Salt rock nonlinear damage creep constitutive model construction method considering temperature effect
By constructing a salt rock creep model consisting of an elastic body, a nonlinear viscous body, and a generalized Kelvin body, and introducing temperature-related damage variables, the problem that traditional models cannot describe temperature effects is solved, and accurate prediction of salt rock creep behavior is achieved, especially the nonlinear characteristic description of the decay creep stage.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-25
- Publication Date
- 2026-03-24
AI Technical Summary
Existing salt rock creep models are insufficient to accurately describe creep behavior under temperature and loading conditions, especially in describing the correlation between temperature and decay creep and steady-state creep stages. Traditional linear models cannot clearly reflect the temperature effect.
A component combination model consisting of an elastomer, a nonlinear viscous body, and a generalized Kelvin body was constructed. Damage variables related to time and temperature were introduced, and a constitutive equation for nonlinear damage creep of salt rock was established. The model parameters were determined through graded loading tests, and their quantitative functional relationship with temperature was established.
It achieves an accurate description of the creep behavior of salt rock under temperature-load coupling conditions. The model parameters explicitly represent the temperature dependence, enabling continuous and quantitative prediction of the creep behavior of salt rock, thus improving the physical meaning and prediction accuracy of the model.
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Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of geotechnical engineering and rock mechanics, and particularly relates to a constitutive model construction method for describing the creep behavior of salt rock under the coupling action of temperature and load. BACKGROUND
[0002] Salt rock is an ideal medium for underground energy storage due to its good creep characteristics, low permeability and self-healing ability. During the long-term service of salt cavern underground hydrogen storage facilities, as the surrounding rock of the cavern, salt rock will continuously bear the double action of the internal gas pressure of the salt cavern and the temperature field of the stratum. This temperature and load coupling environment will significantly change the creep deformation characteristics of salt rock, directly affecting the long-term stability and safety of the storage facility.
[0003] The key to accurately describing the creep behavior of salt rock under complex environment is to establish a constitutive model that can reflect its mechanical mechanism. The establishment of salt rock creep constitutive model mainly includes three methods: phenomenological method, theoretical method of viscoelastic-plastic model and empirical method. Among them, the theoretical method of viscoelastic-plastic model using combined model is the most widely used method. However, the commonly used traditional element combined model such as Maxwell model, Kelvin model, Burgers model and Shizhao model are linear models, which are difficult to accurately depict the deformation characteristics of salt rock under actual working conditions. In existing research, few creep models consider temperature and loading conditions at the same time. Although some scholars have tried to construct creep models based on fractional calculus or empirical formula, these models either lack clear physical meaning or fail to systematically consider the quantitative influence of temperature on the parameters of each element in the model, especially in describing the correlation between temperature and the decay creep and steady creep stages.
[0004] Therefore, it is crucial to develop a nonlinear damage creep constitutive model that can clearly reflect the temperature effect, has clear physical mechanism and parameters easy to determine, for accurately predicting the long-term deformation behavior of salt cavern storage. SUMMARY
[0005] Therefore, the purpose of the present application is to overcome the shortcomings of the prior art and provide a constitutive model construction method that has clear physical meaning, parameters with clear temperature dependence and can accurately describe the whole process of nonlinear creep of salt rock under temperature and load coupling.
[0006] In order to achieve the above-mentioned purpose, the technical solution provided by the present application is as follows: A salt rock nonlinear damage creep constitutive model construction method considering temperature effect, characterized in that it comprises the following steps: Step 1: Construct an element combined model composed of an elastic body, a nonlinear viscous body and a generalized Kelvin body in series as the basic framework of the salt rock nonlinear damage creep constitutive model considering temperature effect; Step two: introducing a damage variable related to time and temperature; Step three: based on the basic framework of the salt rock nonlinear damage creep constitutive model, the introduced damage variable related to time and temperature, establishing the constitutive equation of the salt rock nonlinear viscous body; Step four: based on the constitutive equation of the salt rock nonlinear viscous body, deriving the total creep strain expression of the salt rock nonlinear damage creep constitutive model considering temperature effect under one-dimensional stress state, the total creep strain is the sum of elastic strain ε e , viscous-plastic strain ε vp and viscoelastic strain ε ve ; Step five: using the hierarchical loading method, carrying out uniaxial compression creep test of salt rock samples at different temperatures, obtaining strain-time data and corresponding creep curves of the salt rock samples at corresponding temperatures, and inversely determining the model parameters in the total creep strain expression; and establishing the quantitative function relationship between the model parameters and temperature T.
