Roadbed soil plastic strain fractional order damage model considering freeze-thaw damage and application

Abstract

The invention relates to a roadbed soil plastic strain fractional order damage model considering freeze-thaw damage and application. The matching result of the model and test data is ideal, when the plastic strain state is in a stable state and a critical state, the strain curve is mainly represented as a deceleration plastic strain stage, an I-constant speed plastic strain stage and a II-constant speed plastic strain stage, and compared with a traditional model, the model effectively improves the simulation precision of nonlinear behaviors in the stages. The model provided by the invention can accurately describe the elastic strain stage, the deceleration stage, the I-constant speed stage, the II-constant speed stage and the accelerated plastic strain stage in the plastic strain in stable, critical and damaged states, not only provides theoretical support for the analysis of the plastic strain of the roadbed soil, but also shows the application value of the model in engineering practice.

Description

Technical Field

[0001] The present invention belongs to the technical field of road engineering design, construction and maintenance, and relates to a fractional-order damage model of plastic strain of subgrade soil considering freeze-thaw damage and its uses. Background Technique

[0003] In recent years, domestic and foreign scholars have carried out a series of studies on the influence of freeze-thaw cycles and traffic loads on the strength and stability performance of subgrade soil. For example, Ma et al. found that the porosity and permeability coefficient of the specimens under freeze-thaw cycles increased significantly, and the attenuation amplitude was the largest during the first cycle, and there was a threshold value during 5 - 20 cycles that made the attenuation amplitude tend to be stable. The research results of Yang et al., Liu et al., and Bragar et al. showed that the internal structure of the soil body (such as the cohesion, internal friction angle and other characterization indexes) would deteriorate to varying degrees under the action of freeze-thaw cycles, thereby inducing the attenuation of strength performance. In addition, Sun Jing et al., Liao Xin et al. carried out research on the dynamic stability performance evolution of the soil body under the action of freeze-thaw cycles, and found that the increase of the number of freeze-thaw cycles and deviator stress was an important inducement for the gradual accumulation of plastic strain. In addition to indoor tests and analysis of influence laws, researchers are committed to constructing mathematical models to deepen the understanding of the plastic strain of subgrade soil. Currently, the models are mainly divided into two categories: constitutive models and empirical models. The constitutive model is based on classical soil mechanics principles and can describe the stress-strain curve during any single loading. For example, Cui Kai et al. described the development law of plastic strain of subgrade soil considering the influence of wetting degree in the three-dimensional stress space through the moving mapping rule and the modified Cambridge model, and verified the model with the test data of 6 times of loading and the fitting effect was good. Li et al. introduced an anisotropic hardening parameter and jointly characterized the development law of plastic strain of subgrade soil by combining the elastoplastic double-surface model and the mapping center moving method. The model had good applicability but could only calculate the strain of 10 times of loading. As can be seen from the above, the constitutive model requires higher computing power, the number of calculations is limited, and it is inconvenient for engineering applications. In contrast, the empirical model is constructed based on test laws, is simple and practical, and is the key to calculating the deformation of the subgrade structure by the layer-wise summation method. For example, Monismith et al. proposed a classical exponential empirical model of plastic strain varying with the number of loadings based on clay triaxial tests, and this model has been continuously optimized and improved in subsequent research. Zhang et al. took typical clay in southern China as the object, used the net body stress to reflect the confinement effect, the octahedral shear stress to reflect the shear effect, and the saturation to reflect the humidity state, and established a corresponding power-type empirical model based on the Monismith model. However, the empirical model is essentially a mathematical fitting of test data and lacks a solid theoretical basis, resulting in its applicability being limited to specific soil types and working conditions and insufficient universality.

