A method for constructing a creep fractional order damage constitutive model

Abstract

This invention discloses a method for constructing a fractional-order creep damage constitutive model, relating to the field of geotechnical engineering technology. The method includes: conducting creep tests on a geogrid under different types of soil constraint stress to obtain creep curves; calculating the creep strain of the geogrid at different time points based on the creep curves and plotting the curve of the geogrid's creep strain versus time; introducing the Khachanov damage variable according to the creep damage law to determine the damage variables during the geogrid creep process; constructing an initial fractional-order creep damage constitutive model based on the damage variables, and determining the parameters of the initial fractional-order creep damage constitutive model through the curve of the geogrid's creep strain versus time, thus obtaining the fractional-order creep damage constitutive model. The fractional-order creep damage constitutive model constructed by this invention can accurately reflect the variation law of the geogrid at different creep stages.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering technology, and in particular to a method for constructing a creep fractional-order damage constitutive model. Background Technology

[0002] Geogrids, as an important geosynthetic material, have been widely used in railway engineering projects such as roadbed reinforcement, slope protection, and tunnel lining and support. Their unique structure effectively enhances the bearing capacity of soil, addressing the problems of large roadbed settlement and deformation in highway and railway engineering. However, in practical engineering, geogrids exhibit creep behavior and stress relaxation phenomena under tension, leading to stress redistribution and affecting the overall stability of reinforced soil structures. Therefore, research on the creep characteristics of geogrids under soil constraints is of great significance to ensure the safety and durability of reinforced soil structures.

[0003] Component-based models are widely used in practical engineering due to the clear physical meaning of their parameters and their ability to intuitively reflect the complex mechanical properties of geogrids. However, traditional component-based models assume that the model parameters are constants, thus only reflecting a single creep stage of the geogrid. Furthermore, traditional component and combination models only consider the deterioration of the geogrid's mechanical properties during the accelerated creep stage, neglecting the influence of external loads applied during the creep process after loading in actual engineering. Therefore, currently used geogrid creep models cannot describe the entire creep process of geogrids including damage. Summary of the Invention

[0004] Therefore, it is necessary to provide a method for constructing a creep fractional-order damage constitutive model to address the aforementioned technical problems.

[0005] The present invention adopts the following technical solution: This invention provides a method for constructing a creep fractional-order damage constitutive model, comprising: Creep tests were conducted on geogrids under different types of soil constraint stress to obtain the creep curves of the geogrids. Based on the creep curve of the geogrid, the creep strain of the geogrid at different time points was calculated, and the curve of the creep strain of the geogrid changing with time was plotted. The curve of the creep strain of the geogrid changing with time under the action of soil constraint stress conforms to the creep damage law. Based on the creep damage law, the Khachanov damage variable is introduced to derive the time when the creep damage of the geogrid reaches the critical failure state, and the damage variable in the creep process of the geogrid is determined based on the time when the critical failure state is reached. Obtain a creep fractional constitutive model consisting of spring elements, soft elements, and nonlinear viscous damper elements; Based on the damage variable and the creep fractional order constitutive model, an initial creep fractional order damage constitutive model is constructed. The parameters of the initial creep fractional order damage constitutive model are determined by the curve of creep strain of geogrid with time, and the creep fractional order damage constitutive model is obtained.

[0006] Optionally, the expression for the damage variable is: ; in, D As a damage variable, t Creep time, t e The time required for creep damage in geogrids to reach the critical failure state. These are parameters related to material properties.

[0007] Optionally, based on the damage variables and the creep fractional-order constitutive model, an initial creep fractional-order damage constitutive model is constructed, specifically including: Based on the stress-strain equivalence principle and damage variables, the effective stress in the creep fractional constitutive model is expressed as: ;in, D As a damage variable, For the stress in the fractional-order constitutive model of creep, The effective stress is the fractional-order constitutive model of creep. Will Substituting into the fractional-order creep constitutive model, we obtain the initial fractional-order creep damage constitutive model.

[0008] Optionally, the creep fractional constitutive model is: ; in, The total strain of the creep fractional-order constitutive model is given by... The effective stress of the creep fractional-order constitutive model is... For order, The viscosity coefficient of the nonlinear damper. A These are parameters related to viscous properties. For fractional calculus operators, t Creep time, E 1 represents the elastic modulus. The viscosity coefficient of the software component; The initial creep fractional-order damage constitutive model is: ; in, D For damage variables.

[0009] Optionally, the parameters of the initial creep fractional-order damage constitutive model are determined by the curve of creep strain of the geogrid versus time, specifically including: The elastic modulus was obtained by fitting the curve of creep strain of geogrid with time using Hooke's law. Multiple sets of strain, time, soil constraint stress and elastic modulus were obtained based on geogrid creep tests, and a data matrix was generated. The data matrix, damage variables, and initial creep fractional-order damage constitutive model are input into MATLAB, and an objective function is defined. The objective function is constructed based on the strain values ​​obtained from the geogrid creep test, the average strain value measured by the creep test, and the strain values ​​predicted by the initial creep fractional-order damage constitutive model. Multiple parameters of the initial creep fractional-order damage constitutive model are initialized, and the initial parameters of the BFGS algorithm are set; the initial parameters include Armijo search parameters, Armijo condition parameters, and convergence tolerance. Calculate the gradient vector of the initial values ​​of the parameters of the initial creep fractional order damage constitutive model, and check the convergence of the calculated gradient vector; After performing a convergence check, the BFGS algorithm is used for multiple iterations until the objective function value is 1, thus obtaining the parameters of the initial creep fractional-order damage constitutive model. The parameters of the initial creep fractional-order damage constitutive model include... ;in, The viscosity coefficient of the software component. The viscosity coefficient of the nonlinear damper. For order, These are parameters related to the material properties of geogrids.

[0010] Optionally, the objective function is: ; in, The objective function value, For the first i The strain values ​​obtained from the second geogrid creep test. The strain is the average value determined by the creep test. The strain value is the strain value predicted by the initial creep fractional-order damage constitutive model. n This represents the total number of geogrid creep tests.

