A method for constructing a thermomechanical coupled fractional creep damage constitutive model for limestone

Abstract

The present invention discloses a method for constructing a thermomechanical coupling limestone fractional creep damage constitutive model, including: S1: establishing a first thermal damage elastic element considering the effects of thermal expansion and thermal damage; S2: establishing a thermomechanical coupling damage viscous element considering the effects of thermal coupling; S3: establishing a thermomechanical coupling limestone fractional damage creep model; S4: obtaining a one-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model; S5: establishing a three-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model. The present invention proposes an elastic element considering the effects of thermal expansion and a thermomechanical coupling damage viscous element based on fractional order, which can accurately describe the three-stage characteristics of rock creep under different temperatures and better reflect the rheological mechanism of rock under thermal coupling. The establishment of this model can provide certain theoretical guidance for the long-term stability analysis of surrounding rock of fire tunnels.

Description

Technical Field

[0001] The invention relates to the technical field of rock creep damage model calculation methods, and in particular to a method for constructing a thermal-mechanical coupling limestone fractional creep damage constitutive model. Background Art

[0002] In recent years, with the continuous development of science and technology, the construction of tunnel projects has been increasing. Tunnels not only bring convenience to transportation life, but also bring huge challenges to construction and operation. During the operation of tunnels, there are many vehicles in traffic, and the risk and probability of fire are extremely high. In addition to causing great harm to personnel, fires will also lead to the deterioration of the mechanical properties of the surrounding rock support system, greatly reducing the safety of the bearing structure. The high temperature effect of fire on the tunnel not only causes instantaneous damage to the internal surrounding rock, but also has a complex impact on the long-term stability of the surrounding rock. Therefore, conducting rock creep tests under the action of thermal-mechanical coupling and studying its time-effect mechanical properties are of great engineering significance for ensuring the long-term stability of the surrounding rock of the tunnel under fire. However, in the existing theoretical derivation and analysis of creep models, although they can well describe the creep mechanical properties of rocks, they have not been analyzed in combination with thermal-mechanical coupling conditions. Summary of the invention

[0003] The present invention provides a method for constructing a thermomechanically coupled limestone fractional creep damage constitutive model to overcome the above technical problems.

[0004] In order to achieve the above object, the technical solution of the present invention is:

[0005] A method for constructing a thermal-mechanical coupled limestone fractional creep damage constitutive model comprises the following steps:

[0006] S1: establishing a first thermally damaged elastic element taking into account the effects of thermal expansion and thermal damage to obtain a constitutive relation equation of the first thermally damaged elastic element;

[0007] S2: establishing a thermomechanical coupling damaged viscous element considering the influence of thermomechanical coupling to obtain a constitutive relation equation of the thermomechanical coupling damaged viscous element;

[0008] S3: establishing a thermomechanical coupling limestone fractional damage creep model including the first thermally damaged elastic element and the thermomechanically coupled damaged viscous element;

[0009] S4: according to the thermomechanical coupling limestone fractional damage creep model, obtaining a one-dimensional constitutive relation equation of the thermomechanical coupling limestone fractional damage creep constitutive model;

[0010] S5: According to the one-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model, a three-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model is established.

[0011] Furthermore, in S1, the mechanical constitutive relation equation of the first thermally damaged elastic element is established as follows:

[0012]

[0013] Where: σ is the applied load value; E1 is the elastic modulus of the first thermally damaged elastic element; D T is the thermal damage caused by temperature; ε is the applied strain value; α T1 is the thermal expansion coefficient of the first thermally damaged elastic element; ΔT is the temperature difference.