[0004] Due to its global relevance and long-range memory, fractional calculus has unique advantages in describing complex nonlinear mechanical behaviors. It not only retains the core theoretical framework of viscoelastic-plastic mechanics but also has the advantages of simple structure and clear physical meaning. For example, Zhang Junhui et al. and Deng Huiyuan et al. respectively used the generalized Bingham model and Burgers model to predict the rheological properties of clay in Lianyungang and Hangzhou. However, traditional models based on the series and parallel connection of linear elements still have limitations in dealing with the nonlinear behaviors of geotechnical materials. To overcome this deficiency, Ren Peng et al. proposed a four-element fractional derivative creep model, and its calculation results fit well with the plastic strain curve of clay; Liu Jiashun et al. established a fractional prediction model using a double Abel dashpot, which can reasonably predict the long-term deformation characteristics of subgrade under intermittent cyclic loading. However, traditional fractional models and the improved fractional models of the above scholars can simply describe the steady state and critical state of soil plastic strain, but cannot accurately describe the critical state simultaneously. Moreover, there are few reports on the application of fractional calculus methods in the freeze-thaw damage and dynamic stability problems of subgrade clay. Summary of the Invention

[0005] To solve the above problems, the present invention provides a fractional damage model of subgrade soil plastic strain considering freeze-thaw damage, and the model is as follows:

[0006]

[0007] where: ε all-2 is the total plastic strain at the failure state; ε RD is the viscoelastic strain; is the viscoplastic strain; is the accelerating viscoplastic strain; β and η3 are the accelerating index and the initial value of the viscosity coefficient of the unsteady viscoplastic element; N load-cri is the critical loading times; σ max is the maximum dynamic stress value; t1 is the loading time within each cycle period; T is the load period; N load is the number of loadings; α and η2 are the fractional order and the viscosity coefficient of the fractional damage element, and 0 < α < 1; c, are the total cohesion and the total internal friction angle of the soil respectively; σ3 is the confining pressure; G1 and η1 are the viscoelastic shear modulus and the viscosity coefficient of the Kelvin (generalized) element respectively; I n is the degree of freeze-thaw damage; e is the mathematical constant with a value of 2.71; Γ(α + 1) is the Gamma function of α.

[0008] The model of the present invention can be used to determine the plastic strain state of subgrade soil under freeze-thaw damage conditions and for the monitoring and prediction of subgrade settlement deformation under freeze-thaw damage conditions.

[0009] The present invention also provides a method for determining the plastic strain state of subgrade soil under freeze-thaw damage conditions. This method uses the above model and includes the following steps:

[0010] (1) Obtain the peak strength, total cohesion c, and total internal friction angle of the subgrade soil through triaxial tests on the subgrade soil in the laboratory Critical loading times N load-cri and degree of freeze-thaw damage I n ; at the same time, obtain the confining pressure σ3 through the subgrade fill height and the deviator stress σ through different traffic loads d ; other model parameters are obtained by regression fitting;

[0011] (2) Substitute the parameters obtained in step (1) into the model formula to obtain the relationship curve between the loading times and plastic strain under fixed deviator stress and confining pressure;

[0012] (3) Based on the relationship curve between the loading times and plastic strain under fixed deviator stress and confining pressure obtained, when:

[0013] When R PD ≈ 0, the plastic strain is in the stage of stage I - constant-rate plastic strain;

[0014] When R PD > 0 and the value of R PD is close to a constant value, the plastic strain is in the stage of stage II - constant-rate plastic strain;

[0015] When R PD > 0 and the value of R PD increases non-linearly, the plastic strain is in the stage of accelerating plastic strain;

[0016] On this basis, looking at the entire plastic strain curve, if only the stage of stage I - constant-rate plastic strain exists, the subgrade is in a stable state;

[0017] If both the stage of stage I - constant-rate plastic strain and the stage of stage II - constant-rate plastic strain exist, or only the stage of stage II - constant-rate plastic strain exists, then the subgrade is in a critical state;

[0018] If the stage of accelerating plastic strain exists, or when the plastic strain is greater than or equal to 10%, the subgrade is in a failure state.

