A Method for Constructing Constitutive Models of Anisotropic Sand Based on Stress Testing
By introducing stress testing and structural tensor, an anisotropic constitutive model of sand was constructed, which solved the deviation problem of existing models when simulating anisotropic sand, and achieved accurate simulation under complex conditions, making it suitable for geotechnical engineering analysis.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- ZHEJIANG UNIV
- Filing Date
- 2025-05-13
- Publication Date
- 2026-04-24
AI Technical Summary
Existing constitutive models neglect the evolution of structures when simulating anisotropic sand, resulting in significant discrepancies between simulation results and actual conditions under complex loading conditions.
Through stress testing, we introduce fabric tensor and fabric evolution rules to quantify the anisotropy of sand and construct an anisotropic constitutive model of sand, including stress testing, introduction of fabric anisotropy variable A, fabric evolution rules, and model derivation based on critical stress state.
It accurately simulates the behavior of anisotropic sand under different initial states and loading conditions, improving the applicability and accuracy of the model, and is suitable for the mechanical analysis of sand in geotechnical engineering.
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Abstract
Description
Technical Field
[0001] This invention relates to the field of geotechnical engineering technology, and more specifically, to a method for constructing a constitutive model of anisotropic sand based on stress testing. Background Technology
[0002] Natural sand typically exhibits a significant anisotropic structure, which has a crucial impact on its mechanical behavior. Traditional constitutive models often neglect structural anisotropy, leading to substantial discrepancies between simulation results and actual conditions. Existing anisotropic constitutive models, when considering structural anisotropy, often overlook the evolution of the structure, resulting in limitations in simulating sand behavior under complex loading conditions. Therefore, this invention proposes an anisotropic sand constitutive model based on stress testing, which can accurately simulate the behavior of anisotropic sand under different initial states and loading conditions. Summary of the Invention
[0003] The purpose of this invention is to address the shortcomings of existing technologies by proposing a method for constructing anisotropic constitutive models of sand based on stress testing.
[0004] Firstly, a method for constructing a constitutive model of anisotropic sand based on stress testing is provided, including:
[0005] Step 1, Stress test: Through the stress test simulated by discrete element method, stress increments in different directions are applied to the sand sample, and the corresponding strain increments are measured to form the strain response envelope;
[0006] Step 2, Introduction of structural anisotropy variable A: The anisotropy of sand is quantified by structural tensor, and structural anisotropy variable A is introduced to characterize the relative relationship between the structure and the loading direction.
[0007] Step 3, Configuration Evolution Rules: The configuration evolution rules are used to describe the evolution process of the configuration tensor;
[0008] Step 4: Derive the constitutive model of anisotropic sand based on the critical stress state;
[0009] Step 5: Model Validation: The effectiveness of the anisotropic sand constitutive model is verified by comparing the experimental results and the model prediction results.
[0010] Preferably, in step 1, samples with different porosity ratios are prepared by adjusting the interparticle friction coefficient μ, and the samples are consolidated to different stress states.
[0011] Preferably, step 2 includes:
[0012] Step 2.1: For a representative three-dimensional volume element, the configuration tensor is calculated using the following formula:
[0013]
[0014] Where N is the normalization coefficient, which is usually taken as N. c N c It is the total number of contacts within the RVE. It is the component of the unit vector i in the direction of the k-th contact direction. It is the component of the unit vector j in the direction of the k-th contact direction;
[0015] Step 2.2: Introduce the structural anisotropy variable A, specifically expressed as:
[0016]
[0017] in, It is the deviatoric stress unit tensor specifying the loading direction. It is the unit direction tensor of structural anisotropy. It is a quantity that measures the relative orientation of the structural direction and the loading direction.
[0018] Preferably, in step 3, the configuration evolution rule is expressed as:
[0019]
[0020] Where μ is the model parameter controlling the evolution rate, and L is the loading factor.
[0021] Preferably, in step 4, the constitutive model of the anisotropic sand includes: the elastic relationship of the anisotropic sand, the dilatation coefficient of the anisotropic sand, the flow direction of the anisotropic sand, the hardening law of the anisotropic sand, and the plastic modulus of the anisotropic sand.
[0022] Preferably, in step 5, the model verification includes: anisotropic elastic relationship prediction verification, anisotropic dilatation coefficient prediction verification, anisotropic plastic flow direction prediction verification, anisotropic yield surface prediction verification, anisotropic plastic model prediction verification, and verification of the influence of structural anisotropy on stiffness, dilatation, and non-coaxial response.