[0014] Furthermore, in S2, the constitutive equation of the thermal-mechanical coupling damaged viscous element is established as follows:

[0015] The calculation process of simplified fractional calculus is as follows:

[0016]

[0017] Where: δ is the fractional order; Г(δ) is the Gamma function; f(t) is the fractional function with t as the independent variable; f(ξ) is the fractional function with ξ as the independent variable;

[0018] Where: Gamma function Г(δ) is:

[0019]

[0020] The constitutive equation of the integer-order viscosity pot is transformed into a fractional-order viscosity pot:

[0021]

[0022] Where: σ(t) is the function of stress changing with time, ε(t) is the function of strain changing with time; t is time; η is the viscosity coefficient;

[0023] When the stress is σ(t) = σ, keep the stress unchanged and integrate formula (2), we get:

[0024]

[0025] Considering the damage effect of temperature on rock elastic modulus, we can get:

[0026]

[0027] Where: E T is the elastic modulus of rock at temperature T; E0 is the elastic modulus of rock at room temperature 20℃;

[0028] Assuming that the random variable follows an exponential distribution with parameter λ, then:

[0029]

[0030] Where: λ is the coefficient that characterizes the evolution law of aging damage; t is time; D c is the load damage caused by the load; N F is the number of micro-element bodies destroyed under high temperature; N is the total number of micro-element bodies;

[0031] then

[0032]

[0033] Where: D is the damage variable; D T Thermal damage caused by temperature; D c Load damage caused by load; D T D c is the coupling term;

[0034] The constitutive relation of the thermal-mechanical coupling damaged viscous element is:

[0035]

[0036] Where: η2 is the thermal-mechanical coupling viscosity coefficient; D is the damage variable.

[0037] Furthermore, the thermal-mechanical coupling limestone fractional damage creep model in S3 includes the first thermal damage elastic element, thermal damage viscoelastic body and thermal-mechanical coupling damage viscoplastic body;

[0038] The first thermally damaged elastic element, the thermally damaged viscoelastic body and the thermally coupled damaged viscoplastic body are connected in series;

[0039] The thermally damaged viscoelastic body comprises a second thermally damaged elastic element and a thermally damaged viscous element; the second thermally damaged elastic element is connected in parallel with the thermally damaged viscous element;

[0040] The thermal-mechanical-coupled-damaged-viscoplastic body comprises a thermal-mechanical-coupled-damaged-viscous element and a thermal-mechanical-coupled-damaged-plastic element, and the thermal-mechanical-coupled-damaged-viscous element and the thermal-mechanical-coupled-damaged-plastic element are connected in parallel.

[0041] Furthermore, the one-dimensional constitutive relation equation of the thermal-mechanical coupled limestone fractional damage creep constitutive model is established as follows:

[0042] Total strain ε 总 The following conditions must be met:

[0043] ε 总 =ε e +ε ve +ε vp (10)

[0044] Where: ε e is the thermoelastic strain; ε ve is the thermo-viscoelastic strain; ε vp is the thermo-mechanical coupled viscoplastic strain; the stress-strain relationship of the first thermally damaged elastic element is:

[0045]

[0046] Where: E1 is the elastic modulus of the first thermally damaged elastic element; α T1 is the thermal expansion coefficient of the first thermally damaged elastic element;

[0047] The stress-strain relationship of the thermally damaged viscoelastic body is:

[0048]

[0049] Where: ε H is the strain of the thermally damaged elastic element; σ H is the stress of thermally damaged elastic elements; ε N is the strain of thermally damaged adhesive element; σ N is the stress of the thermally damaged viscous element; η1 is the shear viscosity coefficient of the thermally damaged viscous element; α T2 is the thermal expansion coefficient of the second thermally damaged elastic element; E2 is the elastic modulus of the second thermally damaged elastic element; is the integral operation of the strain of the damaged viscous element over time t;

[0050] Solving equation (12), we get:

[0051]

[0052] Wherein: k is an integer not less than 0;

[0053] Combining equation (5) and equation (8), we get:

[0054]

[0055] Where: s is the yield stress threshold;

[0056] Therefore, the one-dimensional constitutive equation is obtained as follows:

[0057]

[0058] Furthermore, the three-dimensional constitutive relation equation of the thermal-mechanical coupling limestone fractional damage creep constitutive model in S5 is established as follows:

[0059] In the three-dimensional state, the total axial strain ε ij总 for:

[0060]

[0061] Where: is the total axial thermoelastic strain; is the total axial thermo-viscoelastic strain; is the total axial thermo-viscoplastic strain;

[0062] The three-dimensional constitutive relationship of the thermally damaged elastomer is:

[0063]

[0064] Where: S ij is the stress deviator; e ij is the strain deviator; σ ii is the first invariant of the stress tensor; ε ii is the first invariant of the strain tensor; G m is the thermoelastic shear modulus; K is the elastic bulk modulus;

[0065] Therefore, the total axial thermoelastic strain of the thermally damaged elastomer is expressed as:

[0066]

[0067] Where: is the total axial thermoelastic strain, σ m is the spherical stress tensor, δ ij is the Kronecker function;

[0068] The three-dimensional constitutive relation of the thermally damaged viscoelastic body is:

[0069]

[0070] Where: is the total axial thermo-viscoelastic strain; G K is the thermo-viscoelastic shear modulus, η1 is the shear viscosity coefficient of the thermally damaged viscous element;

[0071] The three-dimensional constitutive relation of the thermomechanical coupling damaged viscoplastic body is:

[0072]

[0073] Where: is the total axial thermo-viscous plastic strain; F is the yield function; F0 is the initial yield function value, generally F0 = 1; Q is the plastic potential function, is a power function; 〈〉 is the yield condition judgment function, that is: σ ij is the stress of the thermomechanical coupled damaged viscoplastic body in three-dimensional state;

[0074]

[0075] When F ≥ 0, formula (21) is expressed as:

[0076]

[0077]

[0078] Where: J2 is the second invariant of stress deviator; σ1 is the axial stress; σ3 is the confining pressure;

[0079] Substituting equations (17) to (23) into equation (16), the three-dimensional constitutive relationship equation of the thermomechanical coupled limestone fractional damage creep constitutive model under three-dimensional state is obtained as follows:

[0080]

[0081] Under the conventional triaxial compression test conditions, σ2=σ3, then:

[0082]

[0083]

[0084] Substituting equations (25) to (26) into (24), setting the initial yield function F0 = 1, the axial creep equation is obtained as follows:

[0085]

[0086] Beneficial effects: The method for constructing a thermomechanically coupled limestone fractional creep damage constitutive model of the present invention proposes an elastic element considering thermal expansion and a fractional thermomechanically coupled damage viscous element, and constructs a fractional damage creep constitutive model considering thermomechanical coupling based on the traditional Nishihara model. The calculation process of the present invention can accurately describe the three-stage characteristics of rock creep under different temperatures, and better reflect the rheological mechanism of rock under thermomechanical coupling. The establishment of this model can provide certain theoretical guidance for the long-term stability analysis of surrounding rock of fire tunnels. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0088] Figure 1 Schematic diagram of a creep damage constitutive model in one embodiment of the present invention;

[0089] Figure 2 A diagram of a thermoelastic damage element in one embodiment of the present invention;

[0090] Figure 3 A diagram of a damaged viscous element under thermal-mechanical coupling in one embodiment of the present invention;

[0091] Figure 4 This is a diagram of the original Nishihara model in one embodiment of the present invention;

[0092] Figure 5a A comparison diagram of a limestone creep test curve at 80°C and a theoretical model fitting curve in one embodiment of the present invention;

[0093] Figure 5b It is a comparison diagram of a limestone creep test curve at 150° C. and a theoretical model fitting curve in one embodiment of the present invention;

[0094] Figure 6 The present invention provides a flow chart of a method for establishing a thermal-mechanically coupled limestone fractional creep damage constitutive model. DETAILED DESCRIPTION

[0095] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0096] This embodiment provides a method for constructing a thermomechanical coupled limestone fractional creep damage constitutive model. Figure 6 As shown, the following steps are included:

[0097] S1: Under high temperature, the mineral particles inside the limestone undergo thermal expansion. Since different temperatures will produce differences in the thermal expansion coefficient of the rock, it is easier to form local thermal stress concentration between the mineral particles, resulting in the weakening of the cohesion between the particles, which leads to the weakening of the overall mechanical properties of the limestone, making it easier for the limestone to produce rheological deformation under high temperature conditions. Establish the first thermal damage elastic element that considers the effects of thermal expansion and high temperature damage, as shown in the attached figure. Figure 2 As shown; to obtain the constitutive relation equation of the first thermally damaged elastic element;

[0098] The mechanical constitutive relation equation of the first thermally damaged elastic element is established as follows:

[0099]

[0100] Where: σ is the applied load value; E1 is the elastic modulus of the first thermally damaged elastic element; D T is the thermal damage caused by temperature; ε is the applied strain value; α T1 is the thermal expansion coefficient of the first thermally damaged elastic element; ΔT is the temperature difference.