[0019] The model of the present invention shows an ideal agreement with the test data. When the plastic strain state is in the shakedown and critical states, the strain curve mainly exhibits the decelerating, Stage I - constant velocity, and Stage II - constant velocity plastic strain stages. Compared with the traditional model, the model of the present invention effectively improves the simulation accuracy of the nonlinear behavior in the above - mentioned stages. When the plastic strain state is in the failure state, the strain curve shows an accelerating plastic strain stage, while the fitting of the traditional model has shown obvious deviations in the constant velocity plastic strain stage. The model of the present invention can better reflect the various development stages of plastic strain in the shakedown, critical, and failure states.

[0020] The model of the present invention can accurately describe the elastic strain, decelerating, Stage I - constant velocity, Stage II - constant velocity, and accelerating plastic strain stages in plastic strain under shakedown, critical, and failure states. It not only provides theoretical support for the analysis of subgrade soil plastic strain but also demonstrates its application value in engineering practice. Brief Description of the Drawings

[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0022] Figure 1 It is a schematic diagram of the plastic strain stage and state in the present invention;

[0023] Figure 2 It is a schematic diagram of cyclic load and constant load in the present invention;

[0024] Figure 3 It is a schematic diagram of the Kelvin (generalized) element in the present invention;

[0025] Figure 4 It is a schematic diagram of the fractional - order damage element in the present invention;

[0026] Figure 5 It is a schematic diagram of the unsteady viscoplastic element in the present invention;

[0027] Figure 6 It is a fractional - order damage model considering the influence of freeze - thaw damage in the present invention;

[0028] Figure 7 It is a sensitivity analysis of the model parameters in the present invention: where (a) is the fractional - order; (b) is the acceleration index;

[0029] Figure 8 It is the experimental instrument used in Example 3 of the present invention, where (a) is a programmable thermo - humidity test chamber; (b) is a triaxial test system;

[0030] Figure 9 For the N in Embodiment 3 of the present invention FT Comparison of triaxial test data and model fitting curves under the condition of N = 0: where: (a) is a confining pressure of 30; (b) is a confining pressure of 60; (c) is a confining pressure of 90;

[0031] Figure 10 For the comparison of triaxial test data and model fitting curves with different N under the condition of σ3 = 60 kPa in Embodiment 3 of the present invention FT where: (a) is N FT = 0; (b) is N FT = 1; (c) is N FT = 3; (d) is N FT = 6; (e) is N FT = 10;

[0032] Figure 11 Relationship curve of number of loadings and plastic strain under fixed deviator stress and confining pressure obtained in Embodiment 4 of the present invention. Specific embodiments

[0033] Next, in combination with the embodiments of the present invention, the technical solutions in the embodiments of the present invention will be described clearly and completely. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0034] Embodiment 1

[0035] This application provides a fractional-order damage model for plastic strain of subgrade soil considering freeze-thaw damage, and the model is:

[0036]

[0037] In the formula: ε all-2 is the total plastic strain at the failure state; ε RD is the viscoelastic strain; is the viscoplastic strain; is the accelerated viscoplastic strain; β, η3 are the acceleration index and the initial value of the viscosity coefficient of the unsteady viscoplastic element; N load-cri is the critical number of loadings; σ max is the maximum dynamic stress value; t1 is the load application time within each cycle period; T is the load period; N load is the number of loadings; α, η2 are the fractional-order order and the viscosity coefficient of the fractional-order damage element, and 0 < α < 1; c, are the total cohesion and total internal friction angle of the soil mass, respectively; σ3 is the confining pressure; G1 and η1 are the viscoelastic shear modulus and viscosity coefficient of the Kelvin (generalized) element, respectively; I n is the degree of freeze-thaw damage; e is the mathematical constant with a value of 2.71; Γ(α + 1) is the Gamma function of α.

[0038] The model of the present invention is used to determine the plastic strain state of subgrade soil under freeze-thaw damage conditions and to monitor and predict the subgrade settlement deformation under freeze-thaw damage conditions.