[0011] Optionally, the BFGS algorithm is used for multiple iterations until the objective function value is 1, thereby obtaining the parameters of the initial creep fractional-order damage constitutive model, specifically including: In each iteration of the solution process, the gradient vector of the parameters in the current iteration is calculated, and it is checked whether the norm of the gradient vector is less than the convergence tolerance. If the norm of the gradient vector is less than the convergence tolerance, then stop the iteration; If the norm of the gradient vector is greater than or equal to the convergence tolerance, the search direction for parameter updates is determined based on the product of the inverse of the approximate Hessian matrix of the current round and the gradient vector. The step size for parameter updates is determined by Armijo line search; Based on the updated step size and search direction, update the parameters and calculate the objective function value corresponding to the updated parameters; By utilizing the parameter difference and gradient difference before and after the update in the current iteration, the inverse of the approximate Hessian matrix in the current iteration is corrected based on the BFGS algorithm to maintain the positive definiteness of the approximate Hessian matrix. The parameters corresponding to the objective function value of 1 are determined as the parameters of the initial creep fractional-order damage constitutive model.

[0012] Optionally, the entire process of geogrid creep test includes deceleration creep, steady-state creep, and accelerated creep.

[0013] Optionally, creep strain includes elastic strain and creep strain.

[0014] Optionally, the parameters of the initial creep fractional-order damage constitutive model vary with the soil constraint stress.

[0015] This invention provides an apparatus for constructing a creep fractional-order damage constitutive model, comprising: The test module is used to conduct geogrid creep tests under different types of soil constraint stress to obtain the creep curve of the geogrid. The plotting module is used to calculate the creep strain of the geogrid at different time points based on the creep curve, and plot the curve of the creep strain of the geogrid changing with time; the curve of the creep strain of the geogrid changing with time under the action of soil constraint stress conforms to the creep damage law. The derivation module is used to introduce the Khachanov damage variable based on the creep damage law, derive the time when the creep damage of the geogrid reaches the critical failure state, and determine the damage variable in the creep process of the geogrid based on the time when the critical failure state is reached. The acquisition module is used to acquire the creep fractional constitutive model composed of spring elements, soft elements, and nonlinear viscous damper elements; The determination module is used to construct an initial creep fractional-order damage constitutive model based on damage variables and a creep fractional-order constitutive model, and to determine the parameters of the initial creep fractional-order damage constitutive model by using the curve of creep strain of geogrid over time, thus obtaining the creep fractional-order damage constitutive model.

[0016] The present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the above-described method for constructing a creep fractional-order damage constitutive model.

[0017] The present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the above-described method for constructing a creep fractional-order damage constitutive model.

[0018] The above-mentioned at least one technical solution adopted in this invention can achieve the following beneficial effects: Creep tests were conducted on geogrids under different types of soil constraint stress to obtain creep curves. By simulating the soil constraint conditions on the geogrid in actual engineering projects, the obtained creep curves more closely resembled real-world conditions. Based on the creep curves, the creep strain of the geogrid at different time points was calculated, and the creep strain versus time curve was plotted. According to the creep damage law, the Khachanov damage variable was introduced to derive the time for the geogrid to reach the critical failure state. Based on the time to reach the critical failure state, the damage variable in the geogrid creep process was determined. The introduction of the damage variable and its combination with the critical failure time... This invention transforms the abstract concept of damage during creep into a calculable, time-evolving quantitative indicator, enabling the model to capture the cumulative damage and performance degradation patterns, thus improving the accuracy of characterizing the later stages of creep. A fractional-order creep constitutive model, composed of spring elements, soft elements, and nonlinear viscous damper elements, is obtained. This model retains the physical intuitiveness of traditional element models while enhancing the fitting flexibility for nonlinear and viscoelastic behavior during creep through the fractional-order structure. An initial fractional-order creep damage constitutive model is constructed based on damage variables, and the parameters of this model are determined using the time-varying curve of the geogrid's creep strain, resulting in the final fractional-order creep damage constitutive model. The fractional-order creep damage constitutive model constructed in this invention accurately reflects the changes in the geogrid at different creep stages. Attached Figure Description

[0019] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:

[0020] Figure 1 A schematic diagram of the construction method of a creep fractional-order damage constitutive model provided by the present invention; Figure 2 The strain-time curve development pattern under different soil constraints is provided for the present invention. Figure 3Electron microscopy images of geogrid creep provided for this invention; Figure 4 This is a schematic diagram of the component combination of the creep damage constitutive model provided by the present invention; Figure 5 The stress-strain curve of geogrid tension provided by the present invention; Figure 6 The damage variable D development curves under different soil constraint stresses provided by this invention; Figure 7 The Poisson's ratio sensitivity analysis curves provided by this invention; wherein, (a) shows the different values ​​at 50 kPa. The time-strain curves corresponding to the values, (b) shows the different values ​​at 100 kPa. The time-strain curves corresponding to the values, (c) shows the time-strain curves at 150 kPa. The time-strain curves corresponding to the values, (d) shows the values ​​at 200 kPa. The time-strain curve corresponding to the value; Figure 8 The viscosity coefficient of the software component provided by this invention Sensitivity analysis curves; (a) The figure shows the different values ​​at 50 kPa. (a) The time-strain curves corresponding to the values; (b) The figure shows the strain curves at 100 kPa. The time-strain curves corresponding to the values, (c) shows the time-strain curves at 150 kPa. The time-strain curves corresponding to the values, (d) shows the values ​​at 200 kPa. The time-strain curve corresponding to the value; Figure 9 The viscosity coefficient of the series nonlinear damper element provided by this invention Sensitivity analysis curves; (a) The figure shows the different values ​​at 50 kPa. (a) The time-strain curves corresponding to the values; (b) The figure shows the strain curves at 100 kPa. The time-strain curves corresponding to the values, (c) shows the time-strain curves at 150 kPa. The time-strain curves corresponding to the values, (d) shows the values ​​at 200 kPa. The time-strain curve corresponding to the value; Figure 10 Flowchart for constructing the creep fractional-order damage constitutive model provided by this invention; Figure 11 A schematic diagram of a device for constructing a creep fractional-order damage constitutive model provided by the present invention; Figure 12 A schematic diagram of a computer device for constructing a creep fractional-order damage constitutive model provided by the present invention. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0022] Devices such as desktop computers, servers, and laptops are capable of executing the present invention. For ease of explanation, the following description will focus on servers as the executing entity.