[0101] S2: Establish a thermomechanical coupling damage viscous element considering the influence of thermomechanical coupling, such as Figure 3 As shown in FIG. 1 , the constitutive equation of the thermal-mechanical coupling damaged viscous element is obtained;

[0102] The constitutive equation of the thermal-mechanical coupling damaged viscous element is established as follows:

[0103] Riemann-Liouville type (RL type for short) fractional calculus is the most commonly used definition of fractional calculus. Therefore, by introducing the definition of RL fractional calculus, the calculation process of fractional calculus can be simplified. When the fractional order is δ, the RL fractional function can be expressed as:

[0104] The calculation process of simplified fractional calculus is as follows:

[0105]

[0106] Where: δ is the fractional order; Г(δ) is the Gamma function; f(t) is the fractional function with t as the independent variable; f(ξ) is the fractional function with ξ as the independent variable;

[0107] Where: Gamma function Г(δ) is:

[0108]

[0109] The constitutive equation of the integer-order viscosity pot is transformed into a fractional-order viscosity pot:

[0110]

[0111] Where: σ(t) is the function of stress changing with time, ε(t) is the function of strain changing with time; t is time, at this time δ∈(0,1); η is the viscosity coefficient;

[0112] According to the RL type fractional calculus operator, when the stress is σ(t) = σ, keep the stress unchanged and integrate formula (2), we get:

[0113]

[0114] In the process of rock creep loading under high temperature, various parameters are not fixed. The creep parameters of rock change with the heating temperature. Considering the rock creep mechanical parameters as unsteady can more directly and objectively reflect the nonlinear time-dependent characteristics of high-temperature rock.

[0115] Considering the damage effect of temperature on rock elastic modulus, we can get:

[0116]

[0117] Where: E T is the elastic modulus of rock at temperature T; E0 is the elastic modulus of rock at normal temperature 20℃.

[0118] Secondly, based on the statistical damage principle, the damage development of rock during long-term loading is considered to be a continuous process. The final destruction of rock micro-elements with continuous accumulation of microcracks and pores will reduce the strength of the rock and cause damage. Define load damage D c is the number of micro-elements destroyed under high temperature N F The ratio of the total number of micro-units N, and assuming that the random variable obeys an exponential distribution with parameter λ, then:

[0119]

[0120] Where: λ is the coefficient that characterizes the evolution law of aging damage; t is time; D c is the load damage caused by the load; N F is the number of micro-element bodies destroyed under high temperature; N is the total number of micro-element bodies;

[0121] When the applied load reaches or exceeds a certain stress level, load damage will occur inside the rock. Since the damage caused by the load in the deceleration and stabilization stages is relatively small, the load damage is only considered in the accelerated creep stage in the embodiment. However, due to the influence of temperature, the coupled damage effect caused by temperature and load should be considered in the acceleration stage. Therefore, the damage variable D is introduced to describe the damage degradation of creep parameters in the creep process. The coupling equation is:

[0122]

[0123] Where: D is the damage variable; D T Thermal damage caused by temperature; D c Load damage caused by load; D T D c is the coupling term;

[0124] The constitutive relation of the thermal-mechanical coupling damaged viscous element is:

[0125]

[0126] Where: η2 is the thermomechanical coupling viscosity coefficient; D is the damage variable, 0≤D<1;

[0127] S3: establishing a thermomechanical coupling limestone fractional damage creep model including the first thermally damaged elastic element and the thermomechanically coupled damaged viscous element;

[0128] Under high temperature, limestone successively experiences deceleration creep stage, stable creep stage and acceleration creep stage. As for the model, the classic Nishihara model can well describe the deceleration creep and stable creep stages of rock, as shown in the attached figure. Figure 4 As shown, it is difficult to describe the characteristics of accelerated creep. Therefore, this embodiment establishes a fractional damage creep model of limestone under thermal-mechanical coupling based on the Nishihara model, as shown in Figure 1 shown.