[0007] Further, in step one, the combination mode of each element is series, and the same axial stress is applied.
[0008] Further, in step two, the introduced damage variable related to time and temperature has the expression:
[0009] In the formula: D is the damage variable, t is the time, T is the temperature, and m(T) is the material damage coefficient related to temperature T.
[0010] Further, in step four, the constitutive equation of the salt rock nonlinear viscous body is established; it is:
[0011] In the formula: σ0 is the applied axial stress, η(T) is the viscosity coefficient related to temperature, and ε' is the strain rate.
[0012] Further, in step four, the expression of the total creep strain is:
[0013] In the formula: E M (T) is the elastic modulus of the elastic body in series with the generalized Kelvin body related to temperature, η M (T) is the viscosity coefficient of the nonlinear viscous body in series with the generalized Kelvin body related to temperature, E K (T) is the elastic modulus of the elastic body in the generalized Kelvin body related to temperature, and η K (T) is the viscosity coefficient of the nonlinear viscous body in the generalized Kelvin body related to temperature.
[0014] Further, in step five, the model parameters E M (T), η M (T), E K (T), η K (T), m(T) and temperature T are determined by fitting the test data, in the form of a linear function or an exponential function.
[0015] Further, in step five, the model parameters E M (T), η M (T), E K (T), η K (T), m(T) are determined by the least square method, by fitting the model parameters to the test data, and the algorithm used in the process is the Levenberg-Marquardt optimization algorithm.
[0016] Further, in step five, by linear or nonlinear regression:
[0017] In the formula, the value range of parameter a1 is -0.01 MPa / ℃ to -0.1 MPa / ℃, the value range of parameter b1 is 1 MPa to -10 Mpa; the value range of a2 is 1 MPa to 100 MPa, the value range of parameter b2 is -1 3， The value range of b4 is -1 3， b4<0.
[0018] The application also provides a long-term deformation prediction method for a salt cavern underground gas storage cavern structure, and the salt rock nonlinear damage creep constitutive model considering temperature effect constructed by the method is applied to numerical analysis of the salt cavern underground gas storage cavern structure to predict the creep deformation under specific temperature and load conditions.
[0019] The application has the following beneficial effects: the model is based on the classical element combination theory, the temperature-dependent damage variable and the nonlinear viscous pot are introduced, the physical meaning is clear, and the microscopic mechanism that temperature rise leads to internal damage of salt rock and increased creep is better reflected. Through the damage variable and the specific constitutive relationship, the nonlinear characteristics of salt rock creep are successfully described, especially the nonlinear attenuation process of strain rate in the attenuation creep stage. All key parameters in the model are expressed as explicit functions of temperature, so that the model can continuously and quantitatively predict the creep behavior of salt rock under different temperature conditions, rather than just fitting discrete temperature points. BRIEF DESCRIPTION OF DRAWINGS
[0020] Figure 1 is a step block diagram of the application.
[0021] Figure 2 is a schematic diagram of the nonlinear element combination model considering damage adopted by the present application.
[0022] Figure 3 is a comparison diagram of the model fitting curve and the test data at different temperatures in an embodiment of the present application.
[0023] Figure 4 is a fitting diagram of the model parameters E M (T) and E K (T) and temperature in an embodiment of the present application.
[0024] Figure 5 is a fitting diagram of the model parameters η M (T) and η K (T) and temperature in an embodiment of the present application.
