and the results of axial fitting and test curves are in Fig. 4. Under the axial stress at each stage, both decay and steady-state creep strains are generated. The fitting curves of the axial and radial strains for only the five phases when the stress level is constant at 4 MPa, 8 MPa, 12 MPa, 16 MPa and 20 MPa. The parameters of the proposed model under different stresses are obtained by fitting formular (22) to the creep test curve, as shown in Table 1.

The model formular (22) was used to fit the stress levels at all levels in the salt rock CTCT, and the comparison diagram of radial fitting curves and test results were obtained, as shown in Fig. 5. When the stress level was 4 MPa, the radial deformation
<figure><img src="image_5.png" /><figcaption>**Fig. 4** Comparison of the constant stress axial creep tests (CTCT) results and the model fitting curves of the decay creep phases and the steady creep phase under different stress levels **a** 4 MPa (1st) **b** 8 MPa (2nd) **c** 12 MPa (3rd) **d** 16 MPa (4th) **e** 20 MPa (5th)</figcaption></figure>
<table><caption>**Table 1** Axial model fitting parameters values of the constant stress creep test (CTCT) under different stress levels</caption><thead><tr><th>Stress/MPa</th><th>a</th><th>b</th><th>c</th><th>n</th><th>m</th><th>k</th></tr></thead><tbody><tr><td>4</td><td>11</td><td>1.25</td><td>200</td><td>1.95</td><td>1</td><td>-0.72</td></tr><tr><td>8</td><td>11.5</td><td>1.05</td><td>42</td><td>2.13</td><td>0.7</td><td>-0.68</td></tr><tr><td>12</td><td>6.9</td><td>1.68</td><td>45</td><td>2.1</td><td>1</td><td>-0.61</td></tr><tr><td>16</td><td>6.9</td><td>1.4</td><td>45</td><td>2.1</td><td>1.2</td><td>-0.64</td></tr><tr><td>20</td><td>6.5</td><td>2</td><td>52</td><td>2.2</td><td>0.9</td><td>-0.7</td></tr></tbody></table>

variable was 0. This section only showed the fitting results when the stress level was constant at 8 MPa, 12 MPa, 16 MPa and 20 MPa. The radial creep model parameters under different stresses were obtained, as shown in Table 2.

The model comprehensively considers the influence of time, load, and creep state (steady-state creep

phase) on creep characteristics. The model takes into account the several factors that affect creep characteristics, including time, loads, and the creep state (specifically, the steady-state phase). The fitting curves of all creep phases under the different stresses are good. The fitting curves for five sets of different stress levels and initial state variables closely match the test data

<figure><img src="image_6.png" /><figcaption>**Fig. 5** Comparison of the constant stress radial creep test (CTCT) results and the model fitting curves of the decay creep phases and the steady creep phase under different stress levels (a) 8 MPa (2nd) b 12 MPa (3th) c 16 MPa (4th) d 20 MPa (5th)</figcaption></figure>

<figure><img src="image_7.png" /></figure>
<table><caption>**Table 2** Radial model fitting parameters values of the constant stress creep test (CTCT) under different stress levels</caption><thead><tr><th>Stress/MPa</th><th>a</th><th>b</th><th>c</th><th>n</th><th>m</th><th>k</th></tr></thead><tbody><tr><td>8</td><td>14.8</td><td>1.09</td><td>44.2</td><td>2.2</td><td>0.712</td><td>-0.6</td></tr><tr><td>12</td><td>11.6</td><td>1.68</td><td>40</td><td>2.23</td><td>1</td><td>-0.602</td></tr><tr><td>16</td><td>10.5</td><td>1.6</td><td>45</td><td>2.15</td><td>1.2</td><td>-0.62</td></tr><tr><td>20</td><td>9.8</td><td>2.49</td><td>40</td><td>2.1</td><td>1.19</td><td>-0.62</td></tr></tbody></table>

curves, with high conformity and minimal errors. This indicates that the proposed model can effectively describe the creep characteristics of salt rock under various loading conditions and the model has good applicability.