[0023] Creep models have become a hot research topic in academia due to their important guiding role in describing the creep process and predicting failure. They are not only a core achievement of creep experimental research but also provide solid theoretical support for subsequent numerical simulations. There are many types of creep models, including empirical models, component combination models, and damage rheological models. Most models can only describe the first two stages of geogrid creep (deceleration creep stage and steady-state creep stage), and the large number of parameters in the models leads to non-uniqueness of the fitting results. Therefore, establishing geogrid creep models with fewer parameters and better descriptive effects will become an important direction for scholars to study geogrid creep deformation and failure control. Currently, there are two main commonly used theoretical modeling methods: based on the geogrid creep process, new theoretical models are established by superimposing or modifying the components of classic component combination models (Maxwell model, Kelvin model, Burgers model, etc.) in series or parallel.

[0024] Furthermore, although the patent application number 201410157685.0, "A Novel Method for Establishing a Rock Creep Constitutive Model Based on Variable Fractional Derivatives," uses a unified functional expression to describe the rock creep constitutive model, the models built using these methods are all empirical models. These empirical models cannot be programmed in numerical analysis software, and therefore cannot be used for long-term stability analysis in actual rock engineering. Moreover, the model does not consider the degree of damage during the creep process.

[0025] In order to overcome the shortcomings of the prior art, the purpose of this invention is to provide a method for constructing a fractional-order time-dependent damage constitutive model of geogrid creep throughout the entire process considering soil constraints, so as to solve the problem that the existing technologies are all empirical models, which cannot be programmed in numerical analysis software and therefore cannot be used for long-term stability analysis in actual geotechnical engineering.

[0026] This invention discloses a fractional-order time-dependent constitutive model for geogrid creep damage considering soil constraints and its construction method, belonging to the field of geotechnical engineering technology. Creep tests are conducted on geogrid specimens under soil constraints to obtain geogrid creep curves. Based on these creep curves, the variation curve of geogrid creep strain over time is plotted. The time required for the geogrid creep damage to reach the critical failure state is derived. t e This method establishes a model that describes the damage variable changes during the creep process of geogrids. It replaces the Newtonian body in the Maxwell model with a soft element and connects it in series with a nonlinear damper to obtain a fractional-order creep model considering time-dependent damage. Finally, based on the creep test results of the geogrid under soil constraint, the model parameters are determined. The model established by this method not only describes the three stages of the entire creep process of geogrids under soil constraint with a unified functional expression, but also reflects the changing characteristics of geogrids at different creep stages. Furthermore, it can be easily programmed into numerical analysis software, facilitating its engineering application.

[0027] The technical solutions provided by the various embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0028] Figure 1 This is a schematic diagram of a method for constructing a creep fractional-order damage constitutive model according to the present invention, which specifically includes the following steps: S101: Under different types of soil constraint stress, geogrid creep tests are conducted to obtain the creep curve of the geogrid.

[0029] In one exemplary embodiment, the entire process of the geogrid creep test includes deceleration creep, steady-state creep, and accelerated creep.

[0030] In one exemplary embodiment, creep strain includes elastic strain and creep strain.

[0031] Specifically, soil confinement stress includes , , Creep tests were conducted on geogrids under different types of soil constraint stress to obtain creep curves of the geogrids. Specifically, these tests included tests conducted under soil constraint stress... and Under the action of axial stress, the geogrid underwent accelerated creep within the test time. The creep curves corresponding to accelerated creep included deceleration creep, steady-state creep, and accelerated creep. Under the action of axial stress, the geogrid did not undergo accelerated creep within the test time. The creep curves corresponding to the absence of accelerated creep included deceleration creep and steady-state creep.

[0032] Geogrid specimens were subjected to different constraint stresses using soil confinement stress loading. σ 1. σ 2… σ i1 , σ i Creep test of geogrid under action, The creep curve of the geogrid under the corresponding soil constraint stress was obtained.

[0033] The total strain of geogrids under different soil constraint stresses consists of two parts: instantaneous elastic strain and creep strain; axial stress σ i1 and σ i Under the action of soil constraint stress, the geogrid underwent accelerated creep within the test time. The creep curve included three stages: deceleration creep, steady-state creep, and accelerated creep. The creep curve containing all three stages is the whole-process creep curve. Under other soil constraint stresses, the geogrid did not undergo accelerated creep within the test time. The creep curves only included two stages: deceleration creep and steady-state creep.

[0034] Specifically, an electronic creep testing system for geosynthetics was used to test geogrid samples under different soil constraint stresses. σ 1 = 50 kPa σ 2 = 100 kPa σ 3=150 kPa and σ Under a stress ratio of 45% (4 = 200 kPa), indoor creep tests were conducted on the geogrid to obtain its creep curve under the corresponding soil constraint stress. Figure 2 The indoor creep curves of geogrids under different soil constraint stresses provided by this invention are as follows: Figure 2 As shown, the creep of geogrids is affected by different factors and can be mainly divided into three stages: decelerating creep, steady-state creep, and accelerated creep. Under low soil constraint stress... σ 1 = 50 kPa σ 2 = 100 kPa and σ At 3 = 150 kPa, the total strain of the geogrid consists of two parts: the time-independent instantaneous elastic strain generated during the loading process and the creep strain that gradually increases with time. Furthermore, the geogrid did not undergo accelerated creep during the test, and the creep curve only included two stages: decelerated creep and steady-state creep. Under high soil constraint stress... σ When 4 = 200 kPa, the total strain of the geogrid is composed of two parts: the instantaneous elastic strain that is independent of time generated during the loading process and the creep strain that gradually increases with time. However, the geogrid underwent accelerated creep during the test time.