[0039] Example 2

[0040] Based on the model in Example 1, the present application provides its detailed derivation process as follows:

[0041] 1 Establishment of fractional-order damage model considering freeze-thaw damage effects

[0042] 1.1 Concepts of fractional calculus and strain types

[0043] The plastic strain curve of soil under cyclic loading can be divided into three types: the shakedown state curve, the critical state curve, and the failure state curve. As Figure 1 shown, where:

[0044] Shakedown state curve: When the yield stress (σ s ) < the maximum dynamic stress value (σ max ) ≤ the critical stress (σ cri ), the curve includes an instantaneous elastic strain stage, a viscoelastic strain stage, a decelerating plastic strain stage (the strain rate R PD from high to low), and an I - constant plastic strain stage (R PD ≈0);

[0045] Critical state curve: When σ max ≈σ cri , the curve includes an instantaneous elastic strain stage, a viscoelastic strain stage, a decelerating plastic strain stage, and a II - constant plastic strain stage (R PD >0, and R PD is close to a constant value);

[0046] Failure state curve: When σ max >σ cri , the curve includes an instantaneous elastic strain stage, a viscoelastic strain stage, a variable plastic strain stage, a II - constant plastic strain stage, and an accelerating plastic strain stage.

[0047] When dealing with nonlinear problems, fractional calculus can effectively make up for the inherent limitations of integer-order component models. Among them, the Riemann-Liouville model is particularly prominent due to its wide applicability in engineering problem modeling, and its specific definition is as follows:

[0048] If the function f is continuously integrable in any finite sub-interval within (0, +∞), then for t > 0 and Re(a) ≥ 0, it is defined as:

[0049]

[0050] Equation (1) is the R-L type fractional integral of the function f(t), where Γ is the Gamma function and a is the fractional order.

[0051] Performing the inverse operation on Equation (1) gives the fractional derivative Da equation of the function f(t) as:

[0052]

[0053] The constitutive relationship of the Abel dashpot is as follows:

[0054]

[0055] Where: σ(t) is the stress on the dashpot, ε(t) is the strain generated by the dashpot, t is the stress action time on the dashpot, η is the viscosity coefficient of the dashpot, and the meaning of a is the same as above.

[0056] Since the function f(t) is integrable when approaching t = 0, when a ∈ [0, 1], the Laplace transform of fractional calculus is:

[0057]

[0058] For a constant stress, according to the R-L type fractional theory, performing the forward and inverse Laplace transforms on Equation (3) using Equation (4) gives the strain expression of the Abel dashpot:

[0059]

[0060] When a = 0, the fractional Abel dashpot is a spring element, representing an ideal fluid; when a = 1, the fractional Abel dashpot is a Newtonian body element, representing an ideal solid.

[0061] 1.2 Combination of Kelvin (generalized) element and fractional damage element

[0062] 1.2.1 Elastic strain (one-dimensional) - Kelvin (generalized) element

[0063] Classical models such as the Maxwell model, Burgers model, and Kelvin model are strictly only applicable to describing the one-dimensional strain characteristics of materials under constant loads, and all show limitations in calculating three-dimensional strains under cyclic loads. Therefore, in the present invention, the cyclic load in the triaxial test is equivalently replaced with a constant load, and a strain calculation model in one dimension is derived. Then, the one-dimensional strain calculation model is extended to the three-dimensional case to achieve an accurate characterization of the plastic strain behavior of subgrade soil.

[0064] The functional expression of the vehicle load under the half-sine wave type is shown in Equation (6):

[0065]

[0066] In the formula: σ t is the cyclic stress value at time t, σ max is the maximum dynamic stress value, t1 is the load application time within each cycle period, T is the load period, and N load is the number of loading times.

[0067] According to the stress impulse equivalence principle, the areas enclosed by the cyclic load and the equivalent constant load σ0 respectively with the horizontal coordinate axis are equal. From this, the expression of σ0 can be obtained, as shown in Equation (7):

[0068]

[0069] Meanwhile, the present invention selects the Kelvin (generalized) element to describe the instantaneous elastic strain and viscoelastic strain stages, as Figure 3 shown. Among them, the creep compliance J(t) of the Kelvin (generalized) model can be expressed as:

[0070]

[0071] In the formula: J(t) is the creep compliance, E0 is the instantaneous elastic modulus, and E1 and η1 are respectively the viscoelastic modulus and viscosity coefficient of the Kelvin (generalized) element, and the meanings of other symbols are the same as above.

