Constitutive modeling method of elastic-plastic damage based on full stress-strain relationship of rock materials

Abstract

The present invention discloses a constitutive modeling method for elastic-plastic damage of the full stress-strain relationship of rock materials, comprising selecting a strength criterion in stress space that can describe the peak strength envelope of the material; introducing a unified, continuous and smooth hardening / softening function with generalized shear strain as a variable into the strength criterion to construct a yield function; selecting an appropriate plastic potential function based on material properties and flow laws; establishing a damage criterion for coupling plastic deformation and damage; calculating a plastic scalar multiplier and a damage increment multiplier; calculating a plastic strain increment and a damage increment; and deriving a elastoplastic damage stiffness tensor. The present invention starts with the strength and deformation of rock materials, constructs a yield function based on the strength criterion, introduces a unified, continuous and smooth hardening / softening function, ensures that the yield function at the peak has the same form as the strength criterion, and establishes an elastic-plastic damage constitutive model that can reflect the full stress-strain relationship of rock materials. The model has wide applicability and engineering application prospects.

Description

Technical Field

[0001] The present invention belongs to the field of geotechnical engineering, and in particular relates to a constitutive modeling method of elastic-plastic damage of a rock material based on the full stress-strain relationship. Background Art

[0002] Rock materials, such as soil, rock, and concrete, are ubiquitous and involve many engineering applications. The basic mechanical properties of rock materials, including hydrostatic pressure, strength nonlinearity, strain hardening or softening, and volume dilatancy, are hot topics of research under complex stress states. Establishing a suitable constitutive model is an important way to study the mechanical behavior of rock materials. Strength criteria reflect the failure behavior of materials under loading conditions and are a basic component of geomechanics research. To date, many researchers have successfully proposed hundreds of strength criteria. Common ones include the linear Mohr-Coulomb criterion, the parabolic Mohr-Coulomb criterion, the Drucker-Prager criterion, and the Hoek-Brown criterion. These criteria have been widely used in engineering applications such as water conservancy and hydropower, geological disposal of high-level radioactive waste, strategic oil reserves, and comprehensive development of oil shale. Establishing a nonlinear constitutive model of rock materials based on strength criteria and fully describing their strength characteristics in the process of establishing the constitutive model has become a problem of concern to many scholars. In this regard, many phenomenological constitutive models have been established. These models can be roughly divided into two categories:

[0003] Elastic-plastic strain softening constitutive model based on strength theory. Wang Junxiang and Jiang Annan, in their paper “Establishment of rock strain softening constitutive model and study on its solution by NR-AL method” (Wang Junxiang, Jiang Annan. Establishment of rock strain softening constitutive model and study on its solution by NR-AL method [J]. Rock and Soil Mechanics, 2015, (2): 393-402), based on Drucker-Prager strength criterion, used piecewise linear softening function to establish rock strain softening constitutive model and solved it by NR-AL method; Li Wenting, based on Mohr-Coulomb criterion, in their paper “Study on mechanical behavior of rock post-peak strain softening based on Mohr-Coulomb criterion” (Li Wenting, Li Shuchen, Feng Xianga, et al. Study on mechanical behavior of rock post-peak strain softening based on Mohr-Coulomb criterion [J]. Chinese Journal of Rock Mechanics and Engineering, 2011, 30(7):1460-1466) studied the post-peak strain softening mechanical behavior of rock; Zhang Yu and Wang Jingyin disclosed a method for constructing an elastic-plastic mechanical constitutive model of rock materials using thermodynamic elastic-plastic theory in their patent "A method for constructing an elastic-plastic mechanical constitutive model of rock materials" (201510674185.9) for cataclasite rocks; Lu Dechun introduced the hardening / softening parameters into the nonlinear unified strength criterion in the paper "Three-dimensional elastic-plastic constitutive model of concrete materials" (Lu Dechun, Du Xiuli, Yan Jingru, et al. Three-dimensional elastic-plastic constitutive model of concrete materials [J]. Science China: Technological Sciences, 2014, 44(8):847-860), and then established a nonlinear unified elastic-plastic model of concrete materials. The above models can describe the nonlinear mechanical behavior of rock materials, but fail to consider the influence of damage caused by the development of rock microcracks on the mechanical properties of the material, and cannot reflect the degree of degradation of the elastic modulus and the coupling process of plastic deformation and damage. In addition, using piecewise linear softening functions to describe strain softening behavior cannot guarantee that the constitutive model has first-order continuity and differentiability, which causes difficulties in numerical application.