[0023] Secondly, a system for constructing anisotropic constitutive models of sand based on stress testing is provided, for performing any of the methods described in the first aspect, including:
[0024] The test module is used to apply stress increments in different directions to sand samples through stress testing simulated by discrete element method, measure the corresponding strain increments, and form the strain response envelope.
[0025] A module is introduced to quantify the anisotropy of sand through the fabric tensor, and a fabric anisotropy variable A is introduced to characterize the relative relationship between the structure and the loading direction.
[0026] The description module is used to describe the evolution process of the configuration tensor using configuration evolution rules;
[0027] The derivation module is used to derive the constitutive model of anisotropic sand based on the critical stress state;
[0028] The verification module is used to verify the effectiveness of the anisotropic sand constitutive model by comparing experimental results with model prediction results.
[0029] Thirdly, a computer storage medium is provided, wherein a computer program is stored therein; when the computer program is run on a computer, the computer causes the computer to perform any of the methods described in the first aspect.
[0030] Fourthly, an electronic device is provided, comprising:
[0031] Memory, used to store computer programs;
[0032] A processor for executing the computer program to implement the method as described in any of the first aspects.
[0033] The beneficial effects of this invention are:
[0034] 1. This invention can accurately obtain the incremental stress-strain relationship of anisotropic sand through stress testing, providing reliable experimental data for constructing constitutive models.
[0035] 2. This invention introduces structural anisotropy variables (FAV) and structural evolution rules, which can effectively reflect the structural anisotropy of sand and its evolution process, thus improving the applicability of the model.
[0036] 3. The constitutive model proposed in this invention can accurately simulate the behavior of anisotropic sand under different initial states and loading conditions, and is applicable to the mechanical analysis of sand in geotechnical engineering. Attached Figure Description
[0037] Figure 1 An exploded view of the anisotropic plastic flow direction provided by the present invention;
[0038] Figure 2 The present invention provides α at different deposition angles F Elastic response envelope diagram of anisotropic specimen;
[0039] Figure 3 The prediction diagram of the anisotropic dilatation coefficient model provided by this invention;
[0040] Figure 4 This invention provides a prediction diagram of the anisotropic plastic flow direction.
[0041] Figure 5 An anisotropic yield surface prediction diagram provided by the present invention;
[0042] Figure 6 This invention provides a prediction diagram of the anisotropic plastic flow direction.
[0043] Figure 7 The figure shows a comparison between the test results and model predictions of the undrained triaxial test at b=0.5 provided by the present invention. Detailed Implementation
[0044] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.
[0045] Example 1:
[0046] Embodiment 1 of this application provides a method for constructing a constitutive model of anisotropic sand based on stress testing, which can accurately simulate the behavior of anisotropic sand under different initial states and loading conditions. Specifically, the method includes:
[0047] Step 1, Stress test: Through the stress test simulated by discrete element method (DEM), stress increments in different directions are applied to the sand sample, and the corresponding strain increments are measured to form the strain response envelope.
[0048] In step 1, PFC is used. 3D Discrete element method simulation was performed. By adjusting the interparticle friction coefficient μ, samples with different porosity ratios were prepared and the samples were consolidated to different stress states, including different confining pressures p, stress ratios η = q / p, and intermediate principal stress coefficients b = (σ2-σ3) / (σ1-σ3) (p and q represent the mean stress and shear stress, respectively; σ1, σ2, and σ3 represent the major principal stress, intermediate principal stress, and minor principal stress, respectively). Samples with different stress states and structural anisotropy were obtained through the above method.
[0049] Step 2, Introduction of Fabric Anisotropy Variable (FAV) A: The anisotropy of sand is quantified by fabric tensor, and fabric anisotropy variable A is introduced to characterize the relative relationship between structure and loading direction.
[0050] Step 2 includes:
[0051] Step 2.1: For a three-dimensional representative volume element (RVE), the configuration tensor is calculated using the following formula:
[0052]
[0053] Where N is the normalization coefficient, which is usually taken as N. c N c It is the total number of contacts within the RVE. It is the component of the unit vector i in the direction of the k-th contact direction. It is the component of the unit vector j in the direction of the k-th contact direction.
[0054] The deviatoric stress configuration tensor is derived from equation (1):
[0055]
[0056] Among them, F and F ij The norm and unit direction. F' c (θ σ ) represents F before normalization ij The critical norm ensures that regardless of the stress Lode angle θ σ How to change the critical value of F so that it is always 1, δ ij The Kronecker delta symbol is used when i = j. ij =1, when i≠j, δ ij =0.
[0057] For transversely isotropic soil F with deposition direction x1 axis ij It can be represented as:
[0058]
[0059] Where F0 represents the initial structure norm.