[0035] S102: Based on the creep curve of the geogrid, calculate the creep strain of the geogrid at different time points and plot the curve of the creep strain of the geogrid changing with time; the curve of the creep strain of the geogrid changing with time under the action of soil constraint stress conforms to the creep damage law.

[0036] Specifically, based on the creep curve of the entire creep process of the geogrid exhibiting accelerated creep, the creep strain of the geogrid at different times is calculated, and the curve showing the change of creep strain of the geogrid with time is plotted. The microstructure development of the geogrid during creep is related to factors such as creep strain, load level, and creep time. Therefore, it is necessary to establish reasonable damage variables to describe the deterioration effect inside the geogrid material.

[0037] Figure 3 The geogrid creep electron microscope scanning images provided by the present invention, such as Figure 3 As shown in the microscopic morphology diagrams under different soil constraints, it can be seen that pores and cracks appear inside the geogrid material, the fibers are rearranged, and the spacing between the longitudinal ribs is slightly expanded.

[0038] S103: Based on the creep damage law, the Khachanov damage variable is introduced to derive the time when the creep damage of the geogrid reaches the critical failure state, and the damage variable in the creep process of the geogrid is determined based on the time when the critical failure state is reached.

[0039] Based on the fact that the creep strain of geogrid under soil constraint conforms to the creep damage law in fracture mechanics, the Karchanov damage variable is introduced to derive the time required for the geogrid to reach the critical failure state due to creep damage. t e And determine the expression describing the damage changes during the creep process of geogrid.

[0040] Damage variables are introduced during the creep process of geogrids. The derivative of the damage variables with respect to time is given by formula (1): (1); in, D As a damage variable, M and These are all parameters related to material properties. t This refers to the creep time.

[0041] Integrating the damage variable yields the time required for creep damage to reach the critical failure state. t e For formula (2): (2); in, t e The time required for creep damage to reach the critical failure state.

[0042] The expression for the damage variable describing the damage change is derived as formula (3): (3); S104: Obtain the creep fractional constitutive model consisting of spring elements, soft elements, and nonlinear viscous damper elements.

[0043] By replacing the Newtonian elements in the conventional Maxwell model with soft elements and connecting them in series with nonlinear damping elements, and expressing the strain of the soft elements according to the RL calculus operator theory, a creep fractional constitutive model consisting of spring elements, soft elements and nonlinear viscous damping elements is established.

[0044] By combining spring elements, soft elements, and nonlinear viscous damper elements, a fractional-order constitutive model of creep is constructed to describe the entire creep process of geogrid. Figure 4 This is a schematic diagram of the creep fractional-order constitutive model element combination provided by the present invention, as shown below. Figure 4 As shown, the spring element characterizes the instantaneous tensile strain generated during the initial creep of the geogrid. Fractional-order soft elements characterize the viscoelastic strain generated by the stable development of damage and cracks in geogrids during the deceleration and steady-state creep stages. The nonlinear viscous damper element characterizes the viscoplastic strain generated by the rapid propagation and penetration of damage cracks in the geogrid during the accelerated creep stage. .

[0045] The constitutive equation of the spring element in the Maxwell model is Equation (4): (4); in, For the strain of the spring element, For the stress in the fractional-order constitutive model of creep, E 1 represents the elastic modulus of the spring element.

[0046] Introducing fractional calculus into software components The fractional integral of order Riemann-Liouville is defined by formula (5): (5); in, Let be the integrand that varies with creep time. For fractional integral operators, , For the order of fractional integrals, when the order of fractional integrals is... When, satisfy , t For a moment, Let be the integral variable, representing the range from 0 to 1. t At a certain moment within the time period, In order to be in The function value of creep time.

[0047] Accordingly, The first derivative is defined by formula (6): (6); in, n greater than The smallest integer, , In order to be in The function value of creep time.

[0048] When function exist t = is integrable near 0, and At that time, The Laplace transform is denoted as Then the Laplace transform of the fractional integral of type RL is given by formula (7): (7); in, The Laplace transform of an RL-type fractional integral. Let the fractional order of the complex frequency variable be denoted as . for The Laplace transform of .

[0049] Soft components can reflect different stress states. When the stress is at a constant level, according to the RL differential operator theory, the constitutive equation of the soft component is formula (8): (8); in, For software component strain, The viscosity coefficient of the software component. For fractional calculus operators, For order, .

[0050] The constitutive equation of the nonlinear viscous damper element is Equation (9): (9); in, The viscosity coefficient of the nonlinear damper. A These are parameters related to viscous properties.

[0051] The creep fractional constitutive model composed of spring elements, soft elements, and nonlinear viscous damper elements is given by formula (10): (10); in, The total strain of the creep fractional-order constitutive model is given by... For the strain of the spring element, For software component strain, For viscoplastic strain, The effective stress of the creep fractional-order constitutive model is... For order, The viscosity coefficient of the nonlinear damper. A These are parameters related to viscous properties. For fractional calculus operators, t Creep time, E 1 represents the elastic modulus.

[0052] S105: Based on the damage variable and the creep fractional order constitutive model, an initial creep fractional order damage constitutive model is constructed, and the parameters of the initial creep fractional order damage constitutive model are determined by the curve of the creep strain of the geogrid changing with time, thus obtaining the creep fractional order damage constitutive model.

[0053] Based on Lemaitre's equivalent strain principle, the creep damage of geogrid is equivalent to the strain equivalent change. Considering the damage effect, a creep damage constitutive model is obtained.

[0054] In one exemplary embodiment, the parameters of the initial creep fractional-order damage constitutive model vary with the soil constraint.