[0004] Elastic-plastic damage constitutive model. This type of model takes into account the damage evolution caused by microcracks and has important guiding significance for the development of constitutive models. Zhang Yu and Hu Liangqiang disclosed a method for constructing an elastic-plastic damage coupling mechanical constitutive model for mudstone experimental data in the patent "A method for constructing an elastic-plastic-damage coupling mechanical constitutive model for rock materials" (201910947333.8); Chen Liang established a rock elastic-plastic damage coupling model in the paper "Research on elastic-plastic damage model of deep granite in Beishan" (Chen Liang, Liu Jianfeng, Wang Chunping, et al. Research on elastic-plastic damage model of deep granite in Beishan [J]. Chinese Journal of Rock Mechanics and Engineering, 2013, 32 (2): 289-298). The model is based on the yield stress. The function considers the effects of plastic hardening and damage softening simultaneously. Yuan Xiaoping, in his paper "Study on elastic-plastic damage constitutive model of rock based on Drucker-Prager criterion" (Yuan Xiaoping, Liu Hongyan, Wang Zhiqiao. Study on elastic-plastic damage constitutive model of rock based on Drucker-Prager criterion [J]. Rock and Soil Mechanics, 2012, 33(4): 1103-1108), considers both plastic softening and damage softening based on the Drucker-Prager criterion and establishes an elastic-plastic constitutive relationship for rock materials and a corresponding numerical algorithm. This type of model considers the coupling mechanism of plasticity and damage and well describes the pre-peak hardening and post-peak softening behavior of rock materials. In existing elastic-plastic damage models, the plastic yield surface is related to the damage variable, which means that some parameters in the model are difficult to calibrate because it is impossible to determine the damage value corresponding to the peak stress under different loading conditions. Summary of the Invention

[0005] To address the above-mentioned issues, the present invention proposes a method for constructing an elastic-plastic-damage coupled mechanical constitutive model that reflects the full stress-strain curve characteristics of rock materials. Based on a thermodynamic framework and the small deformation assumption, this method comprehensively considers the nonlinear yield criterion, plastic hardening / softening criteria, and non-associated flow laws. Incorporating the damage constitutive theory of irreversible thermodynamics, this method derives the relationship between the incremental stress tensor and the incremental strain tensor during the full deformation process of rock materials, thereby constructing an elastic-plastic-damage coupled mechanical model for rock materials. This model accurately reflects the strength and nonlinear deformation characteristics of rock materials, boasts clear concepts and mechanisms, a small number of easily identifiable parameters, and first-order continuity, making it easy to apply in finite element methods and possessing broad applicability.

[0006] The present invention is achieved through at least one of the following technical solutions.

[0007] The constitutive modeling method of elastic-plastic damage of rock materials based on the full stress-strain relationship includes the following steps:

[0008] (1) Determine the macroscopic stress-strain relationship of rock materials within the framework of continuum mechanics and thermodynamics;

[0009] (2) Selecting a strength criterion that can describe the peak strength envelope of the material in stress space;

[0010] (3) A unified continuous and smooth hardening / softening function with generalized shear strain as a variable is introduced into the strength criterion to construct a yield function that can describe the expansion and contraction of the yield surface;

[0011] (4) Select a plastic potential function that can describe the volume compression / expansion phenomenon of rock materials under triaxial compression conditions, determine the direction of plastic strain flow, and calculate the plastic strain increment using the plastic strain increment expression in combination with the plastic potential function;

[0012] (5) Establish a damage criterion that considers the coupling of plastic deformation and damage;

[0013] (6) Differentiating the macroscopic stress-strain relationship of the rock material in step (1) to obtain an incremental form of the material constitutive relationship;

[0014] (7) Combining the continuity conditions of plasticity and damage, the continuity conditions of the yield criterion and the damage criterion are solved simultaneously to obtain the plasticity scalar multiplier and the damage increment multiplier;

[0015] (8) Obtain the plastic strain increment according to the plastic scalar multiplier and the plastic strain increment expression in step (4), and obtain the damage increment according to the damage increment multiplier and the damage criterion;

[0016] (9) Substituting the plastic strain increment and damage increment obtained in step (8) into the relationship between the stress increment and the strain increment in step (6) to obtain the elastic-plastic damage stiffness tensor and the elastic-plastic damage constitutive model reflecting the full stress-strain curve characteristics of the rock material.

[0017] Furthermore, in step (1), the material strain and strain increment are decomposed into elastic and plastic parts:

[0018] ε=ε e +ε p ,dε=dε e +dε p (19)

[0019] Among them, ε, ε e and ε p Represent the total strain, elastic strain and plastic strain respectively, dε, dε e and dε p represent the total strain increment, elastic strain increment, and plastic strain increment, respectively;

[0020] The scalar damage variable d is introduced to represent the effect of damage on the elastic modulus of the material. Under the assumption of isotropic damage, the elastic stiffness tensor of rock materials affected by damage evolution is expressed as:

[0021]

[0022] in, is the elastic stiffness tensor of the lossless material, and its specific expression is described by the elastic bulk and shear moduli k0 and μ0 of the original material:

[0023]

[0024] By defining the second-order unit tensor δ, the fourth-order tensor partial operator Sum ball operator Respectively expressed as

[0025] The macroscopic stress-strain relationship of rock materials is expressed as follows:

[0026]

[0027] Among them, σ, ε, ε p represent the stress tensor, total strain tensor, and plastic strain tensor, respectively; represents the elastic stiffness tensor affected by damage evolution, and d represents the damage variable.