[0072] According to the Boltzmann superposition principle, the strain of a material at a certain moment is the sum of all load responses during the entire loading process. Therefore, if σ0 stops acting at the Figure 2 t1 moment shown, it is equivalent to applying a stress equal in value and opposite in direction to σ0 at this time. Then, the strain corresponding to time t after unloading is:

[0073]

[0074] The residual viscoelastic strain of the i-th half-sine cyclic load from the time [(N - i)T + t2] to the end of the N load -th half-sine cyclic load period is:

[0075]

[0076] Using the Boltzmann superposition principle for the second time, the total residual viscoelastic strain generated by a total of N cyclic loads is:

[0077]

[0078] To obtain an explicit analytical formula for Equation (11), construct the expression for the residual viscoelastic strain corresponding to the N load -th cyclic load:

[0079]

[0080] Considering the continuous change of N load , integrate Equation (12) (and add the limiting condition that the residual viscoelastic strain is 0 when N load = 0). On this basis, by combining Equation (7), the one-dimensional residual viscoelastic strain can be obtained as:

[0081]

[0082] 1.2.2 Plastic Strain (One-Dimensional) - Fractional Damage Element

[0083] N FT An increase will significantly deteriorate the structural integrity and mechanical properties of subgrade soil, manifested as a gradual decrease in soil stability and a continuous weakening of bearing capacity. When the freeze-thaw damage accumulates to a critical level, engineering disasters such as subgrade settlement, sliding, and even overall collapse are likely to occur, seriously threatening traffic safety. At the same time, existing research has shown that the peak strength is an important indicator for evaluating the stability and bearing capacity of subgrade soil. Therefore, based on the theory of continuous damage mechanics, this invention introduces a quantitative parameter for the degree of peak strength damage and defines the freeze-thaw damage variable D FT as:

[0084]

[0085] where I n is the degree of freeze-thaw damage, D FT is the freeze-thaw damage factor, S P(NFT＝0) is the peak strength when the number of freeze-thaw cycles is 0, and S P(NFT＝i) is the peak strength when the number of freeze-thaw cycles is i, which can be obtained from static triaxial tests.

[0086] Add the influence of the degree of freeze-thaw damage I n to the Abel dashpot and perform the forward and inverse Laplace transforms on its stress-strain relationship to obtain the strain expression of the damage element, specifically:

[0087]

[0088] In the formula: I n is the damage degree not affected by time t, and when D FT = 0, the expression degenerates into the strain calculation formula of the ideal fractional-order viscous pot model; σ max is the cyclic load form of the maximum dynamic stress; σ0 (σ0 = πσ max / 2) is the constant load form of the maximum dynamic stress; σ s is the cyclic load form of the yield stress; σ 0-s (σ 0-s = πσ s / 2) is the constant load form of the yield stress. When σ0 > σ 0-s the viscoplastic strain begins to occur; α and η2 are the fractional orders and viscosity coefficients of the fractional-order damage element, and 0 < α < 1; the meanings of other symbols are the same as above.

[0089] Furthermore, the viscous pot and the yield friction slider (the starting condition is σ0 > σ 0-s ) are connected in parallel to form a fractional-order damage element model, as specifically shown in Figure 4 . At the same time, it is noted that the yield friction slider is not triggered during the load intermittent period, that is, the actual action time of the cyclic load is N load t1. Combining with Equation (7), the one-dimensional viscoplastic strain can be obtained as:

[0090]

