To determine the nonlinear creep characteristics of rocks under cyclic loading and unloading conditions, a nonlinear Kelvin model and damage viscoplastic model are proposed. The models are connected in series with a linear elastic body to establish a nonlinear damage creep model. The differential damage constitutive equations of the proposed creep model under one-dimensional and three-dimensional stress states are derived based on the creep mechanics and elasticity theory. The damage and unloading creep equations are then obtained based on the superposition principle, and a simple and feasible method for determining the model parameters is determined. Finally, the step cyclic loading and unloading creep test data for lherzolite and limestone are used to verify the rationality and feasibility of the nonlinear damage creep model. The results show that the theoretical creep curves of the nonlinear damage creep model are consistent with the experimental curves which indicates that the proposed model can not only determine the creep properties of lherzolite and limestone under cyclic loading and unloading but also determine the nonlinear characteristics of rocks in the transient and steady-state creep stages and particularly within the accelerating creep stage.
Creep characteristics are intrinsic mechanical properties of rocks, which are significant for the analysis of various rock engineering failure problems, such as dam foundation stability, reservoir subsidence, and tunnel support design [
Currently, there are hundreds of creep models, which can be classified into three types: empirical [
In this study, a novel nonlinear Kelvin body and viscoplastic damage body are established and are then connected in series with the elastic body to obtain a novel nonlinear damage creep model. This model effectively determines the loading and unloading creep properties of the lherzolite and limestone under different stress levels and also favorably represents the nonlinear characteristics of accelerating creep, overcoming the shortcomings of the empirical creep models and the component models.
In the rock step cyclic loading and unloading compression creep test, the total strain can be divided into instantaneous elastic strain, instantaneous plastic strain, viscoelastic strain, and viscoplastic strain [
Typical creep curve during the loading and unloading test.
The instantaneous plastic strain only accounts for 15%–20% of the instantaneous strain, which is smaller than the instantaneous elastic strain. Moreover, the increment of the instantaneous plastic strain per unit stress decreases with the increase in stress level. Simultaneously, if the effect of the instantaneous plastic strain is considered, the creep mechanical model and creep equation of the rock will be more complex, which is not conducive for application. Therefore, the novel nonlinear damage creep model established in this study does not consider the effect of instantaneous plastic strain [
The transient and unloading creep curves have obvious nonlinear characteristics, which are difficult to accurately determine using the traditional Kelvin model [
Nonlinear Kelvin model.
The differential constitutive equation of the nonlinear Kelvin model can be expressed as
By solving equation (
Assuming the applied stress is unloaded at
Equation (
When
By substituting equation (
When the applied stress exceeds the long-term strength of the rock, it undergoes unstable creep, in which the strain and strain rate increases rapidly with the creep time. The rock is destroyed within a short period. Because the unstable creep has no unloading process, the strain generated in the unstable creep stage is regarded as viscoplastic strain [
Damage viscoplastic model.
Based on Lemaitre’s strain equivalence principle [
The damage evolution equation proposed by Kachanov [
The creep damage critical failure time can be derived using equation (
The relationship between the damage variable
We can obtain the constitutive equation of the damaged viscous body by substituting equation (
By integrating equation (
When
By substituting equation (
By substituting equation (
In previous research, the creep equation of the damage viscous body was obtained using the Kachanov creep damage rate as follows:
By calculating the derivative of equation (
Comparing equation (
For a damage viscoplastic model under the one-dimensional stress state, the creep equation is obtained by replacing stress
The instantaneous elastic body determining the instantaneous strain, the nonlinear Kelvin model determining the viscoelastic strain, and the damage viscoplastic model determining the viscoplastic strain are connected in series to form a nonlinear damage creep model, as shown in Figure
Nonlinear damage creep model.