[0028] Furthermore, in step (2), under isothermal static load conditions, the strength criterion is expressed as:

[0029] f s =f s (σ ij )=0 (23)

[0030] Among them, f s represents the strength criterion, σ ij represents the stress component;

[0031] The strength criterion is described in terms of principal stresses, stress invariants and mean stress p-deviatoric stress q:

[0032]

[0033] where σ1, σ2, and σ3 represent the first, second, and third principal stresses, respectively; I1 = trσ represents the first invariant of the stress tensor σ, J2 = (s:s)2 represents the second invariant of the deviatoric stress tensor s, and θ σ represents the stress Lode angle.

[0034] Furthermore, the selection of the hardening / softening function must meet the following three requirements:

[0035] 1) The hardening or softening function introduced in the yield function has the characteristics of monotonically increasing before the peak and monotonically decreasing after the peak;

[0036] 2) The hardening or softening function has at least a first-order continuous derivative;

[0037] 3) The yield function and strength criterion corresponding to the peak stress should have the same form, that is, κ = 1 at the peak stress;

[0038] Combined with the strength criterion, under isothermal static loading conditions, the yield function is expressed as:

[0039] f=f(σ ij ,κ)=0 (25)

[0040] Where f represents the yield function, σ ij represents the stress component, κ represents the plastic hardening / softening function;

[0041] The yield function is described in terms of principal stresses, stress invariants, and mean stress p-deviatoric stress q:

[0042]

[0043] where σ1, σ2, and σ3 represent the first, second, and third principal stresses, respectively; I1 = trσ represents the first invariant of the stress tensor σ, J2 = (s:s)2 represents the second invariant of the deviatoric stress tensor s, and θ σ represents the stress Lode angle;

[0044] According to the stress-strain relationship (4), the expression of the yield function in the strain space must include the internal variables κ and ε. p and d.

[0045] Furthermore, in step (4), the plastic potential function g is used to determine the flow direction of the plastic strain. For rock materials, in order to avoid overestimating the shear expansion of the material, the non-associated flow law is selected, that is, g≠f. At this time, the evolution of the plastic strain produces the corresponding flow law, and the plastic strain increment expression is:

[0046]

[0047] Among them, dε p is the plastic strain increment, g is the plastic potential function, f is the yield function, λ p is a non-negative plastic scalar multiplier.

[0048] Furthermore, the damage criterion is the plastic strain ε p And the function of the damage variable d:

[0049] h=h(εp ,d)≤0 (28)

[0050] Where h is the damage evolution criterion, ε p is the plastic strain, and d is the damage variable.

[0051] Furthermore, in step (6), the incremental form of the material constitutive relation is:

[0052]

[0053] Where σ represents the stress tensor, ε, ε e and ε p denote the total strain, elastic strain and plastic strain respectively, represents the elastic stiffness tensor affected by damage evolution, represents the elastic stiffness tensor of the undamaged original material, and d represents the damage variable.

[0054] Furthermore, in step (7), since plasticity and damage are coupled to each other, the plasticity scalar multiplier and the damage increment multiplier can be solved by the continuity conditions of the simultaneous yield criterion and the damage criterion:

[0055]

[0056] Where f represents the yield function, σ represents the stress tensor, κ represents the plastic hardening / softening function, and γ p represents the generalized plastic shear strain, h represents the damage evolution criterion, ε p represents the plastic strain tensor, and d represents the damage variable;

[0057] Substitute the incremental expression of plastic strain in step (4) and the incremental form of the material constitutive relation in step (6) into equation (12) and simplify it to:

[0058]

[0059] Where: H is the plastic hardening modulus, which is expressed as

[0060]

[0061] where q represents the deviatoric stress, represents the elastic stiffness tensor of the undamaged original material, ε e represents the elastic strain, represents the elastic stiffness tensor affected by damage evolution, and g represents the plastic potential function.

[0062] Furthermore, the plastic strain increment and damage increment are obtained as follows:

[0063] Substituting the plastic scalar multiplier into the plastic strain increment expression yields the plastic strain increment:

[0064]

[0065] Among them, ε and ε p are the total strain and plastic strain respectively, f is the yield criterion, represents the elastic stiffness tensor affected by damage evolution, d is the damage variable, H is the plastic hardening modulus, and g is the plastic potential function;

[0066] Substituting the damage increment multiplier into the damage criterion function yields the damage increment:

[0067]

[0068] Where d is the damage variable, λ p is the plastic scalar multiplier, λ d is the damage increment multiplier, h is the damage criterion, ε p is the plastic strain, and g is the plastic potential function.

