Construction Method of Constitutive Equation of Reef Limestone Based on Binary Medium Model

Abstract

The invention discloses a method for constructing a constitutive equation of reef limestone based on a binary medium model, including the following: dividing the medium inside the reef limestone into cementitious elements and frictional elements, and based on the homogenization theory, obtaining the stress-strain relationship of the unit body through the damage rate by the homogenization method. The stress and strain expressions of the unit body are respectively represented by the stress and strain of the cementitious elements and frictional elements. The stress-strain relationship of the cementitious elements is assumed to be poroelastic, consisting of a solid matrix and pores. The effect of cementation is reflected by the values of elastic parameters, and the parameters are determined by the stress-strain in the initial loading section. The frictional elements are elastoplastic, and then the elastic parameters and plastic stress-strain expressions at the mesoscopic scale are obtained through the homogenization method. The method of the invention establishes a constitutive equation for describing the whole process of unidirectional loading of reef limestone from the mesoscopic to the macroscopic scale until failure, realizes a comprehensive and detailed understanding of the mechanical properties of reef limestone, and describes the strain softening phenomenon of reef limestone.

Description

Technical Field

[0001] The present invention relates to the technical field of constitutive equations of reef limestone, and in particular to a construction method of a non-linear elastoplastic constitutive equation of reef limestone based on the application of a binary medium model. Background Technique

[0002] With the rapid exploitation of marine resources, complex reef limestone strata are often encountered in the construction and design processes of many geotechnical engineering projects. Due to the special sedimentary environment and its own biogenic characteristics of reef limestone, reef limestone has porosity, heterogeneity, and cementation characteristics to varying degrees during its formation process. During the loading process, the internal cementation of reef limestone will gradually be damaged, resulting in strain-softening phenomena in the specimens as a whole. When reef limestone is used as the foundation or subgrade of infrastructure, it is necessary to comprehensively understand the mechanical properties and failure mechanisms of reef limestone. Constructing the constitutive equation of reef limestone not only helps to describe its mechanical variation law but also helps to understand its failure characteristics.Some studies have been carried out on the mechanical properties, sedimentary characteristics and shear properties of reef limestone. For example, in the literature "Wan Z, Dai G, and Gong W. Full-scale load testing of two large-diameter drilled shafts in coral-reef limestone formations. Bulletin of Engineering Geology and the Environment, 2018, 77: 1127–1143.", in-situ load tests were conducted on two large-diameter vertical shafts in the reef limestone formation of the Maldives, and the O-cell test results before and after grouting of the 3.2m diameter and 1.5m diameter driven vertical shafts in the reef limestone formation were reported. The test results show that the bearing capacity of the reef limestone has been greatly improved after grouting, and the bearing capacities all meet the design requirements. Another example is in the literature "Tang Q, Zhang J, Feng Y, et al. Numerical Simulation for Shallow Strata Stability of Coral Reef in the Southwest of Yongshu Reef (South China Sea). J. Ocean Univ. China, 2018 17(4): 763-772.", which explored the shallow sedimentary characteristics of reef limestone in the South China Sea islands and reefs, and established a geological model for numerical simulation. The simulation results show that the stability of reef limestone depends on the fluctuating load and seismic zone intensity of reef limestone. Another example is in the literature "Li D, Shi C, Ruan H, et al. Study on shear behavior of coral reef limestone–concrete interface. Marine Georesources & Geotechnology, 2022, DOI: 10.1080 / 1064119X.2021.1906365.", in which multiple direct shear tests were conducted on cemented specimens of different reef limestone and concrete interfaces to explore the influence of the form and cementation characteristics of the reef limestone and concrete interface on the failure characteristics of the specimens.

[0003] The above mechanical property tests on reef limestone help to understand and master the mechanical properties of reef limestone. However, the current research results on reef limestone are still very few, and only specific mechanical tests have been carried out, lacking research results on the constitutive equation of reef limestone. In order to understand the mechanical properties of reef limestone in detail and serve engineering construction and design, it is very necessary to construct the constitutive equation of reef limestone, so as to fill the research gap of the constitutive equation of reef limestone. Summary of the Invention

[0004] The object of the present invention is to provide a method for constructing a constitutive equation of reef limestone based on a binary medium model in view of the deficiencies of the prior art, establish a constitutive equation that describes the whole process of unidirectional loading of reef limestone from the mesoscopic scale to the macroscopic scale until failure, and achieve a comprehensive and detailed understanding of the mechanical properties of reef limestone and describe the strain softening phenomenon of reef limestone.