## 3.2 Creep deformation under stepped loading

For the salt rock stepped loading creep test (SLCT), the comparison diagram between the axial and radial fitting curves was obtained with the model formula (23). The SLCT stress loading path was shown by the blue line in Fig. 6. First, the test was loaded at a loading rate of 1kN/s. When the stress level was 4 MPa, it was maintained for 2 h. Then, according to the same loading rate, each stress level was increased by 4 MPa for stepped loading. The test was maintained for 2 h at each stress level until the specimen was failed. Figure 6 shows the test curves. The salt rock specimens were loaded with a classified loading creep method. Decay and steady-state creep strains were generated at the entire stress levels. When the stress was applied in the range of 20–24 MPa, the curve showed

an accelerated creep phase, and the specimen became unstable and experiences failure.

The model takes into account all the factors that can affect creep characteristics, including time, loads, and creep state (both the loading and the creep phases). From the fitting results, the creep and loading phase can be fitted well. For the axial fitting, the tests for the loading phase and creep phase of the 1st, 4th and 5th stress levels had a higher degree of fit with the fitting curve. Although there is some error present, the fitting curves for the loading stage and creep stage of the 2nd and 3rd steps generally follow the same trend as the test curves. Regarding the radial fitting curves, the fitting curves for the three loading phases from 3th, 4th and 5th stress levels and the four creep phases with constant axial stresses from 2nd, 3th, 4th and 5th stress levels were consistent.

Overall, the proposed model can effectively describe both the decay and steady-state creep phases of salt rocks. Due to the absence of consideration for the deformation development process of the accelerated creep phases, the fitting effect for this stage is relatively poor. Although there are some errors

<figure><img src="image_8.png" /><figcaption>**Fig. 6** Comparison of the stepped loading creep test (SLCT) results and the model fitting curves of the first two phases of the model a Axial creep strains curves; b Radial creep strains curves</figcaption></figure>
between the fitting and test curves, the overall fit-
ting result of the model is strong. This indicates that
the proposed creep constitutive model for salt rocks,
which considers the influence of loading history, can
effectively describe the creep characteristic of salt
rocks at graded loading conditions. It can accurately
reflect both the decay creep stage and steady-state
creep stage of salt rocks.

3.3 Creep deformation under stepped unloading

For the salt rock stepped unloading creep test (SUCT), the axial and radial fitting curves obtained by fitting model formular (23) were compared with the test results in Fig. 7. The SUCT stress loading path was shown by the blue line in Fig. 7. The test was loaded at a loading rate of 1 kN/s to a stress limit of 20 MPa and maintained for 2 h. The unloading was then carried out at the same rate, it was reduced one stress level every 4 MPa and maintained 2 h at each stress level. The salt rock specimens adopted the method of descending gradient graded creep. Both variable-rate and steady-state creep strains were generated at all stress level.

The model takes into account all the factors that
can affect creep characteristics, including time,
loads, and creep state (both unloading and creep
phases). The fitting curves fit the creep phase and the
unloading phase the best. For the axial fitting, the fit-
ting curves closely match the test curves for all four
unloading stages and creep stages, with very small

errors. In particular, the test and fitting results for
the 1st and 4th stress levels are completely consist-
ent. Although there is some error present, the differ-
ences between the test and fitting results for the 2nd
and 3rd stages are very small and can be considered
negligible. Regarding the radial fitting results, there
is also a high degree of agreement between the fit-
ting and test results. Thus, the model improved by
introducing state variables and variable-rate creep
and steady-state creep unloading factors in this paper
has a high degree of agreement with the descending
gradient graded creep test data. This indicates that
the salt rocks creep constitutive model, which consid-
ers the influence of unloading history, can effectively
describe the creep characteristics of salt rocks under
uniaxial graded unloading conditions. The compari-
son of axial and radial creep strain and creep rate for
SLCT and SUCT at different stress levels is presented
in Table 3.

According to the results in Table 3, in the SLCT,
as the stress level increases, both the axial and radial
creep rates and creep strains of the salt rocks increase.
In the SUCT, as the stress level decreases, negative
creep deformation occurs in both the axial and radial
direction. Except for the highest stress level, the creep
deformation rate of the salt rocks is much lower than
that in the SLCT at the same stress level.