[0025] Figure 6 is a comparison diagram of model verification in an embodiment of the present application. DETAILED DESCRIPTION
[0026] In order to further illustrate the technical means and effects adopted by the present application to achieve the predetermined purposes, the specific embodiments, structures, features and effects according to the present application are described in detail below in combination with the drawings and preferred embodiments. The salt rock samples from Jiangsu Jintan salt mine are taken as embodiments, and the nonlinear damage creep constitutive model considering temperature effect is constructed by applying the method of the present application, including the following steps: Step one: the present application provides a method for constructing a salt rock nonlinear damage creep constitutive model considering temperature effect, and the detailed step block diagram is as shown in Figure 1 The detailed process of constructing the nonlinear damage creep constitutive model includes the following steps: ① A damage variable D(t, T) related to time and temperature is defined to quantify the cumulative damage of the internal structure of the material in the creep process. The damage variable is expressed in the form of a negative exponential function:
[0027] ② The damage variable defined in ① is introduced into the constitutive relationship of the nonlinear viscous body to establish the constitutive relationship of the nonlinear viscous body. The constitutive equation of the nonlinear viscous body is expressed as:
[0028] ③ The creep equations of each part are derived respectively, and the total creep equation is obtained by summation, and the specific derivation process is as follows: the elastic element obeys Hooke's law, and its constitutive relationship is:
[0029] In the formula: εe Elastic strain produced by the elastic element of the model, E M (T) is the temperature-dependent elastic modulus of the elastic body in series with the generalized Kelvin body.
[0030] Further, substituting equation (1) into equation (2) can obtain the constitutive equation of the nonlinear viscous body:
[0031] In the formula: ε vp Viscoplastic strain produced by the nonlinear viscous body of the model, η M (T) is the temperature-dependent viscous coefficient of the nonlinear viscous body in series with the generalized Kelvin body.
[0032] Further, integrating both sides of equation (9) from 0 to t and from 0 to ε vp :
[0033] Equation (11) is the creep equation of the nonlinear viscous body (representing viscoplastic strain).
[0034] Further, the generalized Kelvin body is composed of an elastic body and a nonlinear viscous body in parallel. Its total stress is the sum of the stresses of the two, and the total strain is the strain of the two. The damage variable in equation (1) is considered in the equation:
[0035] In the formula: ε ve Viscoelastic strain produced by the generalized Kelvin body of the model, σ spring Strain produced by the elastic body in the generalized Kelvin body, σ dashpot Strain produced by the nonlinear viscous body in the generalized Kelvin body, E K (T) is the temperature-dependent elastic modulus of the elastic body in the generalized Kelvin body, η K (T) is the temperature-dependent viscous coefficient of the nonlinear viscous body in the generalized Kelvin body.
[0036] Further, equation (12) is a first-order linear non-homogeneous differential equation about ε ve Rewrite it into the standard form:
[0037] Further, assume:
[0038] Further, the general solution of the differential equation of equation (13) is:
[0039] Further, calculate ∫P(t)dt:
[0040] Further, to simplify the expression, assume:
[0041] Further, calculate ∫Q(t)exp(∫P(t)dt)dt:
[0042] Further, the constant C2 will be simplified in the general solution, and the parameters A, u are substituted into ∫Q(t)exp(∫P(t)dt)dt:
[0043] Further, substitute u back and substitute into the general solution of the constitutive equation, and after simplification, we can get:
[0044] Further, using the initial condition, when t=0, ε ve (0)=0:
[0045] Further, substitute the constant C into equation (24) and simplify to get:
[0046] Further, substitute the original expression of A, i.e. equation (18), into equation (27):
[0047] Equation (28) is the creep equation of the generalized Kelvin body (representing viscoelastic strain).
[0048] Further, the total creep strain of the salt rock nonlinear damage creep constitutive model considering temperature effect is the sum of elastic strain, viscoplastic strain and viscoelastic strain:
[0049] Further, substitute equation (8), equation (11) and equation (28) into equation (29) to get the salt rock nonlinear damage creep constitutive model considering temperature effect under one-dimensional stress state:
[0050] ④ Determine the quantitative function relationship between model parameters and temperature, and obtain test data by conducting salt rock uniaxial compression creep test at different temperatures. Based on the least square method principle and Levenberg-Marquardt optimization algorithm, the parameters EM (T), η M (T), E K (T), η K Inverse fitting is performed on (T) and m(T). Then, an explicit functional relationship between these parameters and temperature T is established, obtained through linear or nonlinear regression:
[0051] In the formula: a 1， b 1， a 2， b 2， c 2， a 3， b 3， c 3， a 4， b 4， c4 is a constant determined through fitting.