[0055] In an exemplary embodiment, an initial creep fractional-order damage constitutive model is constructed based on damage variables and a creep fractional-order constitutive model. Specifically, this includes: expressing the effective stress in the creep fractional-order constitutive model as follows, according to the stress-strain equivalence principle and damage variables: ;in, D As a damage variable, For the stress in the fractional-order constitutive model of creep, The effective stress of the creep fractional-order constitutive model; Substituting into the fractional-order creep constitutive model, we obtain the initial fractional-order creep damage constitutive model.

[0056] Considering the damage effect, based on the Lemaitre equivalent strain principle, the effective stress in the creep fractional constitutive model is given by formula (11): (11); in, D As a damage variable, For the stress in the fractional-order constitutive model of creep, The effective stress is the creep fractional-order constitutive model.

[0057] Substitute formula (11) into formula (10), and consider At As a constant term, the creep fractional order damage constitutive model can be obtained as formula (12): (12); in, D As a damage variable, , These are parameters related to the material properties of geogrids.

[0058] In an exemplary embodiment, the parameters of the initial creep fractional-order damage constitutive model are determined by the curve of the creep strain of the geogrid versus time. Specifically, this includes: fitting the curve of the creep strain of the geogrid versus time using Hooke's law to obtain the elastic modulus; generating a data matrix based on multiple sets of strain, time, soil constraint stress, and elastic modulus obtained from geogrid creep tests; inputting the data matrix, damage variables, and the initial creep fractional-order damage constitutive model into MATLAB and defining an objective function; the objective function is the strain value obtained from the geogrid creep test, the average strain value measured by the creep test, and the initial creep value. The strain values ​​predicted by the variable fractional-order damage constitutive model are constructed; multiple parameters of the initial creep fractional-order damage constitutive model are initialized, and the initial parameters of the BFGS algorithm are set; the initial parameters include Armijo search parameters, Armijo condition parameters, and convergence tolerance; the gradient vector of the initial values ​​of the initial creep fractional-order damage constitutive model parameters is calculated, and the convergence of the calculated gradient vector is checked; after the convergence check, the BFGS algorithm is used for multiple iterations until the objective function value is 1, thus obtaining the parameters of the initial creep fractional-order damage constitutive model; the parameters of the initial creep fractional-order damage constitutive model include ;in, The viscosity coefficient of the software component. The viscosity coefficient of the nonlinear damper. For order, These are parameters related to the material properties of geogrids.

[0059] In an exemplary embodiment, the objective function is formula (13): (13); in, The objective function value, For the first i The strain values ​​obtained from the second geogrid creep test. The strain is the average value determined by the creep test. The strain value is predicted by the fractional-order constitutive model of creep. n This represents the total number of geogrid creep tests.

[0060] In an exemplary embodiment, the BFGS algorithm is used for multiple iterations until the objective function value is 1, obtaining the parameters of the initial creep fractional-order damage constitutive model. Specifically, this includes: in each iteration, calculating the gradient vector of the parameters for the current iteration and checking if the norm of the gradient vector is less than the convergence tolerance; if the norm of the gradient vector is less than the convergence tolerance, stopping the iteration; if the norm of the gradient vector is greater than or equal to the convergence tolerance, determining the search direction for parameter updates based on the product of the inverse of the approximate Hessian matrix and the gradient vector for the current iteration; determining the step size for parameter updates through Armijo line search; updating the parameters based on the update step size and search direction and calculating the objective function value corresponding to the updated parameters; using the parameter difference and gradient difference before and after the update in the current iteration, correcting the inverse of the approximate Hessian matrix for the current iteration based on the BFGS algorithm to maintain the positive definiteness of the approximate Hessian matrix; and determining the parameters corresponding to the objective function value of 1 as the parameters of the initial creep fractional-order damage constitutive model.

[0061] Specifically, Figure 5 The stress-strain curve of the geogrid under tension provided by the present invention is as follows: Figure 5 As shown, elastic modulus E 1. Based on the stress-strain curve obtained from the geogrid tensile test, the elastic modulus was determined by Hooke's law through linear fitting of the curve. E 1. The stress-strain curves obtained from the geogrid tensile test are determined by Hooke's law through linear fitting of the curves.

[0062] Initial creep fractional-order damage constitutive model parameters All solutions are based on the BFGS algorithm. In each iteration, the local minimum of the function is found by progressively improving the fitting second derivative information, and then determined using MATLAB. This method, by combining experimental data with gradient-based optimization algorithms, achieves efficient and stable identification of material constitutive model parameters, and provides a reliable basis for the analysis of the physical meaning and influence of the model parameters.

[0063] Step 1: Determine the elastic modulus E 1.

[0064] The stress-strain curve of the material was obtained through uniaxial tensile testing, and its elastic modulus was calculated based on Hooke's law. E 1. Used as known parameters to input into the subsequent recognition process.

[0065] Step 2: Component optimization matrix.

[0066] The strain, time, constraint force, and elastic modulus obtained in the experiment are used to determine the optimal parameters. E1. The data is integrated into a data matrix. Based on this, the expressions for the damage variables and the creep fractional-order damage constitutive model are substituted into the matrix to establish an objective function describing the difference between the response of the creep fractional-order damage constitutive model and the experimental data. Here, the coefficient of determination is used. R 2 Maximize as the objective, i.e. minimize 1- R 2 The maximum value of R is 1.

[0067] Step 3: Parameter initialization and algorithm settings.

[0068] For the parameters of the fractional-order creep damage constitutive model to be identified, the following initial conditions are set: Based on the experimental results and the expression of the damage variable, the initial solution of the parameters of the fractional-order creep damage constitutive model is given; the maximum number of iterations, Armijo line search parameters, Armijo condition judgment parameters, and convergence tolerance are set; gradient calculation is performed, and the partial derivatives of each model parameter in the objective function need to be calculated for the determination of the subsequent search direction.

[0069] Step 4: Solve using BFGS iteration.