[0069] Furthermore, step (9) is specifically as follows:

[0070] Substituting the plastic strain increment and damage increment obtained in step (8) into the incremental form of the material constitutive relation in step (6), the following is simplified:

[0071]

[0072] Where σ is the stress tensor, ε is the strain tensor, is the elastoplastic damage stiffness tensor:

[0073]

[0074] Where f is the yield function, g is the plastic potential function, h is the damage evolution criterion, represents the elastic stiffness tensor affected by damage evolution, represents the elastic stiffness tensor of the undamaged original material, σ is the stress tensor, and ε e is the elastic strain, ε p is the plastic strain, and d is the damage variable.

[0075] Compared with the existing technology, the beneficial effects of the present invention are:

[0076] 1) A yield function is established based on the strength criterion, maintaining consistency with the strength criterion at peak stress, ensuring accurate strength prediction and efficient identification of strength parameters by the constitutive model. A continuous smooth function of the generalized plastic shear strain is introduced as the hardening / softening parameter, which not only ensures a nonlinear description of the stress-strain relationship but also provides the model with first-order continuous partial derivatives, facilitating the numerical implementation of the constitutive model.

[0077] 2) The modeling method proposed in this paper comprehensively considers the nonlinear yield criterion, plastic hardening / softening criterion, non-associated flow law, and damage evolution law of rock materials, and can well reflect the elastic-plastic-damage coupled mechanical behavior of rock materials. It has clear concepts, clear mechanisms, and a rigorous derivation process, ensuring the uniqueness and accuracy of the constitutive model.

[0078] 3) The modeling method proposed in the present invention is widely applicable to mainstream strength criteria, such as the Mises criterion, Tresca criterion, Mohr-Coulomb criterion, Drucker-Prager criterion, Hoek-Brown criterion and generalized nonlinear strength criterion.

[0079] 4) The modeling method proposed in the present invention has a small number of parameters, all of which can be obtained through indoor test results; and can be compiled into an embedded program of finite element software. Therefore, it is believed that this method is simple and convenient, has high accuracy, and can be easily promoted and applied to actual geotechnical engineering calculations and analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 This is a flow chart of the constitutive modeling method for elastic-plastic damage of rock materials based on the full stress-strain relationship of the present invention;

[0081] Figure 2 This is the generalized Ottosen peak intensity envelope diagram of Kareliya granite in Inventive Example 1;

[0082] Figure 3 is a graph of the hardening / softening function κ defined by the generalized plastic shear strain in this embodiment;

[0083] Figure 4 1 is a comparison chart of the model prediction and the experimental results in this embodiment;

[0084] Figure 5 This is a comparison chart of the Donbass sandstone model prediction and test results of Inventive Example 2;

[0085] Figure 6 This is a comparison chart of the Kuru granite model prediction and test results of Inventive Example 3. DETAILED DESCRIPTION

[0086] The following further illustrates a modeling method of an elastic-plastic damage constitutive model capable of reflecting the full stress-strain curve characteristics of rock materials provided by the present invention in conjunction with the accompanying drawings and specific embodiments.

[0087] Example 1

[0088] The modeling idea of ​​the constitutive modeling method of the full stress-strain relationship of rock materials in this invention is to take the yield function established based on the generalized Ottosen strength criterion as an example and introduce the generalized plastic shear strain γ p A unified hardening or softening function is defined, and on this basis, a linear Drucker-Prager type plastic potential function and a damage criterion coupled by plastic strain and damage variables are proposed, thereby establishing a complete elastic-plastic damage constitutive model that can reflect the full stress-strain curve characteristics of rock materials. This modeling method is used to simulate conventional triaxial compression tests on Kareliya granite, verifying the rationality and effectiveness of the modeling method. The specific modeling method includes the following steps:

[0089] (1) In the framework of continuum mechanics and thermodynamics, based on the small deformation hypothesis and classical elastic-plastic theory, the macroscopic stress-strain relationship of rock materials is derived and determined. Considering that the strain of rock materials has an irreversible part, in this case, based on classical elastic-plastic theory and the small deformation hypothesis, the material strain and strain increment can be decomposed into two parts: elastic and plastic:

[0090] ε=ε e +ε p ,dε=dε e +dε p (1)

[0091] Among them, ε, ε e and ε p Represent the total strain, elastic strain and plastic strain respectively, dε, dε e and dε p represent the total strain increment, elastic strain increment, and plastic strain increment, respectively.