[0052] The model parameter E M (T) and E K The relationship between (T) and temperature T is either linear or exponential. When linear, parameter a1 ranges from -0.01 MPa / ℃ to -0.1 MPa / ℃, reflecting the decreasing trend of elastic modulus with increasing temperature. Parameter b1 ranges from 1 MPa to -10 MPa, corresponding to the observed elastic modulus of salt rock at room temperature, reflecting its initial stiffness. When exponential, parameter a2 ranges from 1 MPa to 100 MPa, reflecting the initial elastic modulus, and parameter b2 ranges from -1 to b2 < 0.
[0053] The model parameter η M (T) and η K The functional relationship between (T) and temperature T is exponentially decreasing, a 3， a4 reflects the typical viscosity coefficient of salt rock at room temperature, reflecting its rheological initiation resistance. The function is an exponentially decaying function, therefore b... 3， The range of values for b4 is -1 < b 3， b4 < 0.
[0054] Step 2: Using a self-developed creep testing system, uniaxial compression creep tests were conducted on the salt rock samples under axial stress of 0.8 MPa at temperatures of 20℃, 50℃, and 80℃. Strain-time data were recorded.
[0055] Step three: According to the creep test data, the creep process is determined to contain instantaneous creep, decaying creep and steady-state creep, the instantaneous creep process is represented by an elastic body, the decaying creep process is represented by a viscoelastic body, i.e., a generalized Kelvin body, and the steady-state creep process is represented by a nonlinear viscoplastic body, and the three elements are connected in series to form the nonlinear element combination model considering damage used in the application, as shown in Figure 2 .
[0056] Step four: The total creep equation derived in the application, i.e., equation (3), is used as the creep model for fitting the test data, and the MATLAB software is used to program the model parameters E M (T), η M (T), E K (T), η K (T) and m(T) are inversely fitted based on the least square method principle and Levenberg-Marquardt algorithm, and the fitted curve can be generated by substituting the inversely fitted parameters into the program, and the comparison between the fitted curve and the test data is shown in Figure 3 , and it can be seen that the nonlinear damage creep constitutive model of salt rock considering temperature effect constructed can well describe the creep process.
[0057] Step five: The functional relationship between the model parameters and temperature is established, and the parameter values at each temperature obtained in the above step two are respectively functionally fitted with temperature T. As shown in Figure 4 , E M (T) and E K (T) approximately linearly or exponentially decay with the increase of temperature, and the specific function forms can be obtained by fitting as follows:
[0058] .
[0059] As shown in Figure 5 , η M (T) and η K (T) exponentially decay with the increase of temperature. For example, the specific function forms can be obtained by fitting as follows:
[0060] .
[0061] m(T) changes little in the test temperature range, and its average value 0.0001 h -1 .
[0062] Step six: verify the nonlinear damage creep constitutive model considering temperature effect constructed by the method of the application using existing salt rock creep test data. The creep test data comes from Zhang Shengli, Xu Suguo, Xiao Ning, et al. (Deep salt rock creep damage model under temperature-stress coupling effect[J]. Journal of Coal Research, 2024, 49(8):3425-3438.). The salt rock creep test data (deviatoric stress 15 MPa, temperature 80℃) disclosed in the literature is used. Substitute temperature T=80℃ into the established model parameters and temperature function to calculate the corresponding model parameter values. Substitute these parameters into the total creep equation (equation 3) to calculate the theoretical creep curve. Compare the curve with the test data in the literature, as shown in Figure 6 , the two are in good agreement, and the maximum error is not more than 1%, proving the effectiveness and universality of the model.