[0005] The main idea of the present invention is as follows: The diagenesis and epigenetic evolution of reef limestone are unique and completely different from conventional terrigenous sediments. The pore types of reef limestone are complex. To facilitate the analysis of problems, the poorly compacted cement is simplified as a cementation property problem, and the poorly compacted filling between diagenetic particles is simplified as a porosity problem. The pores existing between cements result in a large difference in the cementation strength of reef limestone, and the pores existing between diagenetic particles further exacerbate the discreteness of the strength of reef limestone. Based on the above understanding, in order to accurately describe the bearing characteristics of reef limestone, the stress-bearing unit of reef limestone is decomposed and the following assumptions are made: (i) At the mesoscopic scale, it is regarded as a mixture of pores and matrix, where the matrix is in a damaged state. By homogenizing the mesoscopic damage state to the representative unit state, a macroscopic constitutive model can be established. According to the failure mechanism and microscopic structure analysis of reef limestone, it can be seen that reef limestone has a large pore distribution and a strong cementation effect between particles. Therefore, the idea of a binary medium can be used to establish a constitutive model; (ii) Reef limestone is divided into different types according to the particle size. Analyzing the pore distribution characteristics of the specimen shows that the larger the particles, the more uneven the pore distribution. From the perspective of the entire formation, the particle-pore distribution is uniform. Therefore, it is assumed in this paper that the particle-pore distribution is uniform; (iii) The samples of reef limestone are basically located on the same horizontal plane, the positions are relatively concentrated, and the cementation difference of the reef limestone specimens is small.

[0006] The method for constructing a constitutive equation of reef limestone based on a binary medium model provided by the present invention mainly includes the following contents:

[0007] Establish the stress-strain relationship of the representative unit. The medium inside the reef limestone is divided into cementation elements and friction elements. Based on the homogenization theory, the stress-strain relationship of the unit is obtained by the homogenization method through the damage rate. The expressions of the stress and strain of the unit are represented by the stress and strain of the cementation elements and friction elements respectively.

[0008] Determine the stress-strain relationship of the cementation element. The stress-strain relationship of the cementation element is assumed to be poroelastic, consisting of a solid matrix and pores. The cementation effect is reflected by the value of the elastic parameter, and the parameters are determined by the stress-strain of the initial loading section;

[0009] Determination of the stress-strain relationship of the cementitious element. The frictional element is elastoplastic and is microscopically assumed to be an elastoplastic porous material composed of a solid soil skeleton and pores. Then, the elastic parameters and the stress-strain expressions of plasticity at the mesoscopic scale are obtained through the homogenization method; the strain increment of the frictional element is divided into elastic and plastic;

[0010] Determination of the structural parameters. The structural parameters include the damage rate and the local strain coefficient. The optimal parameters are selected by the trial-and-error method and the expressions are given.

[0011] The method for constructing the constitutive equation of reef limestone based on the binary medium model provided by the present invention includes the following contents:

[0012] Establish the stress-strain relationship of the representative elementary volume. Based on the homogenization theory, the stress-strain relationship of the representative elementary volume is as follows:

[0013]

[0014]

[0015] Among them, σ ij , ε ij are the stress and strain of the representative elementary volume respectively, are the stress and strain of the cementitious element respectively, are the stress and strain of the frictional element respectively; χ is defined as the damage rate, indicating the volume ratio occupied by the frictional element (volume damage rate), and it is assumed that the damage rate is a function of the strain. The superscripts b and f indicate that the quantity is the cementitious element and the frictional element respectively.

[0016] Obtained from equations (1) and (2),

[0017]

[0018]

[0019] Among them, χ 0 is the current damage rate, and are the current stresses of the cementitious element and the frictional element respectively, and are the current strains of the cementitious element and the frictional element respectively.

[0020] The stiffness matrices of the cementitious element and the frictional element are represented by and respectively, then the stress-strain relationships of the cementitious element and the frictional element can be obtained as follows:

[0021]

[0022]

[0023] Obtained from Equations (1) to (6)

[0024]

[0025] Introduce the local strain coefficient C ijkl , which can establish the relationship between the strain of the cementitious element and the strain of the representative elementary volume, and satisfies The incremental form of this equation is In the formula is the current local strain coefficient matrix. Substitute into the expression in Equation (7), and after arrangement, we get

[0026]

[0027] Considering the current stress-strain state, arrange Equation (8) to obtain the expression of the stress increment under the general stress state:

[0028]

[0029] Equation (9) is the expression of the constitutive equation of the reef limestone obtained, and the cementitious element, friction element, and structural parameters involved are determined by the following method.