[0129] The thermal-mechanical coupling limestone fractional damage creep model includes the first thermal damage elastic element, a thermal damage viscoelastic body and a thermal-mechanical coupling damage viscoplastic body;

[0130] The first thermally damaged elastic element, the thermally damaged viscoelastic body and the thermally coupled damaged viscoplastic body are connected in series;

[0131] The thermally damaged viscoelastic body comprises a second thermally damaged elastic element and a thermally damaged viscous element; the second thermally damaged elastic element is connected in parallel with the thermally damaged viscous element;

[0132] Preferably, in this embodiment, the derivation process of the constitutive relationship equation of the second thermally damaged elastic element is the same as that of the first thermally damaged elastic element, and the only difference is that the actual values ​​of the parameters in the constitutive relationship equation are different according to the specific situation.

[0133] The thermal-mechanical-coupled-damaged-viscoplastic body comprises a thermal-mechanical-coupled-damaged-viscous element and a thermal-mechanical-coupled-damaged-plastic element, and the thermal-mechanical-coupled-damaged-viscous element and the thermal-mechanical-coupled-damaged-plastic element are connected in parallel.

[0134] S4: according to the thermomechanical coupling limestone fractional damage creep model, obtaining a one-dimensional constitutive relation equation of the thermomechanical coupling limestone fractional damage creep constitutive model;

[0135] The one-dimensional constitutive relation equation of the thermal-mechanical coupled limestone fractional damage creep constitutive model is established as follows:

[0136] In one-dimensional state, the total strain ε 总 The following conditions must be met:

[0137] ε 总 =ε e +εve +ε vp (10)

[0138] Where: ε e is the thermoelastic strain; ε ve is the thermo-viscoelastic strain; ε vp is the thermo-mechanical coupled viscoplastic strain.

[0139] The stress-strain relationship of the first thermally damaged elastic element is:

[0140]

[0141] Where: E1 is the elastic modulus of the first thermally damaged elastic element; α T1 is the thermal expansion coefficient of the first thermally damaged elastic element;

[0142] In the thermally damaged viscoelastic body, since the viscous element and the linear spring are in parallel, the strain of the elastic element is equal to the strain of the viscous element, and the stress is the sum of the two, that is, the stress-strain relationship of the thermally damaged viscoelastic body is:

[0143]

[0144] Where: ε H is the strain of the thermally damaged elastic element; σ H is the stress of thermally damaged elastic elements; ε N is the strain of thermally damaged adhesive element; σ N is the stress of the thermally damaged viscous element; η1 is the shear viscosity coefficient of the thermally damaged viscous element; α T2 is the thermal expansion coefficient of the second thermally damaged elastic element; E2 is the elastic modulus of the second thermally damaged elastic element; is the integral operation of the strain of the damaged viscous element over time t;

[0145] Combined with the initial condition t = 0, ε ve =0, solve equation (12) and get:

[0146]

[0147] Wherein: k is an integer not less than 0;

[0148] The thermomechanical coupling damage viscoplastic body is composed of a thermomechanical coupling damage viscous element and a thermomechanical coupling damage plastic element in parallel. According to the combined model theory, when σ<σ s When the plastic element does not work, ε vp =0; when σ≥σ s When , the plastic element starts, and combining equation (5) and equation (8) we get:

[0149]

[0150] Where: s is the yield stress threshold;

[0151] Therefore, considering the strains of thermally damaged elastic elements, thermally damaged viscoelastic bodies, and thermally coupled damaged viscoplastic bodies, the one-dimensional constitutive equation is obtained:

[0152]

[0153] S5: According to the one-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model, a three-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model is established.