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Comparison of the stepped unloading creep test (SUCT) results and the model fitting curves of the first two phases of the model a Axial creep strains curves; b Radial creep strains curves</figcaption></figure>
<table><caption>**Table 3** The axial and radial creep strain and creep rate test results comparison under different stress levels of stepped loading creep test (SLCT) and stepped unloading creep test (SUCT)</caption><thead><tr><th rowspan="3">Stress/MPa</th><th colspan="4">Stepped loading</th><th colspan="4">Stepped unloading</th></tr><tr><th colspan="2">Axial</th><th colspan="2">Radial</th><th colspan="2">Axial</th><th colspan="2">Radial</th></tr><tr><th>Creep strain/%</th><th>Creep rate/h</th><th>Creep strain/%</th><th>Creep rate/h</th><th>Creep strain/%</th><th>Creep rate/h</th><th>Creep strain/%</th><th>Creep rate/h</th></tr></thead><tbody><tr><td>4</td><td>0.057</td><td>0.0285</td><td>0</td><td>0</td><td>-0.012</td><td>-0.006</td><td>-0.005</td><td>-0.0025</td></tr><tr><td>8</td><td>0.180</td><td>0.090</td><td>0.016</td><td>0.008</td><td>-0.002</td><td>-0.001</td><td>0</td><td>0</td></tr><tr><td>12</td><td>0.304</td><td>0.152</td><td>0.433</td><td>0.2165</td><td>-0.003</td><td>-0.0015</td><td>0.008</td><td>0.004</td></tr><tr><td>16</td><td>0.460</td><td>0.230</td><td>0.689</td><td>0.3445</td><td>0.029</td><td>0.0145</td><td>0.084</td><td>0.042</td></tr><tr><td>20</td><td>0.930</td><td>0.465</td><td>1.545</td><td>0.7725</td><td>1.202</td><td>0.601</td><td>2.013</td><td>1.0065</td></tr></tbody></table>

## 3.4 Long-term creep deformation

For the proposed creep model, three sets of uniaxial creep tests are selected: the German Salt Rock Uniaxial Creep Test (GUCT), the Chinese Salt Rock Uniaxial Creep Test (CUCT), and the Turkish Salt Rock Uniaxial Creep Test (TUCT). These tests are used to fit the model under different creep times and conditions to verify the rationality and applicability of the model.

### 3.4.1 Germany salt rock uniaxial creep test

This test took approximately 3.5 years in a German laboratory with a loading stress of 14 MPa (Zeng et al. 2019). During the test, the specimens only experienced two phases of creep: decay creep and steady-state creep. The test only recorded the variation of axial strain over time. The fitting and test result for axial strain were shown in Fig. 8.

From the figure, it is evident that the fitting results are in high agreement with the test results, with small errors. This indicates that the model can effectively fit the decay creep and steady-state creep phases of long-term creep tests.

### 3.4.2 China salt rock uniaxial creep test

Liu et al. (2021) conducted six groups of uniaxial creep tests with a time of 21,000 h (875 days) under axial stress in the laboratory of Sichuan University, China. The changes in salt rock axial and radial creep strains with time were recorded in the test. Under different stress levels as shown in Table 4,

<figure><img src="image_10.png" /><figcaption>**Fig. 8** Comparison of the Germany uniaxial creep test (GUCT) results and the model fitting curve of first two phases of the model</figcaption></figure>

the salt rock deformations in the axial and radial directions were different. When the axial stress is less than 15 MPa, the creep evolution curve of the salt rock can only be divided into the decay and steady-state creep phases. The test and fitting results for axial strain and radial strain are shown in Fig. 9.

From the figure, it can be observed that with the increase of axial stress, both axial and radial strain increase significantly, indicating a notable strengthening effect of axial stress on the creep of salt rock. Both axial and radial decay creep strain and steady-state creep strain evolve over time can be well fitted. However, since this model only aims at the first two phases of creep deformation, the fitting of the accelerated creep phase at an axial stress of 15 MPa is inaccurate.
### 3.4.3 Turkey salt rock uniaxial creep test

In this test, several salt rock specimens were prepared from ores collected from Turkey's Tuzköy rock salt mine. Uniaxial creep tests are conducted under different constant loads for 3000 h (125 days) (Özşen et al. 2014). The test recorded the variation of axial and radial strains over time at an axial stress of 18.7 MPa.