[0030] In the above method, further, determine the stress-strain relationship of the cementitious element according to the following method. The stress-strain relationship of the cementitious element is poroelastic, composed of a solid matrix and pores. The effect of cementation is reflected by the values of elastic parameters, and the parameters are determined by the stress-strain of the initial loading section. The stress-strain relationship of the cementitious element is as follows:

[0031]

[0032] where K b , G b are the bulk modulus and shear modulus of the cementitious element respectively.

[0033] In the cementitious element, the porosity is Then the volume occupied by the solid matrix is By the micromechanics method, the bulk modulus and shear modulus of the cementitious element can be obtained as

[0034]

[0035]

[0036] In the formula, K s , G s are the bulk modulus and shear modulus of the reef limestone matrix respectively. α and β are parameters obtained from the inclusion theory, where

[0037] In the above method, further, the stress-strain relationship of the friction element is determined according to the following method. The friction element is elastoplastic and is assumed to be an elastoplastic porous material composed of a solid soil skeleton and pores at the microscale. Then, the elastic parameters and the plastic stress-strain expressions at the mesoscale are obtained through the homogenization method. The strain increment {dε} of the friction element f is divided into an elastic {dε e} f and a plastic {dε f} f , which is expressed as follows

[0038] {dε} f ={dε e} f +{dε f} f . (13)

[0039] The elastic part of the friction element. The elastic parameters of the bulk modulus K M and shear modulus G M of the solid soil skeleton in the friction element can be obtained by a derivation similar to that of the cementation element. The bulk modulus K fe and shear modulus G fe of the cementation element are respectively:

[0040]

[0041]

[0042] where is the porosity of the friction element.

[0043] The plastic part of the friction element. The yield criterion at the mesoscale is: In the formula, α′ represents the friction coefficient between the broken particles; T is the tensile strength (compression is positive), which is related to the cohesion and friction angle; is the stress of the solid phase, the generalized shear stress where δ ij is the Kronecker symbol.

[0044] From the micromechanics method, the macroscopic yield criterion expressed by the mesoscopic yield can be obtained:

[0045]

[0046] where: α′ is a model parameter,

[0047] The plastic potential is expressed as

[0048]

[0049] where ξ is a model parameter, which is 1.0 when the associated flow rule is adopted.

[0050] According to the plastic theory, the plastic strain increment is expressed as

[0051]

[0052] where dλ is the plastic multiplier.

[0053] In the above method, further, the structural parameters are determined according to the following method. The structural parameters include the damage rate χ and the local strain coefficient C ijkl , and the optimal parameters are selected by the trial-and-error method. The damage rate reflects the degree of transformation of the damaged cementitious elements in the unit body into frictional elements. According to the initial damage state of the reef limestone (the influence of porosity) and the damage degree during the loading process, the following expression is adopted:

[0054]

[0055] where a1, b1, c1 are model parameters; ε1 is the axial strain. The local strain coefficient C ijkl Establish the relationship between the strain of the cementitious element and the strain of the unit body, which is taken as a scalar C here, and is specifically expressed as follows:

[0056]

[0057] where c0 is a model parameter, and the strain of the cementitious element at the initial loading is the strain of the frictional element C = 1.0.

[0058] Determination of the parameters of the cementitious element under the triaxial stress state. Under the triaxial stress state, the stress-strain relationship of the cementitious element is simplified to the following formula:

[0059]

[0060] Determination of the parameters of the frictional element under the triaxial stress state. Under the triaxial stress state, the elastic stress-strain relationship of the frictional element can be simplified to

[0061]

[0062] This formula can also be transformed into Substituting the above two formulas respectively, the elastic incremental expression is obtained:

[0063]

[0064]

[0065] Considering the consistency condition: The following is obtained for dλ:

[0066]

[0067] In the formula Thus, the incremental stress-strain relationship of the friction element can be obtained:

[0068]

[0069] In the formula:

[0070] Determination of structural parameters under triaxial stress state. Under triaxial stress conditions, the expressions for the damage rate and the local strain coefficient are as follows respectively:

[0071]

[0072]