[0154] Since the real tunnel surrounding rock is generally in a complex two-way or three-way stress environment, and uniaxial stress rarely occurs, the one-dimensional creep constitutive model is difficult to apply to actual engineering, and the test conditions of this embodiment are also triaxial tests. In order to make the derived creep model solve actual engineering problems, it is necessary to transform the one-dimensional creep model into a three-dimensional creep model. Studies have shown that the creep constitutive model is consistent in the three-dimensional state and the one-dimensional state. Therefore,

[0155] The three-dimensional constitutive relationship equation of the thermal-mechanical coupled limestone fractional damage creep constitutive model is established as follows:

[0156] In the three-dimensional state, the total axial strain ε ij总 for:

[0157]

[0158] Where: is the total axial thermoelastic strain; is the total axial thermo-viscoelastic strain; is the total axial thermo-viscoplastic strain;

[0159] According to the generalized Hooke's law, the three-dimensional constitutive relation of the thermally damaged elastomer is:

[0160]

[0161] Where: S ij is the stress deviator; e ij is the strain deviator; σ ii is the first invariant of the stress tensor; ε ii is the first invariant of the strain tensor; G m is the thermoelastic shear modulus; K is the elastic bulk modulus;

[0162] Therefore, the total axial thermoelastic strain of the thermally damaged elastomer is expressed as:

[0163]

[0164] Where: is the total axial thermoelastic strain, σ m is the spherical stress tensor, δ ij is the Kronecker function;

[0165] Assuming that the volume change of rock is elastic, the creep property is mainly manifested in shear deformation. Therefore, the three-dimensional constitutive relation of the thermally damaged viscoelastic body is:

[0166]

[0167] Where: is the total axial thermo-viscoelastic strain; G K is the thermo-viscoelastic shear modulus, η1 is the shear viscosity coefficient of the thermally damaged viscous element;

[0168] The three-dimensional constitutive relation of the thermomechanical coupling damaged viscoplastic body is:

[0169]

[0170] Where: is the total axial thermo-viscous plastic strain; F is the yield function; F0 is the initial yield function value, generally F0 = 1; Q is the plastic potential function, is a power function; 〈〉 is the yield condition judgment function, that is: σ ij is the stress of the thermomechanical coupled damaged viscoplastic body in three-dimensional state;

[0171]

[0172] Adopting the relevant flow criterion, the initial yield function value and power index are taken as 1. When F ≥ 0, equation (21) is expressed as:

[0173]

[0174] Under medium and high temperature conditions, it is generally believed that the stress sphere tensor has little effect on creep, and the stress deviator plays a major role in the creep process of rock. Therefore, the yield function is expressed as:

[0175]

[0176] Where: J2 is the second invariant of stress deviator; σ1 is the axial stress; σ3 is the confining pressure;

[0177] Substituting equations (17) to (23) into equation (16), the three-dimensional constitutive relationship equation of the thermomechanical coupled limestone fractional damage creep constitutive model under three-dimensional state is obtained as follows:

[0178]

[0179] Under the conventional triaxial compression test conditions, σ2=σ3, then:

[0180]

[0181]

[0182] Substituting equations (25) to (26) into (24), setting the initial yield function F0 = 1, the axial creep equation is obtained as follows:

[0183]

[0184] Specifically, in this embodiment, the Boltzmann superposition method is used to transform the creep curve of graded loading into the creep curve under separate loading conditions. According to the triaxial creep test data of limestone under different temperatures, the 1stOpt8.0 mathematical analysis software is used to identify the parameters of the fractional damage creep model using the Levenberg-Marquardt and general global optimization algorithms. The model parameter identification results under 80°C and 150°C are also listed, as shown in Table 1.

[0185] Table 1 Identification parameters of the thermomechanical coupled limestone fractional creep damage constitutive model

[0186]

[0187] The comparison between the limestone creep test curve at 80℃ and 150℃ and the fitting curve of the theoretical model was obtained by fitting, as shown in Figure 2. Figure 5a and 5b As can be seen from the figure, the two are in good agreement, and the model fitting curve can well reflect the three-stage characteristics of deceleration creep, stable creep and acceleration creep of limestone under different temperatures, indicating the correctness and applicability of the fractional-order creep constitutive model established in this paper.