[0060] Step 2.2: To quantify the relative relationship between the structural orientation and the loading orientation, an anisotropy variable A is introduced, specifically expressed as:
[0061]
[0062] in, It is the deviatoric stress unit tensor specifying the loading direction. It is the unit direction tensor of structural anisotropy. It is a quantity that measures the relative orientation of the structural direction and the loading direction.
[0063] Step 3, Structure Evolution Rules: The structure evolution rules are used to describe the evolution process of the structure tensor.
[0064] In step 3, the configuration evolution rule is expressed as:
[0065]
[0066] Where μ is the model parameter controlling the evolution rate, and L is the loading factor.
[0067] Step 4: Derive the constitutive model of anisotropic sand based on the critical stress state.
[0068] Step 5: Model Validation: The effectiveness of the anisotropic sand constitutive model is verified by comparing the experimental results and the model prediction results.
[0069] Example 2:
[0070] Based on Example 1, Example 2 of this application provides a more specific method for constructing anisotropic sand constitutive models based on stress testing, including:
[0071] Step 1, Stress test: Through the stress test simulated by discrete element method, stress increments in different directions are applied to the sand sample, and the corresponding strain increments are measured to form the strain response envelope.
[0072] Step 2, Introduction of structural anisotropy variable A: The anisotropy of sand is quantified by structural tensor, and structural anisotropy variable A is introduced to characterize the relative relationship between the structure and the loading direction.
[0073] Step 3, Structure Evolution Rules: The structure evolution rules are used to describe the evolution process of the structure tensor.
[0074] Step 4: Derive the constitutive model of anisotropic sand based on the critical stress state.
[0075] In step 4, the constitutive model of anisotropic sand includes: the elastic relationship of anisotropic sand, the dilatation coefficient of anisotropic sand, the flow direction of anisotropic sand, the hardening law of anisotropic sand, and the plastic modulus of anisotropic sand.
[0076] Specifically, step 4 includes:
[0077] (1) The elastic relationship of anisotropic sand is as follows:
[0078]
[0079]
[0080] in, It is the partial elastic strain increment, δ ij This is the Kronecker delta symbol, which is 1 when i = j and 0 otherwise.
[0081] It is the volumetric elastic strain increment, ds ij dp is the deviatoric stress increment, G is the average stress increment (hydrostatic pressure increment), G is the shear modulus, G0 is the reference shear modulus, e is the void ratio, P is the average effective stress, P0 is the atmospheric pressure, K is the bulk modulus, and ν is Poisson's ratio.
[0082] (2) The expression for the shear dilatation coefficient of anisotropic sand is:
[0083] Introducing the fabric anisotropy variable (FAV) A to reflect the influence of anisotropy on the dilatation behavior of sand, the expression for the dilatation coefficient is obtained as follows:
[0084]
[0085] Where ξ is the dilatation state parameter, e A d0 and m are model parameters, R is the stress ratio, and M is the stress ratio. c ξ is the critical stress ratio, and ψ is the state parameter. ξ is a function of void ratio e, confining pressure p, and fabric anisotropy A, reflecting the density state of sand under the combined influence of void ratio, confining pressure, and fabric anisotropy.
[0086] In formula (7), M = M c g(θ) represents the critical stress ratio, and g(θ) depends on the Lode angle θ. σ Interpolation function:
[0087]
[0088] Where c = M e / M c These are model parameters, M e and M c These are the critical stress ratios under triaxial tension and compression, respectively, θ σ It is Lode angle.
[0089] (3) The derivation of the flow direction of anisotropic sand is as follows:
[0090] Construction tensor F ij and plastic flow direction n ij Decomposed into The proportional and non-proportional parts are shown in equation (8):
[0091]
[0092] Among them, F ij It is a structural tensor, n ij It is the direction of plastic flow. It is an isotropic reference direction. and They are F ijProportional and non-proportional parts, and They are n ij The proportional and non-proportional parts; the proportional part (superscript pr): with Proportional components are obtained through dot product projection (e.g.) The non-proportional part (superscript np): the remaining part, i.e., the original tensor minus the proportional part (e.g., ...). ).
[0093] Figure 1 This is a schematic diagram of the decomposition. Figure 1 In this context, φ represents and The angle between the directions is defined as the first joint invariant:
[0094]
[0095] in, It is the dot product of the two non-proportional parts. is the product of the magnitudes of the two vectors, used for normalization, and k1 is the scaling factor.
[0096] The direction of plastic flow in anisotropic structures is derived using formulas (8a) and (9b):
[0097]
[0098] Will Substituting into equation (10), we get:
[0099]
[0100] Because n ij It is a unit norm tensor, with Substituting into equation (11), we get:
[0101]
[0102] In formulas (9) to (12), k1 to k4 are stress state variables (such as σ). ij F ij Scalar functions (e.g., A, etc.).