[0073] Compared with the prior art, the present invention has the following beneficial effects:

[0074] Based on the concept of a binary medium model, the present invention regards reef limestone as a binary medium material composed of cementing elements and friction elements, and establishes a constitutive model of reef limestone based on the homogenization theory. The established constitutive model simultaneously considers the cementation effect and the characteristics of pore distribution of reef limestone. The uniaxial compression test of reef limestone shows that reef limestone has strain-softening characteristics, and a method for determining the parameters of the binary medium constitutive model is given through the uniaxial compression test, and the rationality and accuracy of the constitutive equation of the present invention are verified by comparing with the test results. Through the parameter sensitivity analysis of the constitutive model, it shows that the change of the main parameters of the model can reflect the strain-softening degree of reef limestone, thereby providing a basis for the design of reef limestone foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 It is a schematic diagram for constructing the reef limestone - theoretical model described in the present invention;

[0076] Figure 2 It is the verification of the uniaxial compression stress-strain curve of reef limestone, where a) is specimen 1; b) is specimen 2; c) is specimen 3;

[0077] Figure 3 It is the influence of parameter changes on the uniaxial compression stress-strain curve of reef limestone, where a) porosity; b) T; c) c1; d) b1; e) c0. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0078] The construction method of the constitutive equation of reef limestone based on the binary medium model of the present invention will be further described below through specific embodiments.

[0079] The diagenesis and epigenetic evolution of reef limestone are unique and completely different from conventional terrigenous sediments. The pore types of reef limestone are complex. For the convenience of analyzing problems, the incompactly filled cement is simplified as a cementation property problem, and the incompactly filled space between diagenetic particles is simplified as a porosity problem. The pores existing between cements result in a large difference in the cementation strength of reef limestone, and the pores existing between diagenetic particles further exacerbate the discreteness of the strength of reef limestone. Based on the above understanding, in order to accurately describe the bearing characteristics of reef limestone, the stress-bearing unit of reef limestone is decomposed and the following assumptions are made: (i) On the mesoscopic scale, it is regarded as a mixture of pores and matrix, where the matrix is in a damaged state. By homogenizing the mesoscopic damage state to the representative unit state, a macroscopic constitutive model can be established. According to the failure mechanism and microscopic structure analysis of reef limestone, it can be known that reef limestone has a large pore distribution and a strong cementation effect between particles. Therefore, the constitutive model can be established using the idea of binary medium, as Figure 1 shown; (ii) Reef limestone is divided into different types according to the particle size. Analyzing the pore distribution characteristics of the specimen shows that: the larger the particles, the more uneven the pore distribution. From the perspective of the entire formation, the particles-pores are evenly distributed. Therefore, it is assumed in this paper that the particles-pores are evenly distributed; (iii) The samples of reef limestone are basically located on the same horizontal plane, the positions are relatively concentrated, and the cementation difference of the reef limestone specimens is small.

[0080] The construction method of the constitutive equation of reef limestone based on the binary medium model is as follows:

[0081] Establish the stress-strain relationship of the representative unit cell. Based on the homogenization theory, the stress-strain relationship of the representative unit cell is as follows:

[0082]

[0083]

[0084] where, σ ij and ε ij are the stress and strain of the representative unit cell respectively, are the stress and strain of the cement element respectively, are the stress and strain of the friction element respectively; χ is defined as the damage rate, representing the volume fraction of the friction element (volume damage rate), and it is assumed that the damage rate is a function of strain. The superscripts b and f indicate that the quantity is the cement element and the friction element respectively.

[0085] Obtained from equations (1) and (2),

[0086]

[0087]

[0088] where χ 0 is the current damage rate, and are the current stresses of the cementitious element and the frictional element respectively, and are the current strains of the cementitious element and the frictional element respectively.

[0089] The stiffness matrices of the cementitious element and the frictional element are represented by and respectively, then the stress-strain relationships of the cementitious element and the frictional element can be obtained as follows:

[0090]

[0091]

[0092] From equations (1) to (6), we get

[0093]

[0094] Introduce the local strain coefficient C ijkl , which can establish the relationship between the strain of the cementitious element and the strain of the representative elementary volume, and satisfies The incremental form of this equation is In the formula is the current local strain coefficient matrix. Substitute into the expression in equation (7), and after rearrangement, we get

[0095]

[0096] Considering the current stress-strain state, rearrange equation (8) to obtain the expression of the stress increment under the general stress state:

[0097]

[0098] Equation (9) is the expression of the constitutive equation of the reef limestone obtained, and the cementitious element, the frictional element and the structural parameters involved are determined by the following method.