The test and fitting results for axial strain and radial strain are shown in Fig. 10.

Form the Fig. 10, it can be observed that at the axial stress is 18.7 MPa, the specimens experienced three phases during the test time: decay, steady-state and accelerated creep. However, there were certain errors in fitting the individual test data for the axial and radial strains. Overall, the proposed model had

<table><caption>**Table 4** The stress level used in the China uniaxial creep test (CUCT)</caption><thead><tr><th>Test number</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><th>Stress level/MPa</th><td>7.5</td><td>8.75</td><td>10</td><td>11.25</td><td>12.5</td><td>15</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Fig. 9** Comparison of the China Uniaxial creep test (CUCT) results and the model fitting curves of the first two phases a Axial creep strains curves; b Radial creep strains curves</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10** Comparison of the Turkey uniaxial creep test (TUCT) results and the model fitting curves of the first two phases of the model a Axial creep strains curves; b Radial creep strains curves</figcaption></figure>

<figure><img src="image_13.png" /></figure>
an excellent fitting effect for decay creep and steady-state creep, indicating its reasonable validity. Similarly, around two-thirds of the test duration, the error between the test and fitting results began to increase. This is because the test enters the accelerated creep phase at this point.

The fitting parameters of the axial and radial models for the above three sets of tests from different countries under different stress conditions and test times were shown in Tables 5 and 6. Please note that the radial deformation data were not available for the GUCT.

The comparison of the three tests indicated that when the axial stress was similar, the creep stages experienced in salt rocks from different regions, as shown by the test results, are very close. In the CUCT, the test time was 875 days. During the test period, the salt rocks test only entered the accelerated creep stage when the axial stress exceeded 15 MPa. In contrast, in a long-term indoor GUCT, at an axial stress of 14 MPa (less than 15 MPa), the salt rock test did not enter the accelerated creep phase throughout the duration of 3.5 years. Similarly, in the TUCT, at an axial stress of 18.75 MPa (greater than 15 MPa), the salt rocks specimen entered the accelerated creep phase within 125 days of testing.

## 3.5 Model improvement

Although the fit between the test and the model curves was excellent, Figs. 6, 9, and 10 showed that there was a noticeable deviation in the later parts of the curves. This indicates that the model failed to characterize the accelerated features of the third creep phase. The third creep phase is fracture expansion, and the fracture expansion factor *d* is introduced here. The crack formation is mainly caused by dislocation accumulation. The stress field generated by a large accumulation of dislocations, combined with the stress field from external forces, reaches a fracture toughness, resulting in crack initiation (or nucleation), which initiates the formation of fractures (fracture initiation): the initiation of fissures occurs in a rapid, instantaneous process. So far, the evolution process has not been observed testly. Assuming the initial nucleation length is *d*₀ and it forms instantaneously. Its formation timing is related to the externally applied stress and accumulated plastic deformation.

$$K_c = \Gamma(\sigma, \varepsilon_p) \qquad (28)$$

$K_c$ represents the material's fracture toughness, $\varepsilon_p$ is the plastic strain, which is an inherent property of