[0092] The scalar damage variable d is introduced to consider the effect of damage on the elastic modulus of the material. Under the assumption of isotropic damage, the elastic stiffness tensor of rock materials affected by damage evolution can be expressed as:

[0093]

[0094] in, is the elastic stiffness tensor of the undamaged original material, and its specific expression can be described by the elastic bulk and shear moduli k0 and μ0 of the original material:

[0095]

[0096] By defining the second-order unit tensor δ, the fourth-order tensor partial operator Sum ball operator Can be expressed as

[0097] Therefore, the macroscopic stress-strain relationship of rock materials can be expressed as follows:

[0098]

[0099] (2) The strength criterion (Ottosen strength) is generally a function of the stress state. In the stress space, a strength criterion that can describe the peak strength envelope of the material is selected. The generalized Ottosen strength criterion takes into account the influence of the stress Rhodes angle on the yield surface and can more accurately describe the nonlinear failure characteristics of rock materials. Therefore, the yield function is constructed based on the generalized Ottosen strength criterion:

[0100]

[0101] Where p is the mean stress, q is the deviatoric stress, and J2 represents the second invariant of the deviatoric stress tensor s. a, b, and c0 are parameters related to the strength of rock materials and can be determined by fitting the triaxial compression strength envelope. κ is the introduced hardening / softening function. θ is the stress Rhodes angle, and r(θ) is the shape function describing the yield surface, which uses the Shi-Yang form here:

[0102]

[0103] Where k is the internal friction angle Parameters:

[0104] (3) According to the requirements of hardening / softening function, such as Figure 3 As shown, the following hardening / softening function based on generalized plastic shear strain is introduced:

[0105]

[0106] Where κ0 defines the initial plastic yield threshold; the model parameter N controls the evolution rate of κ in the hardening / softening mechanism; γ p is the generalized plastic shear strain:

[0107]

[0108] is the generalized plastic shear strain at the peak stress, which is called the generalized shear strain critical value.

[0109] (4) Based on the material properties and flow laws, a plastic potential function that can accurately describe the volume compression / expansion phenomenon of rock materials under triaxial compression conditions is selected to determine the direction of plastic strain flow, and the plastic strain increment is calculated using the plastic strain increment expression in combination with the plastic potential function. If the associated flow law is used for materials such as rocks, the shear dilatancy of the material will be overestimated. Therefore, in order to avoid this problem, the non-associated flow law is often used in the constitutive modeling of rock materials, that is, a plastic potential function g that is different from the yield function f is introduced. For simplicity, the widely used linear Drucker-Prager plastic potential function is used in the present invention:

[0110] g=q-ηp (9)

[0111] Where η is a parameter related to the rock expansion angle:

[0112] At this time, the evolution of plastic strain produces the following flow law, that is, the expression of plastic strain increment:

[0113]

[0114] Where g is the plastic potential function, λ p is a non-negative plastic scalar multiplier, ε p is the plastic strain.

[0115] (5) Plastic deformation and damage occur simultaneously in rock materials, and their evolution laws are coupled with each other. From this perspective, a damage criterion that considers the coupling of plastic deformation and damage is established.

[0116] Under compressive stress conditions, most microcracks are in a closed state, and damage evolution is mainly caused by friction sliding along the microcracks, which means that damage evolution is essentially related to plastic shear strain. Based on the Mazars damage criterion, the following damage evolution criterion considering plastic volume strain is constructed:

[0117]

[0118] Where: h is the damage evolution criterion, d c is the asymptotic damage value, b′ controls the speed of damage evolution, is the plastic volume strain.

[0119] (6) Differentiating the macroscopic stress-strain relationship (4) yields the incremental form of the material constitutive relationship:

[0120]

[0121] Where σ represents the stress tensor, ε, ε e and ε p denote the total strain, elastic strain and plastic strain respectively, represents the elastic stiffness tensor affected by damage evolution, represents the elastic stiffness tensor of the undamaged original material, and d represents the damage variable.

[0122] (7) Since plasticity and damage are coupled to each other, the plasticity scalar multiplier and the damage increment multiplier can be solved by combining the continuity conditions of the yield criterion and the damage criterion:

[0123]

[0124] Where f represents the yield function, σ represents the stress tensor, κ represents the plastic hardening / softening function, and γ p represents the generalized plastic shear strain, h represents the damage criterion, ε p represents the plastic strain tensor, and d represents the damage variable. Substituting equations (10) and (12) into (13), we can obtain:

[0125]

[0126] Where: H is the plastic hardening modulus, which is expressed as

[0127]

[0128] (8) Substituting the plastic scalar multiplier of Equation (14) into Equation (10) yields the plastic strain increment:

[0129]

[0130] Substituting the damage increment multiplier of formula (14) into formula (11) yields the damage increment:

[0131]

[0132] (9) Substituting formulas (16) and (17) into formula (12), we obtain:

[0133]

[0134] in, is the elastoplastic damage stiffness tensor:

[0135]

[0136] The elastic-plastic damage constitutive model (Formulas (1) to (19)) established based on the generalized Ottosen strength criterion contains a total of 12 parameters, all of which can be determined through indoor mechanical tests. The parameter determination method is as follows:

[0137] 1) Elastic parameters E and v can be determined based on the slope of the elastic section of the stress-strain line in indoor uniaxial compression or triaxial compression tests;