[0063] Through the above steps, the construction of the nonlinear damage creep constitutive model considering temperature effect for a specific salt rock is completed. The model and its parameter determination method can be directly used to predict the long-term creep deformation of the salt rock under different constant temperature and constant load conditions, providing key input for related engineering design.
[0064] The above is only a preferred embodiment of the application, and does not limit the application in any form. Any simple modification, equivalent change and modification made according to the technical essence of the application to the above embodiment are still within the protection scope of the application.
Claims
1. A method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects, characterized in that, Includes the following steps: Step 1: Construct a component combination model consisting of an elastic body, a nonlinear viscous body, and a generalized Kelvin body connected in series as the basic framework for a constitutive model of nonlinear damage creep in salt rock that considers temperature effects. Step 2: Introduce a damage variable that is related to time and temperature; Step 3: Based on the basic framework of the nonlinear damage creep constitutive model of salt rock and the introduced time- and temperature-related damage variables, establish the constitutive equation of the nonlinear viscous body of salt rock; Step 4: Based on the constitutive equation of the nonlinear viscous salt rock, derive the total creep strain expression for the nonlinear damage creep constitutive model of salt rock under one-dimensional stress state considering temperature effects. The total creep strain is the elastic strain ε. e viscoplastic strain ε vp and viscoelastic strain ε ve sum; Step 5: Using a graded loading method, conduct uniaxial compression creep tests on the salt rock sample at different temperatures to obtain strain-time data and corresponding creep curves of the salt rock sample at the corresponding temperatures, and inversely determine the model parameters in the total creep strain expression; and establish a quantitative functional relationship between the model parameters and temperature T.
2. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 1, characterized in that, In step one, the components are connected in series and bear the same applied axial stress.
3. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 1, characterized in that, In step two, the time- and temperature-related damage variables introduced are expressed as follows: In the formula: D is the damage variable, t is time, T is temperature, and m(T) is the material damage coefficient related to temperature T.
4. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 3, characterized in that, In step four, the constitutive equation for the nonlinear viscous body of salt rock is established; it is: In the formula: σ0 is the applied axial stress, η(T) is the temperature-dependent viscosity coefficient, and ε' is the strain rate.
5. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 4, characterized in that, In step four, the expression for the total creep strain is: In the formula: E M (T) is the temperature-dependent elastic modulus of the elastomer connected in series with the generalized Kelvin model, η M (T) is the temperature-dependent nonlinear viscous viscosity coefficient connected in series with the generalized Kelvin model, E K (T) represents the elastic modulus of the elastic body in the generalized Kelvin scale, which is related to temperature, and η represents the elastic modulus. K (T) is the viscosity coefficient of a nonlinear viscous body in a temperature-dependent generalized Kelvin model.
6. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 5, characterized in that, In step five, the model parameter E in the constitutive model M (T), η M (T), E K (T), η K The functional relationships between m(T) and temperature T are determined by fitting experimental data, and are in the form of linear or exponential functions.
7. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 5, characterized in that, In step five, the model parameters E in the constitutive model are determined. M (T), η M (T), E K (T), η K The methods (T) and m(T) are based on the least squares method, which fits and inverts the model parameters through experimental data. The algorithm used in this process is the Levenberg-Marquardt optimization algorithm.
8. The method for constructing a constitutive model of nonlinear damage creep in salt rock considering temperature effects according to claim 5, characterized in that, In step five, the following is obtained through linear or nonlinear regression: In the formula, the value range of parameter a1 is -0.01MPa / ℃ to -0.1MPa / ℃, the value range of parameter b1 is 1MPa to -10MPa; the value range of a2 is 1MPa to 100MPa, and the value range of parameter b2 is -1 < b2 < 0; b 3， The range of values for b4 is -1 < b 3， b4 < 0.
9. A method for predicting the long-term deformation of a salt cavern underground gas storage structure, characterized in that, The nonlinear damage creep constitutive model of salt rock considering temperature effects, constructed by the method according to any one of claims 1 to 8, is applied to the numerical analysis of the underground gas storage cavern structure of the salt cavern to predict its creep deformation under specific temperature and load conditions.