<table><caption>**Table 5** The axial model fitting parameters values of the China uniaxial creep test (CUCT), Germany uniaxial creep test (GUCT) and Turkey uniaxial creep test (TUCT)</caption><thead><tr><th>Test</th><th>Stress/MPa</th><th>a</th><th>b</th><th>c</th><th>n</th><th>m</th><th>k</th></tr></thead><tbody><tr><td rowspan="6">CUCT</td><td>7.5</td><td>10.1</td><td>2.25</td><td>150</td><td>1.95</td><td>0.8</td><td>-0.67</td></tr><tr><td>8.75</td><td>9.9</td><td>2.2</td><td>150</td><td>1.95</td><td>0.8</td><td>-0.51</td></tr><tr><td>10</td><td>9.3</td><td>2.25</td><td>130</td><td>1.93</td><td>1.05</td><td>-0.45</td></tr><tr><td>11.25</td><td>7</td><td>2.2</td><td>350</td><td>2</td><td>0.6</td><td>-0.7</td></tr><tr><td>12.5</td><td>8</td><td>2.15</td><td>200</td><td>2</td><td>0.6</td><td>-0.66</td></tr><tr><td>15</td><td>5.7</td><td>2.3</td><td>200</td><td>2.1</td><td>1.3</td><td>-0.61</td></tr><tr><td>GUCT</td><td>14</td><td>9</td><td>6.4</td><td>250</td><td>2</td><td>1</td><td>-0.64</td></tr><tr><td>TUCT</td><td>18.7</td><td>7</td><td>7</td><td>20</td><td>2.1</td><td>1.2</td><td>-0.6</td></tr></tbody></table>

<table><caption>**Table 6** The radial model fitting parameters values of the China uniaxial creep test (CUCT) and Turkey uniaxial creep test (TUCT)</caption><thead><tr><th>Test</th><th>Stress/MPa</th><th>a</th><th>b</th><th>c</th><th>n</th><th>m</th><th>k</th></tr></thead><tbody><tr><td rowspan="6">CUCT</td><td>7.5</td><td>4.6</td><td>2</td><td>270</td><td>2.12</td><td>0.8</td><td>-0.55</td></tr><tr><td>8.75</td><td>4.7</td><td>1.8</td><td>230</td><td>2.2</td><td>1.8</td><td>-0.47</td></tr><tr><td>10</td><td>3.05</td><td>2</td><td>220</td><td>1.93</td><td>1.05</td><td>-0.6</td></tr><tr><td>11.25</td><td>3.4</td><td>2.05</td><td>300</td><td>2.1</td><td>0.5</td><td>-0.62</td></tr><tr><td>12.5</td><td>2.2</td><td>2.3</td><td>130</td><td>1.98</td><td>1.1</td><td>-0.58</td></tr><tr><td>15</td><td>2.7</td><td>2.2</td><td>300</td><td>2.25</td><td>1.1</td><td>-0.52</td></tr><tr><td>TUCT</td><td>18.7</td><td>9.9</td><td>2.05</td><td>35</td><td>2.2</td><td>1</td><td>-0.78</td></tr></tbody></table>
the material. Subsequently, each new crack formation is related to the rate of deformations development.

$$ \dot{d} = \mu_d \dot{\epsilon} \qquad (29) $$

where $\mu_d$ is the fracture factor.

$$ d = a\mu_d \int \sigma_{eff}^n \left\{ 1 - \frac{bck\sigma_{eff}^m (t + \sigma_0^{-*})^{k-1}}{[1 + c\sigma_{eff}^m (t + \sigma_0^{-*})^k]^2} \right\} dt + d_0 \qquad (30) $$

The effective stress $\sigma_{eff}^n$ can be obtained by

$$ \sigma_{eff}^n (1 - d) + d\lambda \frac{\sigma}{\sigma_c} = \frac{\sigma}{\sigma_c} \qquad (31) $$

where $\sigma_c$ is compressive strength.

Formular (31) can be rewritten as

$$ \sigma_{eff}^n = \frac{1 - d\lambda}{1 - d} \sigma \qquad (32) $$

$\lambda$ is the dynamic friction coefficient of the crack surfaces. When entering the third stage, the deformations are as follows:

$$ \epsilon = a \int \sigma_{eff}^n \left\{ 1 - \frac{bck\sigma_{eff}^m (t + \sigma_0^{-*})^{k-1}}{[1 + c\sigma_{eff}^m (t + \sigma_0^{-*})^k]^2} \right\} dt + \epsilon_0 \qquad (33) $$

The test results and fitting curves of the axial and radial in SLTC are shown in Fig. 6. The axial and radial creep curves of CUCT at an axial stress of 15 MPa are shown in Fig. 9. The axial and radial creep variation results for TUCT are shown in Fig. 8. As the specimen enters the accelerated creep stage, the curve is concave and the strain increases faster, breaking away from the original linear law of the steady state creep phase. This is because the salt rocks specimens begin to crack at this phase, significantly increasing the deformations. The model of formular (23) is only aimed at the decay and steady-state creep phases, and it does not account for the accelerated stage. Therefore, the fitting effect of the entire creep phases is poor. Formular (33) is fitted to the above three tests. The test and fitting result were shown in Fig. 11.