[0091] 1.2.3 Elastic-plastic strain (three-dimensional) - combined element

[0092] Research shows that the element model shows significant similarities in mathematical structure from one-dimensional to three-dimensional forms. Therefore, in the framework of the Kelvin (generalized) element theory, this invention replaces σ0 and E1 in the one-dimensional case of Equation (13) with the stress deviator tensor S ij and the viscoelastic shear modulus G1 in the three-dimensional case, and then derives the three-dimensional residual viscoelastic strain as:

[0093]

[0094] In the fractional-order damage element, due to the existence of the plastic friction plate, it cannot be directly analogized to the one-dimensional form. After introducing the yield function and the plastic potential function, the three-dimensional viscoplastic strain can be obtained as:

[0095]

[0096] In the formula: F is the yield function of the soil; F0 is the initial value of the soil yield function, F0 = 1; ψ is a power function (the power is usually taken as 1); Q is the plastic potential function.

[0097] Based on the associated flow rule, at this time, the yield function F is equal to the plastic potential function Q, so Equation (18) can be transformed into:

[0098]

[0099] The yield function adopts the Drucker-Prager yield criterion. Since it is smooth and has no cusp on the π plane and is easy to determine the elastoplastic constitutive relationship of soil, it is widely used. The expression under triaxial compression is:

[0100]

[0101] where: I1 is the first invariant of the stress tensor, J2 is the second invariant of the stress deviator tensor; c, are the total cohesion and total internal friction angle of the soil, respectively, which can be obtained from the static triaxial test.

[0102] In the dynamic triaxial test, the confining pressure σ3 is equal to the intermediate principal stress σ2, so there is:

[0103]

[0104]

[0105] The present invention analyzes with the first principal strain ε 11 Combining Equations (17), (19) to (24), the model combination expression of the Kelvin (generalized) element and the fractional-order damage element can be obtained, which can describe the stabilization and critical state of the plastic strain of subgrade soil. Specifically, it is shown in Equation (25):

[0106]

[0107] where: ε all-1 is the total plastic strain in the stabilized or critical state; ε RD is the viscoelastic strain; is the viscoplastic strain; σ max is the maximum dynamic stress value; σ3 is the confining pressure; G1, η1 are the viscoelastic shear modulus and viscosity coefficient of the Kelvin (generalized) element; N load is the number of loading times; α, η2 are the fractional-order order and viscosity coefficient of the fractional-order damage element; the meanings of other symbols are the same as above.

[0108] 1.3 Construction of the unsteady viscoplastic element

[0109] When σ max > σ criWhen the plastic strain transitions from the critical state to the failure state, it exhibits significant strain acceleration characteristics. In the ideal viscoplastic model, this process is manifested as the slip initiation of the friction element, which then causes the viscous pot element to bear the main load (as shown in Figure 5 ). As time increases, the material damage gradually intensifies. The continuous generation and expansion of cracks make the viscosity coefficient exhibit unsteady characteristics, that is, the viscosity coefficient decreases with time instead of remaining constant.

[0110] Therefore, the present invention constructs an unsteady viscoplastic element to describe the variation relationship of the viscosity coefficient of subgrade soil with time, specifically as shown in Equation (26):

[0111]

[0112] In the formula: η ae is the viscosity coefficient of the acceleration element; η3 is the initial value of the viscosity coefficient coefficient of the acceleration element; β is the acceleration index, reflecting the rate of accelerating creep; t is the loading time; t cri is the reference failure time, defined as the time corresponding to when the plastic strain enters the failure state from the critical state; the meanings of other symbols are the same as above.

[0113] Based on Equation (26), the strain calculation expression of the unsteady viscoplastic element in one-dimensional case after Laplace forward and inverse transformation can be obtained, specifically:

[0114]

[0115] Note that the plastic friction plate in the fractional-order mechanical element model does not trigger during the load intermittent period, that is, the actual action time of the cyclic load is N load t1. From Equations (7) and (27), the one-dimensional unsteady viscoplastic strain can be obtained, specifically as shown in Equation (28):

[0116]

[0117] In the formula: σ cri is the cyclic load form of the critical dynamic stress; σ 0-cri (σ 0-cri =πσ cri / 2) is the constant load form of the critical dynamic stress; the meanings of other symbols are the same as above.