Based on the stress-strain relationship of the model in the series-parallel connection, we can obtain the following equations:
The constitutive equations for each part are expressed as
When it is one-dimensional, the constitutive equation is as follows: when when
The axial creep equations of the nonlinear damage rheological model can be obtained based on the superposition principle. If If
Because actual rock mass engineering often involves rocks in a complex three-dimensional stress condition, it is necessary to establish the axial creep and unloading equations of the nonlinear damage creep model under three-dimensional stress conditions to reflect the creep properties of rocks more accurately. Assuming that the rock specimen is a continuous and uniform material, the total strain of rock under triaxial compression creep state,
For an instantaneous elastic body, the strain under three-dimensional stress conditions is obtained according to the general Hooke's law [
For the nonlinear Kelvin model, the creep equation under three-dimensional stress conditions can refer to the creep equation under one-dimensional stress conditions as follows:
The three-dimensional constitutive relationship of the damage viscoplastic model is
The Mohr–Coulomb and Von Mises yield criteria are the most applicable rock yield criteria. However, the Mohr–Coulomb criterion does not consider the influence of the intermediate principal stress on the yield failure of the rocks, and the Von Mises yield criterion ignores the influence of the spherical stress on the creep properties of the rocks, especially soft rocks. Therefore, the Drucker–Prager yield criterion [
Here,
The following equations can be obtained in the conventional triaxial step cycle loading and unloading compression creep test:
According to equations (
Assuming the applied stress is unloaded at
When
Subtracting equation (
Suppose
The fitting parameters
Numerous studies have been conducted to determine the creep model parameters; these studies can be divided into two types. The first is a graphical method, which determines the creep parameters based on the relation between the creep test curve and physical significance of the creep parameters. The second is an optimization analysis algorithm method, such as the regression analysis method and the least squares method (LSM). In this study, the widely used Levenberg–Marquardt nonlinear least squares method [
To verify the rationality and feasibility of the nonlinear damage creep model, the proposed model was evaluated by multilevel loading and unloading cycles compression creep test data for lherzolite and limestone [
Basic parameters of the rock specimens.
Rock specimens | Confining pressure (MPa) | Compressive strength (MPa) | Elastic modulus (GPa) | Poisson's ratio | Sticky cohesion (MPa) | Internal friction angle (°) |
---|---|---|---|---|---|---|
Lherzolite | 3 | 78.63 | 19.07 | 0.227 | 8.62 | 52.64 |
Limestone | 4 | 59.20 | 7.79 | 0.251 | 8.12 | 35.10 |
As shown in Table
Test scheme for triaxial creep.
Rock specimen | Confining pressure (MPa) | Deviatoric stress level (MPa) | |||||
---|---|---|---|---|---|---|---|
First level | Second level | Third level | Fourth level | Fifth level | Sixth level | ||
Lherzolite | 3 | 13.3 | 22.1 | 26.6 | 28.8 | 31.0 | 39.9 |
Limestone | 4 | 29.6 | 33.8 | 38.0 | 42.2 | 46.4 | 50.6 |
The model parameters of lherzolite and limestone can be obtained based on the experimental data and parameter identification method (as shown in Tables
Model parameters under triaxial creep for lherzolite.
Stress levels (MPa) | |||||||||
---|---|---|---|---|---|---|---|---|---|
13.3 | 11.64 | 7.77 | 4.68 | 19.69 | 0.75 | 0.24 | — | — | — |
22.1 | 3.17 | 13.81 | 1.09 | — | — | — | |||
26.6 | 2.91 | 17.87 | 1.35 | — | — | — | |||
28.8 | 2.68 | 22.12 | 1.57 | — | — | — | |||
31.0 | 2.17 | 6.12 | 0.67 | — | — | — | |||
39.9 | 2.89 | 9.94 | 0.73 | 655 | 4.82 | 20.45 |
Model parameters under triaxial creep for limestone.
Stress levels (MPa) | |||||||||
---|---|---|---|---|---|---|---|---|---|
29.6 | 5.21 | 3.11 | 12.55 | 32.27 | 0.91 | 0.18 | — | — | — |
33.8 | 11.04 | 118.89 | 1.60 | — | — | — | |||
38.0 | 9.66 | 94.04 | 1.78 | — | — | — | |||
42.2 | 6.08 | 68.58 | 1.87 | — | — | — | |||
46.4 | 4.11 | 14.94 | 0.49 | — | — | — | |||
50.6 | 4.94 | 24.38 | 0.78 | 676 | 3.35 | 25.18 |
Comparisons between the nonlinear damage creep model and experimental curves at various deviatoric stresses for lherzolite specimen: (a) first stress (13.3 MPa) and second stress (22.1 MPa); (b) third stress (26.6 MPa) and fourth stress (28.8 MPa); (c) fifth stress (31.0 MPa) and sixth stress (39.9 MPa).
Comparisons between the nonlinear damage creep model and experimental curves at various deviatoric stresses for limestone specimen.
A comparison between the experimental and theoretical curves of the nonlinear damage creep model (Figures
In this study, a novel nonlinear damage creep model for rocks incorporating instantaneous elasticity, viscoelasticity, and viscoplasticity was proposed based on the elastic body, nonlinear Kelvin model, and damage viscoplastic model.
The creep equations and unloading equations for rocks under one-dimensional and three-dimensional stress conditions were derived by introducing the creep mechanics theory, respectively. The parameters of the nonlinear damage creep model were determined using the nonlinear least squares method.
The proposed nonlinear damage creep model was used to fit lherzolite and limestone creep test data; the results showed that the model accurately determines the loading and unloading creep properties of lherzolite and limestone under different stress levels and also favorably represents the nonlinear characteristics of accelerating creep, thus validating our model.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (51374010 and 51474004).