[0138] 2) Strength parameters a, b, c0 and The peak strength of conventional triaxial compression tests under different confining pressures is determined by fitting the generalized Ottosen strength criterion, as follows: Figure 2 As shown;

[0139] 3) Plastic parameters κ0, N, and η:κ0 can be obtained by fitting the yield stress envelope of the initial yield surface; the parameter N controls the strain hardening and softening rates; The parameters correspond to the generalized plastic shear strain at the peak stress; η controls the volume expansion and can be determined based on the volume compression-expansion transition point;

[0140] 4) Damage parameter d c and b′: damage threshold d c is the limit of the damage variable. If there is a loading and unloading cycle in the residual stage, we can get d c =1-E c E, where E c is the elastic modulus of the residual stage; the parameter b′ controls the damage evolution rate.

[0141] According to the parameter calibration method introduced above and combined with the indoor test data of Kareliya granite, the numerical simulation parameters shown in Table 1 were obtained, and then the conventional triaxial compression test of Kareliya granite under different confining pressure conditions was simulated. Figure 4 The simulation results of the stress-strain relationship for different confining pressure values ​​are shown in Figure 2 (Note: The experimental data are from Unteregger D, Fuchs B, Hofstetter G (2015) A damage plasticity model for different types of intact rock. International Journal of Rock Mechanics and Mining Sciences 80: 402-411). There is good consistency between the simulated values ​​and the experimental values, indicating that the model can well reflect the macroscopic mechanical behavior of Kareliya granite, including nonlinear stress-strain relationships (pre-peak hardening and post-peak softening effects), hydrostatic pressure effects, volume expansion effects, etc. In addition, the model has a good prediction of the peak strength and residual strength of the rock. The above results show that the modeling method of the elastic-plastic damage constitutive model provided by the present invention can reflect the characteristics of the full stress-strain curve of rock materials and has a wide range of applicability.

[0142] Table 1 Mechanical parameters of Kareliya granite model

[0143]

[0144] Example 2

[0145] In this example, the experimental data used Donbass sandstone from D. Unteregger's paper (Unteregger D, Fuchs B, Hofstetter G (2015) A damage plasticity model for different types of intact rock. International Journal of Rock Mechanics and Mining Sciences 80:402-411) was used to verify the model. Elastic parameters (E, v): The elastic modulus E and Poisson's ratio v of the rock can be obtained from the elastic segment of the stress-strain curve of the uniaxial or triaxial compression test; the strength parameter According to the peak strength of the conventional triaxial compression test data of rock under different confining pressures, the failure envelope is drawn in the pq plane, and the failure envelope is fitted based on the generalized Ottosen strength criterion, and then the parameters a, b, c0 and Plasticity parameters Based on the yield strength of rock triaxial compression test data under different confining pressures, the initial yield envelope is drawn in the pq plane. κ0 is obtained by fitting the envelope through the initial yield surface. The parameter N is determined according to the strain hardening and softening rates in the test data. The parameters can be obtained from the generalized plastic shear strain at the peak stress. The parameter η that controls the volume expansion is obtained according to the volume compression-expansion transition point. The damage parameter (d c , b′): Damage threshold d c is the limit of the damage variable. If there is a loading and unloading cycle in the residual stage, we can get d c =1-E c E, where E c is the elastic modulus in the residual stage, and parameter b′ controls the damage evolution rate. The above calibration method is used to fit the Donbass sandstone test data to obtain the model parameters of the embodiment, as shown in Table 2.

[0146] Table 2 Mechanical parameters of Donbass sandstone model

[0147]

[0148] Based on the elastic-plastic damage constitutive model and the above model parameters in the above embodiment, the stress-strain relationship of rock under different confining pressures can be simulated, such as Figure 5As shown, the simulated and experimental values ​​show good consistency, demonstrating that the model can well reflect the macroscopic mechanical behavior of Donbass sandstone, including nonlinear stress-strain relationships (pre-peak hardening and post-peak softening effects), hydrostatic pressure effects, and volume expansion effects. Furthermore, the model provides excellent predictions of both the peak and residual strengths of the rock. These results demonstrate that the elastic-plastic damage constitutive modeling method provided by the present invention can reflect the characteristics of the full stress-strain curve of rock materials and has broad applicability.