[0103] k4 = k np (1-A) / R (13)
[0104] Where, k np A is a positive model parameter, A is the anisotropy, and R is the stress state parameter. Equation (13) shows that as A and R increase, the value of k4 decreases.
[0105] (4) The expression for the yield surface of anisotropic sand is:
[0106] f = RH (14)
[0107] Where R is a stress-related quantity and H is a hardening parameter.
[0108] (5) The expression for the hardening law of anisotropic sand is:
[0109] Hardening Law:
[0110] pm ij dr ij -K p L=0 (15)
[0112] Where, m ij It is a partial unit tensor, representing the yield surface at r. ij The normal direction at the location, dr ij It is the stress increment tensor, P is the proportionality constant, and K is the stress increment tensor. p It is the plastic modulus, and L is the cumulative plastic deformation parameter.
[0113] (6) The expression for the plastic modulus of anisotropic sand is:
[0114]
[0115] Where h1, h2, h3, α, θ, β, ξ, and n are model parameters, A is the anisotropy, e is the porosity, R is the stress ratio, P is the mean effective stress, P0 is the atmospheric pressure, and M is the mean effective stress. c It is the critical stress ratio, and g(θ) is the Lode angle function.
[0116] Step 5: Model Validation: The effectiveness of the anisotropic sand constitutive model is verified by comparing the experimental results and the model prediction results.
[0117] In step 5, the model verification includes: anisotropic elastic relationship prediction verification, anisotropic dilatation coefficient prediction verification, anisotropic plastic flow direction prediction verification, anisotropic yield surface prediction verification, anisotropic plastic model prediction verification, and verification of the influence of structural anisotropy on stiffness, dilatation and non-coaxial response.
[0118] Specifically, step 5 includes:
[0119] (1) Validation of anisotropic elastic relationship prediction
[0120] like Figure 2 The left figure shows α at different deposition angles. F The elastic response envelope of anisotropic specimens, from Figure 2 As shown in the left figure, anisotropy has almost no effect on the elastic response of the soil. Figure 2The right figure shows the elastic envelope predicted by formula (6). Figure 2 The model prediction in the right figure matches the DEM results well, indicating that the elastic relationships in formulas (6a) and (6b) are sufficient to describe the elastic behavior of anisotropic sand.
[0121] (2) Validation of the prediction of anisotropic shear dilatation coefficient
[0122] Figure 3 The values of the dilatation coefficient predicted by formula (7) are shown. Under the same conditions, when the loading direction is consistent with the deposition direction of the soil structure, A increases, which leads to a decrease in D calculated by formula (7), which is in good agreement with the experimental trend. It can be seen that formula (7) effectively captures the dilatation behavior of anisotropic sand under different experimental conditions, verifying the effectiveness of formula (7).
[0123] (3) Validation of the prediction of anisotropic plastic flow direction
[0124] Figure 4 The direction of plastic flow predicted by formula (13) is shown. Figure 4 It can be seen that anisotropic soil follows the associated flow law on the deviated plane, and anisotropy will significantly affect the flow direction of soil on the deviated plane, verifying the effectiveness of formula (13).
[0125] (4) Verification of anisotropic yield surface prediction
[0126] Figure 5 Symmetrical stress testing was conducted on samples with different deposition angles to obtain the relationship between the magnitude of the plastic strain increment and the direction of the stress increment. From Figure 5 As can be seen from this, plastic strain will occur in the sample only when the stress ratio changes, and this characteristic is independent of the sample deposition angle. Therefore, the yield surface in formula (14) is still valid for anisotropic sand.
[0127] (5) Validation of anisotropic plasticity model prediction
[0128] Figure 6 This demonstrates the plasticity model K predicted by formula (16) under the same conditions. p Values are represented by solid lines. The model predictions and experimental results are in good agreement, indicating that formula (16) can reasonably reflect the influence of structural anisotropy on the plastic hardening of sand.
[0129] (6) The influence of structural anisotropy on stiffness, dilatation and non-coaxial response
[0130] Figure 7 The results of the undrained triaxial test at b=0.5 are compared with the model predictions. Different principal stress directions α were considered during the experiment. σThe initial void ratio of the soil sample was e0 = 0.821–0.828. The model accurately predicted the effects of structural anisotropy on stiffness, dilatation, and non-coaxial response.
[0131] It should be noted that the parts in this embodiment that are the same as or similar to those in Embodiment 1 can be referred to each other, and will not be repeated in this application.