[0099] The stress-strain relationship of the cementitious element. The stress-strain relationship of the cementitious element is poroelastic, consisting of a solid matrix and pores. The effect of cementation is reflected by the values of elastic parameters, and the parameters are determined by the stress-strain of the initial loading section. The stress-strain relationship of the cementitious element is as follows:

[0100]

[0101] Among which K b and G b are the bulk modulus and shear modulus of the cementitious element respectively.

[0102] In the cementitious element, the porosity is then the volume occupied by the solid matrix is By the micromechanics method, the bulk modulus and shear modulus of the cementitious element can be obtained as

[0103]

[0104]

[0105] In the formula, K s and G s are the bulk modulus and shear modulus of the reef limestone matrix respectively, and α and β are parameters obtained from the inclusion theory, where

[0106] The stress-strain relationship of the friction element. The friction element is elastoplastic and is assumed to be an elastoplastic porous material composed of a solid soil skeleton and pores at the microscale. Then, the elastic parameters and plastic stress-strain expressions at the mesoscale are obtained through the homogenization method. The strain increment {dε} of the friction element f is divided into the elastic {dε e} f and the plastic {dε f}, which are expressed as f as follows

[0107] {dε} f ={dε e} f +{dε f} f . (13)

[0108] The elastic part of the friction element. The elastic parameters of the solid soil skeleton in the friction element, the bulk modulus K M and the shear modulus G M , by using a similar derivation as that of the cementitious element, the bulk modulus K fe and the shear modulus G fe of the cementitious element can be obtained, which are respectively:

[0109]

[0110]

[0111] where is the porosity of the friction element.

[0112] The plastic part of the friction element. The yield criterion at the mesoscale is: where α′ represents the friction coefficient between the damaged particles; T is the tensile strength (positive for compression), which is related to the cohesion and the friction angle; is the stress of the solid phase, generalized shear stress where δ ij is the Kronecker symbol.

[0113] By the micromechanics method, the macroscopic yield criterion expressed by the mesoscopic yield can be obtained:

[0114]

[0115] where: α′ is a model parameter,

[0116] The plastic potential is expressed as

[0117]

[0118] where ξ is a model parameter, and when the associated flow rule is adopted, ξ is 1.0.

[0119] According to the plastic theory, the plastic strain increment is expressed as

[0120]

[0121] where dλ is the plastic multiplier.

[0122] Determination of the structural parameters. The structural parameters include the damage rate χ and the local strain coefficient C ijkl , and the optimal parameters are selected by the trial-and-error method. The damage rate reflects the degree of conversion of the cementitious elements in the unit body into frictional elements. According to the initial damage state of the reef limestone (the influence of porosity) and the damage degree during the loading process, the following expression is adopted:

[0123]

[0124] where a1, b1, c1 are model parameters; ε1 is the axial strain. The local strain coefficient C ijkl establishes the relationship between the strain of the cementitious element and the strain of the unit body. Here, it is taken as a scalar C, and the specific expression is as follows:

[0125]

[0126] where c0 is a model parameter, and the strain of the cementitious element at the initial loading is the strain of the frictional element C = 1.0.

[0127] Determination of parameters of cementitious elements under triaxial stress state. Under triaxial stress state, the stress-strain relationship of cementitious elements is simplified to the following formula:

[0128]

[0129] Determination of parameters of friction elements under triaxial stress state. Under triaxial stress state, the elastic stress-strain relationship of friction elements can be simplified to

[0130]

[0131] This formula can also be transformed into Substituting the above two formulas respectively, the elastic increment expression is obtained:

[0132]

[0133]

[0134] Considering the consistency condition: The following is obtained for dλ:

[0135]

[0136] In the formula Thus, the increment stress-strain relationship of friction elements can be obtained:

[0137]

[0138] In the formula:

[0139] Determination of structural parameters under triaxial stress state. Under triaxial stress conditions, the expressions of damage rate and local strain coefficient are as follows respectively:

[0140]

[0141]

[0142] Verification of the constitutive equation constructed by the present invention. To clarify the constitutive equation of coral reef limestone, relevant tests are carried out to obtain the parameter values. Among them, the porosity is determined to be By solving the bulk modulus and shear modulus matrix parameters through cementation strength and uniaxial tests, the constitutive model of coral reef limestone in the sample area is formed.