[0188] The present invention has the following beneficial effects:

[0189] 1. The present invention establishes a fractional creep damage constitutive model of limestone under the action of thermal-mechanical coupling, and clarifies the method for determining each parameter of the model. In addition, the parameters required by the model can all be calculated from the creep test results, and they are all conventional mechanical parameters, which are relatively convenient to apply;

[0190] 2. The calculation process of the present invention is highly complex and can accurately describe the three-stage characteristics of rock creep under different temperatures compared to the traditional Nishihara model. The test results fully demonstrate the applicability and rationality of the limestone creep model under thermal-mechanical coupling.

[0191] Therefore, the creep damage model established in the present invention can better reflect the rheological mechanism of rock under the action of thermal-mechanical coupling, and the establishment of this model can provide certain theoretical guidance for the long-term stability analysis of surrounding rock of fire tunnels.

[0192] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for constructing a thermomechanically coupled limestone fractional creep damage constitutive model, characterized in that: The steps include: S1: establishing a first thermally damaged elastic element taking into account the effects of thermal expansion and thermal damage to obtain a constitutive relation equation of the first thermally damaged elastic element; S2: establishing a thermomechanical coupling damaged viscous element considering the influence of thermomechanical coupling to obtain a constitutive relation equation of the thermomechanical coupling damaged viscous element; S3: establishing a thermomechanical coupling limestone fractional damage creep model including the first thermally damaged elastic element and the thermomechanically coupled damaged viscous element; S4: according to the thermomechanical coupling limestone fractional damage creep model, obtaining a one-dimensional constitutive relation equation of the thermomechanical coupling limestone fractional damage creep constitutive model; The one-dimensional constitutive relation equation of the thermal-mechanical coupled limestone fractional damage creep constitutive model is established as follows: Total strain ε 总 The following conditions must be met: e 总 =e e +e ve +e vp (10) Where: ε e is the thermoelastic strain; ε ve is the thermo-viscoelastic strain; ε vp is the thermo-mechanical coupled viscoplastic strain; The stress-strain relationship of the first thermally damaged elastic element is: Where: E1 is the elastic modulus of the first thermally damaged elastic element; α T1 is the thermal expansion coefficient of the first thermally damaged elastic element; The stress-strain relationship of thermally damaged viscoelastic body is: Where: ε H is the strain of the thermally damaged elastic element; σ H is the stress of thermally damaged elastic elements; ε N is the strain of thermally damaged adhesive element; σ N is the stress of thermally damaged adhesive components; η1 is the shear viscosity coefficient of the thermally damaged viscous element; α T2 is the thermal expansion coefficient of the second thermally damaged elastic element; E2 is the elastic modulus of the second thermally damaged elastic element; is the integral operation of the strain of the damaged viscous element over time t; Solving equation (12), we get: Wherein: k is an integer not less than 0; Combining equation (5) and equation (8), we get: Where: s is the yield stress threshold; Therefore, the one-dimensional constitutive equation is obtained as follows: S5: According to the one-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model, a three-dimensional constitutive relationship equation of the thermomechanical coupling limestone fractional damage creep constitutive model is established.

2. The method for constructing a thermomechanical coupled limestone fractional creep damage constitutive model according to claim 1, characterized in that: In S1, the mechanical constitutive relation equation of the first thermally damaged elastic element is established as follows: σ=E1(1-D T )ε-E(1-D T )α T1 △T (1) Where: σ is the applied load value; E1 is the elastic modulus of the first thermally damaged elastic element; D T is the thermal damage caused by temperature; ε is the applied strain value; α T1 is the thermal expansion coefficient of the first thermally damaged elastic element; ΔT is the temperature difference.