[0132] Example 3:
[0133] Based on Example 2, Example 3 of this application provides a system for constructing anisotropic constitutive models of sand based on stress testing, comprising:
[0134] The test module is used to apply stress increments in different directions to sand samples through stress testing simulated by discrete element method, measure the corresponding strain increments, and form the strain response envelope.
[0135] A module is introduced to quantify the anisotropy of sand through the fabric tensor, and a fabric anisotropy variable A is introduced to characterize the relative relationship between the structure and the loading direction.
[0136] The description module is used to describe the evolution process of the configuration tensor using configuration evolution rules;
[0137] The derivation module is used to derive the constitutive model of anisotropic sand based on the critical stress state;
[0138] The verification module is used to verify the effectiveness of the anisotropic sand constitutive model by comparing experimental results with model prediction results.
[0139] Specifically, the system provided in this embodiment is the same as the system provided in embodiment 2. Therefore, the parts in this embodiment that are the same as or similar to those in embodiment 2 can be referred to each other and will not be described again in this application.
Claims
1. A method for constructing an anisotropic constitutive model of sand based on stress testing, characterized in that, include: Step 1, Stress test: Through the stress test simulated by discrete element method, stress increments in different directions are applied to the sand sample, and the corresponding strain increments are measured to form the strain response envelope; Step 2, Introduction of the anisotropic variable A: The anisotropy of sand is quantified by the fabric tensor, and the anisotropic variable A is introduced to characterize the relative relationship between the fabric and the loading direction. Step 2 includes: Step 2.1: For a representative three-dimensional volume element, the configuration tensor is calculated using the following formula: Where N is the normalization coefficient, taken as , It is the total number of contacts within the RVE. It is the component of the unit vector i in the direction of the k-th contact direction. It is the component of the unit vector j in the direction of the k-th contact direction; Step 2.2: Introduce the anisotropic variable A, specifically expressed as: in, It is the deviatoric stress unit tensor specifying the loading direction. It is the unit direction tensor that constitutes anisotropy. It is a quantity that measures the relative orientation of the structural direction and the loading direction; Step 3, Configuration Evolution Rules: The configuration evolution rules are used to describe the evolution process of the configuration tensor; Step 4: Derive the constitutive model of anisotropic sand based on the critical stress state; Step 5: Model Validation: The effectiveness of the anisotropic sand constitutive model is verified by comparing the experimental results and the model prediction results.
2. The method for constructing anisotropic constitutive models of sand based on stress testing according to claim 1, characterized in that, In step 1, the friction coefficient between particles is adjusted. Samples with different porosity ratios were prepared and the samples were consolidated to different stress states.
3. The method for constructing anisotropic constitutive models of sand based on stress testing according to claim 2, characterized in that, In step 3, the configuration evolution rule is expressed as: in, L is the loading factor, which is a model parameter used to control the evolution rate.
4. The method for constructing anisotropic constitutive models of sand based on stress testing according to claim 3, characterized in that, In step 4, the constitutive model of anisotropic sand includes: the elastic relationship of anisotropic sand, the dilatation coefficient of anisotropic sand, the flow direction of anisotropic sand, the hardening law of anisotropic sand, and the plastic modulus of anisotropic sand.
5. The method for constructing anisotropic constitutive models of sand based on stress testing according to claim 4, characterized in that, In step 5, the model verification includes: anisotropic elastic relationship prediction verification, anisotropic dilatation coefficient prediction verification, anisotropic plastic flow direction prediction verification, anisotropic yield surface prediction verification, anisotropic plastic model prediction verification, and verification of the influence of structural anisotropy on stiffness, dilatation and non-coaxial response.
6. A system for constructing anisotropic constitutive models of sand based on stress testing, characterized in that, For performing the method according to any one of claims 1 to 5, comprising: The test module is used to apply stress increments in different directions to sand samples through stress testing simulated by discrete element method, measure the corresponding strain increments, and form the strain response envelope. A module is introduced to quantify the anisotropy of sand through the fabric tensor, and a fabric anisotropy variable is introduced to characterize the relative relationship between the fabric and the loading direction. The description module is used to describe the evolution process of the configuration tensor using configuration evolution rules; The derivation module is used to derive the constitutive model of anisotropic sand based on the critical stress state; The verification module is used to verify the effectiveness of the anisotropic sand constitutive model by comparing experimental results with model prediction results.
7. A computer storage medium, characterized in that, The computer storage medium stores a computer program; when the computer program is run on the computer, it causes the computer to perform the method described in any one of claims 1 to 5.
8. An electronic device, characterized in that, include: Memory, used to store computer programs; A processor for executing the computer program to implement the method as described in any one of claims 1 to 5.