[0149] Example 3

[0150] In this example, the experimental data used in the Kuru granite paper by Tkalich D (Tkalich D, Fourmeau M, Kane A, et al. Experimental and numerical study of Kuru granite under confined compression and indentation [J]. International Journal of Rock Mechanics & Mining Sciences, 2016, 87: 55–68) was used to verify the model. Elastic parameters (E, v): The elastic modulus E and Poisson's ratio v of the rock can be obtained from the elastic section of the stress-strain curve of the uniaxial or triaxial compression test; the strength parameter According to the peak strength of the conventional triaxial compression test data of rock under different confining pressures, the failure envelope is drawn in the pq plane, and the failure envelope is fitted based on the generalized Ottosen strength criterion, and then the parameters a, b, c0 and Plasticity parameters κ0 can be obtained by fitting the yield stress envelope of the initial yield surface. The parameter N controls the strain hardening and softening rates. The parameters conform to the generalized plastic shear strain at the peak stress, η controls the volume expansion and can be determined according to the volume compression-expansion transformation point; the damage parameter (d c , b′): Damage threshold d c is the limit of the damage variable. If there is a loading and unloading cycle in the residual stage, we can get d c =1-E c E, where E c is the elastic modulus of the residual stage, and parameter b′ controls the damage evolution rate. The above calibration method is used to fit the Kuru granite test data to obtain the model parameters of the embodiment, as shown in Table 3.

[0151] Table 3 Mechanical parameters of Kuru granite model

[0152]

[0153]

[0154] Based on the elastic-plastic damage constitutive model and the above model parameters in the above embodiment, the stress-strain relationship of rock under different confining pressures can be simulated, such as Figure 6 As shown, the simulated and experimental values ​​show good consistency, indicating that the model can well reflect the macroscopic mechanical behavior of Kuru granite, including the nonlinear stress-strain relationship (pre-peak hardening and post-peak softening effects), hydrostatic pressure effects, and volume expansion effects. Furthermore, the model provides good predictions of both the peak strength and residual strength of the rock. These results demonstrate that the modeling method for the elastic-plastic damage constitutive model provided by the present invention can reflect the characteristics of the full stress-strain curve of rock materials and has broad applicability.

[0155] The present invention provides a rational and effective method for constructing an elastic-plastic damage constitutive model for rock materials based on a strength criterion. The models established by this method can effectively describe the stress-strain curves of rock materials and have strong applicability. These models can have the following characteristics: 1) accurately reflect the strength properties of rock materials; 2) accurately reflect the nonlinear deformation characteristics of rock materials; 3) have rigorous mathematical derivation and clear physical mechanisms; 4) have a small number of model parameters that are easy to identify; and 5) have at least first-order continuity, facilitating numerical implementation.

[0156] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to the specific embodiments described. Obviously, many modifications and variations are possible based on the content of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. The constitutive modeling method of elastic-plastic damage of rock materials based on the full stress-strain relationship is characterized by: The steps include: (1) Determine the macroscopic stress-strain relationship of rock materials within the framework of continuum mechanics and thermodynamics; (2) Select a strength criterion that can describe the peak strength envelope of the material in stress space; under isothermal static load conditions, the strength criterion is expressed as: f s =f s (s ij )=0 (1) Among them, f s represents the strength criterion, σ ij represents the stress component; The strength criterion is described in terms of principal stresses, stress invariants and mean stress p-deviatoric stress q: where σ1, σ2, and σ3 represent the first, second, and third principal stresses, respectively; I1 = trσ represents the first invariant of the stress tensor σ, J2 = (s:s) / 2 represents the second invariant of the deviatoric stress tensor s, and θ σ represents the stress Lode angle; (3) A unified continuous and smooth hardening / softening function with generalized shear strain as a variable is introduced into the strength criterion to construct a yield function that can describe the expansion and contraction of the yield surface; (4) Select a plastic potential function that can describe the volume compression / expansion phenomenon of rock materials under triaxial compression conditions, determine the direction of plastic strain flow, and calculate the plastic strain increment using the plastic strain increment expression in combination with the plastic potential function; (5) Establish a damage criterion that considers the coupling of plastic deformation and damage; (6) Differentiating the macroscopic stress-strain relationship of the rock material in step (1) to obtain an incremental form of the material constitutive relationship; (7) Combining the continuity conditions of plasticity and damage, the continuity conditions of the yield criterion and the damage criterion are solved simultaneously to obtain the plasticity scalar multiplier and the damage increment multiplier; (8) Obtain the plastic strain increment according to the plastic scalar multiplier and the plastic strain increment expression in step (4), and obtain the damage increment according to the damage increment multiplier and the damage criterion; (9) Substituting the plastic strain increment and damage increment obtained in step (8) into the relationship between the stress increment and the strain increment in step (6) to obtain the elastic-plastic damage stiffness tensor and the elastic-plastic damage constitutive model reflecting the full stress-strain curve characteristics of the rock material.

2. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: In step (1), the material strain and strain increment are decomposed into elastic and plastic parts: εM e +ε p , dεdε e +dε p (3) Among them, ε, ε e and ε p Represent the total strain, elastic strain and plastic strain respectively, dε, dε e and dε p represent the total strain increment, elastic strain increment, and plastic strain increment, respectively; The scalar damage variable d is introduced to represent the effect of damage on the elastic modulus of the material. Under the assumption of isotropic damage, the elastic stiffness tensor of rock materials affected by damage evolution is expressed as: C(d)=(1-d)C0 (4) where C0 is the elastic stiffness tensor of the undamaged original material, and its specific expression is described by the elastic bulk and shear moduli k0 and μ0 of the original material: By defining the second-order unit tensor δ, the fourth-order tensor partial operator K and the spherical operator J are expressed as The macroscopic stress-strain relationship of rock materials is expressed as follows: σ=C(d),(ε-ε p ) (6) Among them, σ, ε, ε p represent the stress tensor, total strain tensor, and plastic strain tensor, respectively; C(d) represents the elastic stiffness tensor affected by damage evolution, and d represents the damage variable.

3. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: The selection of hardening / softening function must meet the following three requirements: 1) The hardening or softening function introduced in the yield function has the characteristics of monotonically increasing before the peak and monotonically decreasing after the peak; 2) The hardening or softening function has at least a first-order continuous derivative; 3) The yield function and strength criterion corresponding to the peak stress should have the same form, that is, κ = 1 at the peak stress; Combined with the strength criterion, under isothermal static loading conditions, the yield function is expressed as: f=f(σ ij ,k)=0 (7) Where f represents the yield function, σ ij represents the stress component, κ represents the plastic hardening / softening function; The yield function is described in terms of principal stresses, stress invariants, and mean stress p-deviatoric stress q: where σ1, σ2, and σ3 represent the first, second, and third principal stresses, respectively; I1 = trσ represents the first invariant of the stress tensor σ, J2 = (s:s) / 2 represents the second invariant of the deviatoric stress tensor s, and θ σ represents the stress Lode angle; According to the stress-strain relationship (4), the expression of the yield function in the strain space must include the internal variables κ and ε. p and d.

4. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: In step (4), the plastic potential function g is used to determine the flow direction of the plastic strain. For rock materials, in order to avoid overestimating the shear expansion of the material, the non-associated flow law is selected, that is, g≠f. At this time, the evolution of the plastic strain produces the corresponding flow law, and the plastic strain increment expression is: Among them, dε p is the plastic strain increment, g is the plastic potential function, f is the yield function, λ p is a non-negative plastic scalar multiplier, and σ represents the stress tensor.

5. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein the damage criterion is the plastic strain ε p And the function of the damage variable d: h=h(ε p ,d)≤0 (10) Where h is the damage evolution criterion, ε p is the plastic strain, and d is the damage variable.

6. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: In step (6), the incremental form of the material constitutive relation is: dσC(d):(d-d p )-C0:ε e dd (11) Where σ represents the stress tensor, ε, ε e and ε p denote the total strain, elastic strain and plastic strain respectively, C(d) denotes the elastic stiffness tensor affected by damage evolution, C0 denotes the elastic stiffness tensor of the damageless material, and d denotes the damage variable.

7. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: In step (7), since plasticity and damage are coupled to each other, the plasticity scalar multiplier and the damage increment multiplier can be solved by the continuity conditions of the simultaneous yield criterion and the damage criterion: Where f represents the yield function, σ represents the stress tensor, κ represents the plastic hardening / softening function, and γ p represents the generalized plastic shear strain, h represents the damage evolution criterion, ε p represents the plastic strain tensor, and d represents the damage variable; Substitute the incremental expression of plastic strain in step (4) and the incremental form of the material constitutive relation in step (6) into equation (12) and simplify it to: Where: H is the plastic hardening modulus, which is expressed as where q represents the deviatoric stress, C0 represents the elastic stiffness tensor of the undamaged original material, and ε e represents the elastic strain, C(d) represents the elastic stiffness tensor affected by damage evolution, and g represents the plastic potential function.

8. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: Get the damage increment as follows: Substituting the plastic scalar multiplier into the plastic strain increment expression yields the plastic strain increment: Among them, ε and ε p are the total strain and plastic strain, respectively, f is the yield criterion, C(d) represents the elastic stiffness tensor affected by damage evolution, d is the damage variable, H is the plastic hardening modulus, and g is the plastic potential function; Substituting the damage increment multiplier into the damage criterion function yields the damage increment: Where d is the damage variable, λ p is the plastic scalar multiplier, λ d is the damage increment multiplier, h is the damage criterion, ε p is the plastic strain, and g is the plastic potential function.

9. The constitutive modeling method for elastic-plastic damage of rock materials based on full stress-strain relationship according to claim 1, wherein: Step (9) is as follows: Substituting the plastic strain increment and damage increment obtained in step (8) into the incremental form of the material constitutive relation in step (6), the following is simplified: dσ=C epd :dε (17) Among them, σ is the stress tensor, ε is the strain tensor, C epd is the elastoplastic damage stiffness tensor: Where f is the yield function, g is the plastic potential function, h is the damage evolution criterion, represents the elastic stiffness tensor affected by damage evolution, C0 represents the elastic stiffness tensor of the original undamaged material, σ is the stress tensor, and ε e is the elastic strain, ε p is the plastic strain, and d is the damage variable.

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