[0070] Gradient calculation and convergence judgment: Calculate the gradient vector at the current parameter point and check if its norm is less than the convergence tolerance. If it is satisfied, stop the iteration; otherwise, continue. Determine the search direction: Use the current approximation. Hessian The product of the matrix inverse and the gradient determines the search direction for parameter updates; Armijo Line search determines the step size: Perform a one-dimensional search along the search direction and select the values ​​that satisfy the condition. Armijo The step size is determined by the condition to ensure the objective function decreases sufficiently; parameters and function values ​​are updated: parameters are updated according to the search direction and step size, and function values ​​at the new parameter points are calculated; the Hessian approximation matrix is ​​updated: using the parameter differences and gradient differences in the current iteration, based on... BFGS The algorithm corrects the inverse of the Hessian matrix to avoid directly calculating the second derivative and maintains the positive definiteness of the matrix; repeated iteration: repeat the above process until the convergence condition is met or the maximum number of iterations is reached, and finally output the five model parameter values ​​that make the objective function optimal.

[0071] Step 5: Parameter sensitivity analysis.

[0072] After obtaining the optimal parameter set, further parameter sensitivity analysis is carried out: keeping other parameters constant, Poisson's ratio, soft element parameters and viscosity coefficient of nonlinear viscous damper are adjusted respectively, the changes in model prediction results are observed, the influence of each parameter on the model output is evaluated, and thus the key parameters affecting constitutive behavior are identified.

[0073] Initial creep fractional-order damage constitutive model parameters All solutions are based on the BFGS algorithm. In each iteration, the local minima of the function are found by progressively improving the fitting second derivative information, and then determined using MATLAB. The steps are as follows: Step 1: Input the strain (as in formula (14), time (as in formula (15), stress (as in formula (16), elastic modulus, and stiffness matrix) from the experiment into MATLAB: (14).

[0074] (15).

[0075] (16).

[0076] Step 2: Input the damage variables and the creep constitutive model, and use the definition of residual sum of squares and total sum of squares. R 2 The objective function is shown in formula (17), and is for... The partial derivatives of the parameter solution are given by formula (17): (17).

[0077] Step 3: Assign initial parameter values ​​based on experimental data and engineering experience. Set the maximum number of iterations to 500, the Armijo search parameter to 0.55, the Armijo condition parameter to 0.4, and the convergence tolerance to epsilon1e-5. Calculate the initial gradient. ,in, This is the gradient vector at the current parameter point. Represents the MATLAB function evaluation operators. This is the gradient function, which returns the partial derivatives of the objective function with respect to each parameter. Viscosity coefficient of software components , The viscosity coefficient of the nonlinear damper , Poisson's ratio , For the constitutive model equations Convergence check ,in, The 2-norm of the gradient vector measures the magnitude of the gradient. This is the convergence threshold of the gradient norm; iteration stops when the gradient is sufficiently small.

[0078] Step 4: Solve the equations and calculate the search direction. ,in, The search direction vector, For approximate Hession matrix, Given the gradient vector, the Armijo line search determines the step size, calculates the new function value, and performs Armijo condition checks.

[0079] Update parameters, , ,in, For the updated parameters, Step size factor It is a contractile factor. The smallest integer that satisfies Armijo's condition For parameter changes, The gradient change The gradient at the new parameter point.

[0080] if but Finally, the optimal parameters are obtained. This is the updated approximate Hession matrix. For parameter change vectors, This is the gradient change vector.

[0081] Step 5: Based on the optimal parameters, change the Poisson ratio. Viscosity coefficients of soft components and nonlinear dampers Parameter sensitivity analysis was performed to obtain the specific impact on creep strain.

[0082] Among them, to study the effect of Poisson's ratio on accelerated creep time, it is necessary to first convert the damage variable in formula (3) into a variable. D Poisson's ratio Solve for the partial derivatives and use the chain rule to obtain formula (18); then solve for the partial derivatives of the strain value with respect to Poisson's ratio in formula (12), and substitute the damage variable equation into it, i.e., formula (19), and perform Poisson's ratio calculation. Sensitivity analysis.

[0083] (18); (19); Study viscosity coefficient The effect of accelerated creep time is calculated by applying the strain value to the viscosity coefficient in formula (11). Solving for the partial derivatives yields the specific expressions, which are given by formulas (20) and (21): (20); (twenty one); The inversion determination results of creep parameters under different axial stresses are shown in Table 1.

[0084] Table 1 Figure 6 The damage variable D development curves provided by this invention under different soil constraint stresses can be obtained by substituting the Poisson's ratio under different soil constraints into formula (3). D The curve as it progresses over time, such as Figure 6 As shown, under different soil lateral confinement conditions, the damage development of geogrids is rapid in the early stage, exhibiting exponential growth, while the damage variable approaches 1 in the later stage. Overall, the creep accumulation damage of geogrids under soil lateral confinement has a limit value; after exceeding a certain threshold, the damage development stabilizes, and numerous micro-defects develop within the geogrid, even partially penetrating it. With increasing soil lateral confinement, the damage increases; when the confinement increases from 50 kPa to 200 kPa, the stabilization time increases, and the damage variable decreases by 43.3%.

[0085] In an exemplary embodiment, in order to study the applicability of each part of the creep fractional-order time-dependent damage constitutive model, the present invention specifically addresses the Poisson's ratio. Viscosity coefficient of software components Viscosity coefficient in nonlinear dampers Sensitivity and stress correlation analysis were performed on parameters such as these.

[0086] Poisson's ratio Sensitivity analysis was conducted on the main geogrid under soil constraints of 50, 100, 150, and 200 kPa, showing the Poisson's ratio. Using 0.1, 0.3, 0.5, 0.7, and 0.9 as baselines respectively, the optimal solution is kept unchanged. First, calculate the partial derivatives of the damage variable formula (3) and the constitutive model formula (12) with respect to Poisson's ratio, as shown in formulas (18) and (19). Then, calculate the sensitivity response matrix. Substitute the curve into formula (19) to obtain the predicted curve, fit it with the experimental discrete points, determine the optimal range of Poisson's ratio, and study the evolution law of the model creep curve. Figure 7 The Poisson's ratio sensitivity analysis curve provided by this invention is as follows: Figure 7 As shown, strain increases sharply with increasing Poisson's ratio, and the rate of increase gradually increases. During creep, the material's Poisson's ratio is around 0.3–0.5, showing the best fit with the experimental results. After 316 h, the experimental strain tends to stabilize, reflecting that the damage development of the grid is relatively stable during the decay and steady-state creep stages. Later-stage damage under soil lateral confinement. D It tends to stabilize, but when the stress is relatively high, an accelerated creep stage is added.