From the axial and radial fitting results with the test in Fig. 11, the three phases of creep deformation in salt rocks can be described by the revised model

that includes the crack factor. Additionally, the high degree of agreement between the test and fitting results indicates that the modified model is applicable to all three phases of creep deformation.

Compared with other models (Wang et al. 2022; Zhang et al. 2020), The constitutive model can not only consider the cumulative effect of historical loading and unloading, but also obtain good results for the fitting of the whole process (three phases) of the test, especially in the accelerated deformation phase.

## 4 Discussion on the effect of key parameters on creep strain

Since the creep tests fitted were all single loading or unloading, and the tests that considered both the loading and the unloading path were not given. The indoor salt rock trapezoidal wave creep test (TWCT) was conducted to verify whether the proposed constitutive model could better fit the salt rock creep path considering historical loading and unloading effects. Moreover, the influence of three key parameters on the development of the salt rock strain curve over time was discussed. The loading and unloading paths of the test are shown in Fig. 12.

### 4.1 Test conditions

The specimens are natural pure salt rock taken in the southern foothills of the Himalayas from Pakistan, buried at a depth of 800-1000 m. The specimens surface appears light red (impurities) or white in colour. Its main component is NaCl. The content of insoluble matter in other argillaceous components is less than 1%. The shape of the specimen is a standard 50×100 mm cylinder (Fig. 13a). Before the test, it is necessary to polish end faces of the specimens to ensure that the flatness of the two end faces is within ±0.02 mm. During the test, the test maintains stable temperature and humidity conditions, room temperature 23±2 °C, relative humidity 60±5%. The tests were proceeded at the State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University. The test utilized the MTS815 rock mechanics testing system manufactured by MTS Corporation, USA, as shown in Fig. 13b.

Creep is a time-dependent deformation process influenced by loading and unloading history,
<figure><img src="image_14.png" /><figcaption>**Fig. 11** Comparison of test and the model fitting results of the decay creep phase, steady creep phase and accelerated creep phase</figcaption></figure>

<caption>a Axial fitting of SLCT b Radial fitting of the SLCT
c Axial fitting of the CUCT d Radial fitting of the CUCT
(e) Axial fitting of the TUCT (f) Radial fitting of the TUCT</caption>
<figure><img src="image_15.png" /><figcaption>**Fig. 12** Salt rock trapezoidal wave creep test (TWCT) loading and unloading path $\sigma_{a,min} = 1$ MPa, $\sigma_{a,max} = 18$ MPa, $N = 15$</figcaption></figure>

path, and strain rate. To validate the applicability of the model and investigate the effect of loading and unloading history on the creep behaviour of salt rocks, the TWCT was conducted at room temperature. The test experienced four phases: load, constant load, unload, constant load in one cycle, which the holding time of the constant load phase is 2 h. The upper limit stress was set to 18 MPa, and the lower limit stress was set to 1 MPa. The one cycle period is 8 h. A total of 15 cycles (time=120 h) were carried out (Yang et al. 2024). The state variables, test results and fitted curves for each phase were shown in Fig. 14. Due to the selection of a relatively high upper stress limit and a low lower stress limit during testing, the salt rocks were in a steady-state deformation phase for a long period. Considering factors such as testing machine operation conditions, the test was stopped after the salt rock reached the steady-state deformation phase.