3. The method for constructing a thermal-mechanical coupled limestone fractional creep damage constitutive model according to claim 2, characterized in that: In S2, the constitutive equation of the thermal-mechanical coupling damaged viscous element is established as follows: The simplified calculation process of fractional calculus is as follows: Where: δ is the fractional order; Г(δ) is the Gamma function; f(t) is the fractional function with t as the independent variable; f(ξ) is the fractional function with ξ as the independent variable; Where: Gamma function Г(δ) is: The constitutive equation of the integer-order viscosity pot is transformed into a fractional-order viscosity pot: Where: σ(t) is the function of stress changing with time, ε(t) is the function of strain changing with time; t is time; η is the viscosity coefficient; When the stress is σ(t) = σ, keep the stress unchanged and integrate formula (2), we get: Considering the damage effect of temperature on rock elastic modulus, we can get: Where: E T is the elastic modulus of rock at temperature T; E0 is the elastic modulus of rock at room temperature 20℃; Assuming that the random variable follows an exponential distribution with parameter λ, then: Where: λ is the coefficient that characterizes the evolution law of aging damage; t is time; D c is the load damage caused by the load; N F is the number of micro-element bodies destroyed under high temperature; N is the total number of micro-element bodies; then Where: D is the damage variable; D T Thermal damage caused by temperature; D c Load damage caused by load; D T D c is the coupling term; The constitutive relation of the thermal-mechanical coupling damaged viscous element is: Where: η2 is the thermal-mechanical coupling viscosity coefficient; D is the damage variable.

4. The method for constructing a thermomechanical coupled limestone fractional creep damage constitutive model according to claim 3, characterized in that: The thermal-mechanical coupled limestone fractional damage creep model in S3 includes the first thermal damage elastic element, thermal damage viscoelastic body and thermal-mechanical coupled damage viscoplastic body; The first thermally damaged elastic element, the thermally damaged viscoelastic body and the thermally coupled damaged viscoplastic body are connected in series; The heat-damaged viscoelastic body comprises a second heat-damaged elastic element and a heat-damaged viscous element; the second heat-damaged elastic element is connected in parallel with the heat-damaged viscous element; The thermal-mechanical-coupled-damaged-viscoplastic body comprises a thermal-mechanical-coupled-damaged-viscous element and a thermal-mechanical-coupled-damaged-plastic element, and the thermal-mechanical-coupled-damaged-viscous element and the thermal-mechanical-coupled-damaged-plastic element are connected in parallel.

5. The method for constructing a thermomechanical coupled limestone fractional creep damage constitutive model according to claim 1, characterized in that: The three-dimensional constitutive equation of the thermal-mechanical coupling limestone fractional damage creep constitutive model in S5 is established as follows: In the three-dimensional state, the total axial strain ε ij总 for: Where: is the total axial thermoelastic strain; is the total axial thermo-viscoelastic strain; is the total axial thermo-viscoplastic strain; The three-dimensional constitutive relationship of the thermally damaged elastomer is: Where: S ij is the stress deviator; e ij is the strain deviator; σ ii is the first invariant of the stress tensor; ε ii is the first invariant of the strain tensor; G m is the thermoelastic shear modulus; K is the elastic bulk modulus; Therefore, the total axial thermoelastic strain of the thermally damaged elastomer is expressed as: Where: is the total axial thermoelastic strain, σ m is the spherical stress tensor, δ ij is the Kronecker function; The three-dimensional constitutive relation of thermally damaged viscoelastic body is: Where: is the total axial thermo-viscoelastic strain; G K is the thermo-viscoelastic shear modulus, η1 is the shear viscosity coefficient of the thermally damaged viscous element; The three-dimensional constitutive relation of the thermomechanical coupling damaged viscoplastic body is: Where: is the total axial thermo-viscous plastic strain; F is the yield function; F0 is the initial yield function value; Q is the plastic potential function, is a power function; 〈 〉 is the yield condition judgment function, that is: σ ij is the stress of the thermomechanical coupled damaged viscoplastic body in three-dimensional state; When F ≥ 0, formula (21) is expressed as: Where: J2 is the second invariant of stress deviator; σ1 is the axial stress; σ3 is the confining pressure; Substituting equations (17) to (23) into equation (16), the three-dimensional constitutive relationship equation of the thermomechanical coupled limestone fractional damage creep constitutive model under three-dimensional state is obtained as follows: Under the conventional triaxial compression test conditions, σ2=σ3, then: Substituting equations (25) to (26) into (24), setting the initial yield function F0 = 1, the axial creep equation is obtained as follows:

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