[0087] Viscosity coefficient of software components Sensitivity analysis was conducted on the viscosity coefficient of the main geogrid under soil constraints of 50, 100, 150, and 200 kPa. Using 20, 50, 80, 100, and 120 as references respectively, the viscosity coefficient of the soft component is first determined by the constitutive model formula (12). The partial derivatives, i.e., formula (20), are then used to determine the sensitivity response matrix. Substituting the other optimal parameters into formula (20) yields the predicted curve, which is then fitted to the experimental discrete points to determine the optimal curve. The optimal range was determined by studying the influence of model parameters on the evolution of the creep strain curve. Figure 8 The viscosity coefficient of the software component provided by this invention Sensitivity analysis curves, such as Figure 8 As shown, the slope of the strain-time curve gradually decreases with the increase of the viscosity coefficient of the soft component, and the viscosity coefficient increases with the increase of the soil lateral confinement. It exhibits a non-linear decreasing trend and a decrease in sensitivity.

[0088] Viscosity coefficient in nonlinear dampers Sensitivity analysis was conducted on the viscosity coefficient of the main geogrid under soil constraints of 50, 100, 150, and 200 kPa. Using 200, 500, 1000, 1500 and 3000 as references respectively, the creep fractional damage constitutive model formula (12) was first applied to the viscosity coefficient of the nonlinear damper. The partial derivatives, i.e., formula (21), are then used to determine the sensitivity response matrix. Substituting the other optimal parameters into formula (21) yields the predicted curve, which is then fitted to the experimental discrete points to determine the optimal curve. The optimal range was determined by studying the influence of model parameters on the evolution of the creep strain curve. Figure 9 The viscosity coefficient of the series nonlinear damper element provided by this invention Sensitivity analysis curves, such as Figure 9 As shown, when the viscosity coefficient is less than 1000, the strain-time curve changes more significantly, and the slope of the curve decreases as the viscosity coefficient increases.

[0089] As Poisson's ratio increases and the viscosity coefficient decreases, the time for the accelerated creep phase to occur decreases. The sensitivity order of the parameters Poisson's ratio and viscosity coefficient to the occurrence time of the accelerated creep phase is: Poisson's ratio... Viscosity coefficient of software components Viscosity coefficient of nonlinear damper .

[0090] In one exemplary embodiment, the present invention provides as follows Figure 10 The flowchart shown is as follows: Figure 10 As shown, indoor creep tests of geogrids were conducted under soil constraints. A fractional-order creep constitutive model, including spring elements, software elements, and nonlinear viscous damping elements, was obtained. Damage effects were considered in the fractional-order creep constitutive model to obtain a fractional-order creep damage constitutive model. The fractional-order time-dependent damage constitutive model for the entire creep process is also known as the fractional-order creep damage constitutive model. The parameters of the fractional-order creep damage constitutive model were determined using the BFGS algorithm. The calculated values ​​of the fractional-order creep damage constitutive model were compared with the experimental results. Sensitivity analysis was performed on the parameters of the fractional-order creep damage constitutive model.

[0091] When applying the method for constructing the creep fractional-order damage constitutive model provided by this invention, it is not necessary to follow the... Figure 1 The steps shown are executed in sequence. The specific execution order of each step can be determined as needed, and this invention does not impose any restrictions on it.

[0092] The above describes a method for constructing a creep fractional-order damage constitutive model according to one or more embodiments of the present invention. Based on the same idea, the present invention also provides a corresponding apparatus for constructing a creep fractional-order damage constitutive model, such as... Figure 11 As shown.

[0093] Figure 11 A schematic diagram of a device for constructing a creep fractional-order damage constitutive model provided by the present invention includes: Test module 1101 is used to conduct geogrid creep tests under different types of soil constraint stress to obtain the creep curve of the geogrid.

[0094] The plotting module 1102 is used to plot the creep curve based on the geogrid, calculate the creep strain of the geogrid at different time points, and plot the curve of the creep strain of the geogrid changing with time; the curve of the creep strain of the geogrid changing with time under the action of soil constraint stress conforms to the creep damage law.

[0095] The derivation module 1103 is used to introduce the Khachanov damage variable according to the creep damage law, derive the time when the creep damage of the geogrid reaches the critical failure state, and determine the damage variable in the creep process of the geogrid based on the time when the critical failure state is reached.

[0096] The acquisition module 1104 is used to acquire the creep fractional constitutive model composed of spring elements, soft elements and nonlinear viscous damper elements.

[0097] The determination module 1105 is used to construct an initial creep fractional-order damage constitutive model based on damage variables and a creep fractional-order constitutive model, and to determine the parameters of the initial creep fractional-order damage constitutive model by using the curve of creep strain of geogrid changing with time, thereby obtaining the creep fractional-order damage constitutive model.

[0098] Specific limitations regarding the construction apparatus for the creep fractional-order damage constitutive model can be found in the limitations on the construction method of the creep fractional-order damage constitutive model above, and will not be repeated here. Each module in the aforementioned construction apparatus for the creep fractional-order damage constitutive model can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device in hardware form, or stored in the memory of a computer device in software form, so that the processor can call and execute the operations corresponding to each module.

[0099] The present invention also provides a computer-readable storage medium storing a computer program that can be used to execute the above-described... Figure 1 A method for constructing a creep fractional-order damage constitutive model is provided.