## 4.2 Test results

As shown in Fig. 14 a, the fitted curves and the test results matched well on the whole. It could be found from Fig. 14 b that in the TWCT loading and unloading test, the changes of state variables and deformation were consistent. The first phase had a large changes range. Except for the first cycle, both state variable and strain value changed decrease. The state variables were used to reflect the interaction between constant stress, loading and unloading during salt rock deformation. What's more, in a cycle period, when it was in the loading, the state variable value would increase. However, when it was in the constant stress or unloading, the state variable values hardly changed. This showed that the loading had a greater effect on the failure of salt rock than constant stress and unloading. From the model prediction/fitting curves, the model had good applicability to salt rock creep considering both loading and unloading. From the comparative results of Fig. 14c and d, it could be observed that the model demonstrated good predictive capability for the influence of loading and unloading on creep of salt rocks under trapezoidal wave loading. Although there was some error in the initial loading phase of the first cycle, the model's predictions aligned perfectly with the actual deformation trend in the steady-state deformation phase, indicating excellent fitting performance.

Based on the above test results and model characteristics. The parameter k, m and c in the constitutive model had a greater effect on the curve fitting, so the three parameters were selected for analysis the effects on the creep strain and state variables during loading and unloading in this section.

<figure><img src="image_16.png" /><figcaption>**Fig. 13** a The processed specimen; b The mechanical testing machine (1. computer control system, 2. mechanical loading system, 3. pressurization device)</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 14** Comparison of the TWCT results and the model fitting curve of the model a Test results and fitting curve; b State variable changes with time c TWCT deceleration test results</figcaption></figure>

and fitting curve, as shown by the blue dashed line box of Fig. 12a; d TWCT stability test results and fitting curve, as shown by the purple dashed line box of Fig. 12a

## 4.3 Influence of the *k* on the creep model

When stress and the other parameters are the same, only changing the value of *k* in the model, a set of different creep strain curves can be obtained, as shown in Fig. 15a and b. When the value of *k* is -0.1, there is no significant gradient change in the whole process of creep strain in salt rock. As the value of *k* continues to decrease, the creep strain curve shows a more obvious gradient change.

The rate of change increased as the value of *k* decreased. The smaller the value of *k*, the larger the creep strain at the same moment. Additionally, the creep curve's curvature radius increases during

variable-rate creep stage. Therefore, it is evident that the variation in the value of *k* has a significant impact on the creep model of salt rock. The value of *k* directly affects the morphology of the creep curve of salt rocks.

The *k* index affects the change of the creep curve by influencing the state variables in the model. Figure 15c and d shows the variation of state variables over time under trapezoidal wave conditions when only the value of *k* in the model is changed. The creep curve and the change of state variable revealed similar trends. During the test, as the value of *k* decreases, the curve gradually shows stage changes in gradient. The smaller the value of *k* is, the quicker the
<figure><img src="image_18.png" /><figcaption>Fig. 15 The creep strain and the state variable variation when the stress and the other parameters are the same except the *k* value a. Axial creep strains curves; b Radial creep strains curves; c Axial state variable curve; d Radial state variable curve</figcaption></figure>

accelerated creep phase appears. The more significant state variables changes, the greater the value of the state variables at the same time, and the higher the degree of strain hardening.

## 4.4 Influence of the *m* on the creep model

When the stress and other parameters are the same, only changing the value of *m* in the model results in a different set of different creep strain curves, as shown in Fig. 16a and b. From the figure, it can be observed that as the value of *m* increases, the total creep deformation also increases, and the creep strain values at the same moment also increase. Additionally, except for the initial cycle, the creep deformation becomes smaller. Moreover,

the gradient changes in the entire process of creep strain in salt rocks become less pronounced. At the same time, the earlier it enters the steady-state stage, the larger the curvature radius of the creep curves during the variable-rate creep stage. In summary, the value of *m* primarily influences the creep deformation except for the initial cycle in salt rocks and the duration of the steady-state creep phase.

The parameter *m* influences the variation of the creep curve by affecting the state variables in the model. Figure 16c and d shows the variation of state variables when only the value of *m* in the model is changed. The creep curve and state variable change graphs have similar trends. As the *m* value increases, the variation range of the state variable decreased. The state variables tend to stabilize earlier, and the
<figure><img src="image_19.png" /><figcaption>**Fig. 16** The creep strain and the state variable variation when the stress and the other parameters are the same except the *m* value a. Axial creep strains curves; b Radial creep strains curves; c Axial state variable curve; d Radial state variable curve</figcaption></figure>

values of state variables at the same moment increase.
This indicates a higher degree of rock hardening.