[0143] Uniaxial compression tests were carried out on reef limestone specimens with a diameter of 50 mm and a height of 100 mm. The displacement control method was used for loading at a loading rate of 0.002 mm / s. Uniaxial compression tests were conducted on 16 specimens to obtain the stress-strain curves of coral reef limestone. The peak strength of the coral reef limestone measured in the tests was between 7 MPa and 56 MPa, with an average strength of 21 MPa. The strain corresponding to the peak strength was between 0.70% and 2.19%, with an average peak strain of 1.34%. The elastic modulus was between 3 GPa and 28 GPa, with an average elastic modulus of 16 GPa. The strength and elastic modulus of coral reef limestone are relatively low and have a wide distribution range, showing strong discreteness, which is related to the unique diagenetic and epigenetic evolution processes of reef limestone. Its strength is mainly determined by the strength of the constituent minerals and the degree of mineral cementation. The compression failure modes of each specimen include brittle failure (specimens 027, 028, 059, 078, 110, 115) and ductile failure (specimens 004, 046, 050, 065, 066, 084, 090, 096, 099, 106). Before reaching the ultimate strength, the axial stress-strain curves of the two failure modes both show a linear growth trend. For brittle failure, the stress drops suddenly after reaching the peak strength, and the bearing capacity is quickly lost. For specimens with ductile failure, after reaching the peak strength, the specimens do not undergo complete failure. Their bearing capacity gradually decreases and the strain further increases. The stress drop rate after the peak is relatively small, which is quite different from the failure mode of brittle rocks, indicating that there is still a relatively large residual strength after the failure of reef limestone.

[0144] To prove the reliability of the constitutive model of reef limestone, the stress-strain curves of reef limestone under uniaxial compression conditions were obtained, and the test results were compared with the model calculation results, as Figure 2 shown. The calculation parameters are as follows: The porosity of specimens 1, 2, and 3 are 0.17, 0.20, and 0.18 respectively; the values of K s , G s , K M , G M are shown in Table 1. Among them, K s , G S are determined from the test results at the initial stage of loading, and K M , G M are determined from the test results at the residual stage of loading; α′ = 0.48, ξ = 2.5, T = 0.2 MPa, and the porosity of the friction element is taken as 1.1 times that of the cementation element; b1 = 0.452, c1 = 23.5, c0 = 0.2. As Figure 2 can be seen, the calculation results are basically consistent with the test results and can reflect the strain softening phenomenon.

[0145] Table 1 gives K s , G s , K M , G MParameter values:

[0146] Table 1 K s , G s , K M , G M Parameter values

[0147]

[0148] Sensitivity analysis of the parameters in the constitutive equation constructed by the present invention. Considering that the constitutive equation of the present invention can also be used to verify existing reef limestone or other geotechnical materials that exhibit strain softening, a sensitivity analysis of the parameters in the constitutive equation is therefore carried out, that is, when the main parameters change, how the stress-strain that the constitutive equation can describe changes. Figure 3 The results of the parameter sensitivity analysis are given, and the effects of the changes in porosity, T, c1, b1, and c0 on the principal stress difference are analyzed. The changes in porosity, T, c1, b1, and c0 will all have an impact on the principal stress difference. As the porosity of the specimen increases, the peak value of the principal stress difference gradually decreases, but the overall change trend is the same, as shown in Figure 3 a; as T increases, the peak value of the principal stress difference decreases, as shown in Figure 3 b; the effect of c0 on the principal stress difference is relatively significant. The peak principal stress difference corresponding to the minimum c1 value is about 5 times that of the maximum c1 value. As c1 increases, the peak value of the principal stress difference gradually decreases, and the decreasing amplitude increases with the increase of deformation, as shown in Figure 3 c; the effect of b1 on the principal stress difference is similar to that of c1, but the decreasing amplitude is smaller than that of c1, as shown in Figure 3 d; the effect of c0 on the principal stress difference is not significant. As c0 increases, the peak value of the principal stress difference gradually decreases, and the decreasing amplitude is relatively small, as shown in Figure 3 e. The parameter sensitivity analysis of the model shows that the degree of strain softening of reef limestone can be reflected by the changes in the main parameters of the model, thus providing a basis for the design of reef limestone foundations.