[0100] The present invention also provides Figure 12 The schematic diagram of the computer device shown is as follows: Figure 12 As shown, at the hardware level, this computer device includes a processor, internal bus, network interface, memory, and non-volatile memory, and may also include other hardware required for business operations. The processor reads the corresponding computer program from the non-volatile memory into memory and then executes it to achieve the above. Figure 1 A method for constructing a creep fractional-order damage constitutive model is provided.

[0101] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the methods described above. Any references to memory, storage, databases, or other media used in the embodiments provided by this invention can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, or optical storage, etc. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc.

[0102] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this invention.

Claims

1. A method for constructing a creep fractional-order damage constitutive model for geogrids, characterized in that, include: Creep tests were conducted on geogrids under different types of soil constraint stress to obtain the creep curves of the geogrids. Based on the creep curve of the geogrid, the creep strain of the geogrid at different time points was calculated, and the curve of the creep strain of the geogrid changing with time was plotted. The curve of the creep strain of the geogrid changing with time under the action of soil constraint stress conforms to the creep damage law. Based on the creep damage law, the Khachanov damage variable is introduced to derive the time when the creep damage of the geogrid reaches the critical failure state, and the damage variable in the creep process of the geogrid is determined based on the time when the critical failure state is reached. Obtain a creep fractional constitutive model consisting of spring elements, soft elements, and nonlinear viscous damper elements; Based on the damage variables and the creep fractional-order constitutive model, an initial creep fractional-order damage constitutive model is constructed. The parameters of the initial creep fractional-order damage constitutive model are determined by the curve of the creep strain of the geogrid changing with time, thus obtaining the creep fractional-order damage constitutive model.

2. The method as described in claim 1, characterized in that, The expression for the damage variable is: ； in, D As a damage variable, t Creep time, t e The time required for creep damage in geogrids to reach the critical failure state. These are parameters related to material properties.

3. The method as described in claim 1, characterized in that, The construction of an initial creep fractional-order damage constitutive model based on the damage variables and the creep fractional-order constitutive model specifically includes: Based on the stress-strain equivalence principle and damage variables, the effective stress in the creep fractional constitutive model is expressed as: ;in, D As a damage variable, For the stress in the fractional-order constitutive model of creep, The effective stress is the fractional-order constitutive model of creep. Will Substituting the creep fractional-order constitutive model into the model yields the initial creep fractional-order damage constitutive model.

4. The method as described in claim 3, characterized in that, The creep fractional constitutive model is as follows: ； in, The total strain of the creep fractional-order constitutive model is given by... The effective stress of the creep fractional-order constitutive model is... For order, The viscosity coefficient of the nonlinear damper. A These are parameters related to viscous properties. For fractional calculus operators, t Creep time, E 1 represents the elastic modulus. The viscosity coefficient of the software component; The initial creep fractional-order damage constitutive model is as follows: ； in, D For damage variables.

5. The method as described in claim 1, characterized in that, The determination of parameters for the initial creep fractional-order damage constitutive model by using the curve of creep strain of geogrid over time specifically includes: The elastic modulus was obtained by fitting the curve of creep strain of geogrid with time using Hooke's law. Multiple sets of strain, time, soil constraint stress and elastic modulus were obtained based on geogrid creep tests, and a data matrix was generated. The data matrix, damage variables, and initial creep fractional-order damage constitutive model are input into MATLAB, and an objective function is defined. The objective function is constructed based on the strain values ​​obtained from the geogrid creep test, the average strain value measured by the creep test, and the strain values ​​predicted by the initial creep fractional-order damage constitutive model. Multiple parameters of the initial creep fractional-order damage constitutive model are initialized, and the initial parameters of the BFGS algorithm are set; the initial parameters include Armijo search parameters, Armijo condition parameters, and convergence tolerance. Calculate the gradient vector of the initial values ​​of the parameters of the initial creep fractional order damage constitutive model, and check the convergence of the calculated gradient vector; After performing the convergence check, the BFGS algorithm is used for multiple iterations until the objective function value is 1, thus obtaining the parameters of the initial creep fractional-order damage constitutive model. The parameters of the initial creep fractional-order damage constitutive model include... ;in, The viscosity coefficient of the software component. The viscosity coefficient of the nonlinear damper. For order, These are parameters related to the material properties of geogrids.

6. The method as described in claim 5, characterized in that, The objective function is: ； in, The objective function value, For the first i The strain values ​​obtained from the second geogrid creep test. The strain is the average value determined by the creep test. The strain value is the strain value predicted by the initial creep fractional-order damage constitutive model. n This represents the total number of geogrid creep tests.

7. The method as described in claim 5, characterized in that, The BFGS algorithm is used for multiple iterations until the objective function value is 1, thus obtaining the parameters of the initial creep fractional-order damage constitutive model, specifically including: In each iteration of the solution process, the gradient vector of the parameters in the current iteration is calculated, and it is checked whether the norm of the gradient vector is less than the convergence tolerance. If the norm of the gradient vector is less than the convergence tolerance, then stop the iteration; If the norm of the gradient vector is greater than or equal to the convergence tolerance, the search direction for parameter updates is determined based on the product of the inverse of the approximate Hessian matrix of the current round and the gradient vector. The step size for parameter updates is determined by Armijo line search; Based on the updated step size and search direction, update the parameters and calculate the objective function value corresponding to the updated parameters; By utilizing the parameter difference and gradient difference before and after the update in the current iteration, the inverse of the approximate Hessian matrix in the current iteration is corrected based on the BFGS algorithm to maintain the positive definiteness of the approximate Hessian matrix. The parameters corresponding to the objective function value of 1 are determined as the parameters of the initial creep fractional-order damage constitutive model.

8. The method as described in claim 1, characterized in that, The entire process of the geogrid creep test includes deceleration creep, steady-state creep, and accelerated creep.

9. The method as described in claim 1, characterized in that, The creep strain includes elastic strain and creep strain.

10. The method as described in claim 1, characterized in that, The parameters of the initial creep fractional-order damage constitutive model vary with the soil constraint stress.

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