## 4.5 Influence of the *c* on the creep model

When the stress and other parameters are the same, changing only *c* in the model could result in a different set of total creep strain curves, as shown in Fig. 17a and b. This figure shows that the smaller the *c* value is, the larger the creep strains deformation, and the larger the creep strains at the same time. The larger the creep deformation is, the faster the curve changes except for the first cycle, and the larger the creep curve curvature radius in

the changing creep rate phase. The *c* value mainly affects the salt rock creep rate and the creep deformation except for in the first cycle.

The parameter *c* affects the change in the creep curve by influencing the state variables in the model. Figure 17c and d. shows the state variables when only *c* was changed in the model. The creep graphs and state variable change graphs have similar trends. The smaller the *c* value is, the larger the deformation of the state variable. The faster the state variable changes in the variable-rate creep phase, the later the state variable tends to be stable. At the same moment, the larger the values of the state variables is, the faster the strains hardening rate and the higher the hardening degree is.
<figure><img src="image_20.png" /><figcaption>**Fig. 17** The creep strain and the state variable variation when the stress and the other parameters are the same except the *c* value (a) Axial creep strain curve; (b) Radial creep strain curve; (c) Axial state variable curve; (d) Radial state variable curve</figcaption></figure>

# 5 Conclusion

Taking into account that loading and unloading history can affect the creep deformations of salt rocks, a new constitutive model for salt rocks creep was established by introducing state variables. The model was validated using test data under different loading and unloading paths. The main conclusions are as follows:

(1) Through the analysis of creep test data under different loading and unloading paths, and it is found that the loading and unloading history significantly affects the creep behaviour of salt rocks. In the stepped loading creep test, the creep deformation increases with the increase of stress

level. In the stepped unloading creep test, except for the highest stress level, the creep deformation rate was much smaller than that in the stepped loading creep test under the same stress levels. In the stepped unloading creep test, zero creep or negative creep occurs when the stress level was relatively low.

(2) Aiming at the influence of different loading and unloading histories on the rheological properties of salt rocks, a new three-dimensional creep constitutive model for salt rocks considering the effect of its loading and unloading history is established and revised by introducing state variables to characterize the strain hardening degree of rock. The proposed creep constitutive model
is verified by three different stress loading paths to achieve different creep deformations—stepped loading creep deformation, stepped unloading creep deformation and typical creep tests adopted under different stress loadings. The results of the test and the fitting indicate that the model can characterize the creep characteristics of salt rock considering the loading and unloading history.

(3) Trapezoidal wave creep tests are conducted on salt rocks specimens, involving both loading and unloading phases. Through the analysis of the influence of different parameter indexes on the state variables of the model, the influence trend of different constant indexes on the salt rock creep was obtained. The shape of the creep curves is primarily influenced by the parameter *k*. The creep deformation except in the first cycle and the length of the steady-state creep phase are influenced primarily by the parameter *m*. The salt rock creep rate and deformations are mainly influenced by the parameter *c* except for the first cycle.

In conclusion, the proposed model can effectively fit the influence of historical loading and unloading effects on rock deformations. However, the data used for model fitting are all derived from laboratory experiments. Due to the limitations of laboratory experiments, there are differences compared to actual engineering operation conditions. Future work will involve embedding the model into numerical simulation software and validating it with field measurement data to ensure the model can be well applied in engineering.

**Acknowledgements** This work was supported by National Natural Science Foundation of China (No. 52274073, 52022014), Chongqing Natural Science Foundation Innovation and Development Joint Fund (Municipal Education Commission) (No. CSTB2024NSCQ-LZX0056) and National Key R&D Program of China (2024YFB4007100). The authors are also grateful for the financial support from the Rut and Sten Brand foundation.

**Author Contributions** JF, FY, ZL, YZ wrote the main manuscript text and DJ, JC, DN prepared figures 1–17. All authors reviewed the manuscript.

**Funding** Open access funding provided by Lulea University of Technology.

**Availability of data materials** No datasets were generated or analysed during the current study.

**Declarations**

**Conflicts of interest** The authors declare no competing interests.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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