Claims

1. A method for constructing a constitutive equation of reef limestone based on a binary medium model, characterized in that, It includes the following contents: Establish the stress-strain relationship of the representative volume element. Based on the homogenization theory, the stress-strain relationship of the representative volume element is as follows: Among them, σ ij , ε ij are the stress and strain of the representative unit cell respectively, are the stress and strain of the cementitious element respectively, are the stress and strain of the friction element respectively; χ is defined as the damage rate, representing the volume ratio occupied by the friction element, and it is assumed that the damage rate is a function of the strain; the superscripts b and f indicate that the quantity is the cementitious element and the friction element in sequence; Obtained from Equations (1) and (2), where χ 0 is the current damage rate, and are the current stresses of the cementitious element and the frictional element respectively, and are the current strains of the cementitious element and the frictional element respectively; The stiffness matrices of the cementitious element and the frictional element are represented by and respectively. Then, the stress-strain relationships of the cementitious element and the frictional element can be obtained as follows: Obtained from Equations (1) to (6) Introduce the local strain coefficient C ijkl , and satisfy The incremental form of this equation is In the formula is the current local strain coefficient matrix; Substitute into the expression in Equation (7), and after arrangement, we get Considering the current stress-strain state, Equation (8) is sorted out to obtain the expression of the stress increment under the general stress state: The stress-strain relationship of the cementitious element involved in Equation (9): The stress-strain relationship of the cementitious element is as follows: where K b and G b are the bulk modulus and shear modulus of the cementitious element, respectively; In the cementitious element, the porosity is Then the volume occupied by the solid matrix is By the micromechanics method, the bulk modulus and shear modulus of the cementitious element can be obtained as where K s and G s are the bulk modulus and shear modulus of the reef limestone matrix respectively, and α and β are parameters obtained from the inclusion theory, where 2. The method according to claim 1, characterized in that, Determine the stress-strain relationship of the friction element involved in Equation (9) according to the following method: Strain increment {dε} of the friction element f is divided into elastic {dε e} f and plastic {dε f} f and is expressed as follows {dε} f ={dε e} f +{dε f} f (13) Elastic part of the friction element: Bulk modulus K of the elastic parameter of the solid soil skeleton in the friction element M , shear modulus G M , through a derivation similar to that of the cementation element, the bulk modulus K fe and shear modulus G fe of the cementation element can be obtained, which are respectively: Among them is the porosity of the friction element; Plastic part of the friction element: The yield criterion at the mesoscopic scale is as follows: where α′ represents the friction coefficient between the damaged particles; T is the tensile strength (positive for compression), which is related to the cohesion and the friction angle; is the stress of the solid phase, generalized shear stress where δ ij is the Kronecker symbol; Based on the micromechanics method, a macroscopic yield criterion expressed by microscopic yield can be obtained: where: α′ is a model parameter, The plastic potential is expressed as where ξ is a model parameter. When the associated flow rule is adopted, ξ is 1.0; According to the plastic theory, the plastic strain increment is expressed as where dλ is the plastic multiplier.

3. According to the method described in claim 2, wherein, Determine the structural parameters involved in Equation (9) according to the following method: The structural parameters include the damage rate χ and the local strain coefficient C ijkl , and the optimal parameters are selected by the trial-and-error method; the damage rate reflects the degree of transformation of the damaged cementitious elements into frictional elements in the unit cell. According to the initial damage state of the reef limestone and the damage degree during the loading process, the following expression is used: where a1, b1, and c1 are model parameters; ε1 is the axial strain; Local strain coefficient C ijkl Establish the relationship between the strain of the cementitious element and the strain of the unit body, which is taken as the scalar C here, and is specifically expressed as follows: where c0 is a model parameter, and the strain of the cementitious element at the initial loading is the strain of the friction element C = 1.

0.

4. According to the method described in any one of claims 1 to 3, wherein, Determination of the parameters of the cementitious element under the triaxial stress state: Under the triaxial stress state, the stress-strain relationship of the cementitious element is simplified to the following formula: Under the triaxial stress state, the elastic stress-strain relationship of the friction element can be simplified to This equation can also be transformed into Substituting the above two equations respectively, the incremental expression of elasticity is obtained: Considering the consistency condition: dλ is obtained as follows: wherein The incremental stress-strain relationship of the friction element is obtained as follows: In the formula: Determination of the structural parameters under the triaxial stress state: Under the triaxial stress condition, the expressions of the damage rate and the local strain coefficient are as follows:

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