[0118] Similarly, due to the existence of the plastic friction plate, the yield function and plastic potential function are introduced, and combined with the relevant flow rule and Drucker-Prager yield criterion (Equations (20) - (24)), the expression of the three-dimensional unsteady viscoplastic strain is obtained, specifically as shown in Equation (29):

[0119]

[0120] 1.4 Establishment of fractional-order damage model and analysis of model parameters

[0121] Combined with the Drucker-Prager yield criterion and Eqs. (25) and (29), the fractional-order damage model considering the influence of freeze-thaw damage can be obtained (as shown in Figure 6 ), which can describe the stable, critical and failure states of the plastic strain of subgrade soil, specifically as shown in Eq. (30):

[0122]

[0123] In the formula: ε all-2 is the total plastic strain in the failure state; is the accelerated viscoplastic strain; β and η3 are the acceleration index and the initial value of the viscosity coefficient of the unsteady viscoplastic element; N load-cri is the critical loading times, and the meanings of other symbols are the same as above.

[0124] As can be seen from the above, the core parameters of the fractional-order damage model are determined as the fractional-order a and the acceleration index β. To systematically analyze the influence of these parameters on the model calculation, in this invention, by the method of controlling variables, under the condition that other parameters remain unchanged, a and β are independently adjusted respectively, and the results are as shown in Figure 7 :

[0125] Figure 7 (a) shows the influence of the fractional-order a on the stable and critical state curves (other parameters are fixed values: σ3 = 0.03 MPa, σ d = 0.1 MPa, G1 = 19.32 MPa, η1 = 466.96 MPa·s, I n = 1, η2 = 0.73 MPa·s). It can be seen that as a increases, the strain rates in the deceleration, I-constant velocity and II-constant velocity plastic strain stages increase with the increase of the loading times N load , and gradually tend to a straight line with an upward slope. This means the key role of a in regulating the proportion of each stage of plastic strain (deceleration, I-constant velocity, II-constant velocity and accelerated plastic strain): the higher the value of a, the shorter the duration of the constant velocity stage, the model enters the accelerated stage earlier, and the range of N load required for the accelerated stage expands. From Figure 7 (b), it can be seen that when β = 1 (other parameters are fixed values: σ3 = 0.06 MPa, σ d = 0.3175 MPa, G1 = 14.3 MPa, η1 = 466.99 MPa·s, I n = 0.17, α = 0.27, η2 = 6.7 MPa·s, N load-cri = 5000, η3 = 2525.74 MPa·s), the strain and N loadThere is a linear correlation, and the strain rate should remain relatively constant. However, as the β value increases, the acceleration rate rises significantly, resulting in a more obvious acceleration effect and an earlier starting time of the accelerated plastic strain stage. This means that by regulating the β value in the unsteady viscoplastic element, this element can effectively cover the series combination of the Kelvin (generalized) element and the fractional-order damage element, and at the same time accurately reproduce the failure curve of subgrade clay under freeze-thaw cycles, thereby realizing the simulation of the entire process of plastic strain.

[0126] Figure 7 The regulation effects of a and β on the plastic strain curve in the fractional-order damage model were systematically analyzed. It can be seen that as a increases, the specimen gradually evolves from a viscoelastic state to a viscoelastic-plastic state, thus driving the plastic strain to transition from a stable state to a critical state. The increase in β further promotes the transition of the plastic strain from the critical state to the failure state, where the unsteady viscoplastic element realizes the precise regulation of the accelerated strain rate and the loading cycle range through β and N load-cri respectively. To sum up, the fractional-order damage model can accurately describe the elastic strain, deceleration, stage I-constant speed, stage II-constant speed, and accelerated plastic strain stages in plastic strain under stable, critical, and failure states. It not only provides theoretical support for the analysis of subgrade soil plastic strain but also demonstrates its potential value in engineering practice.

[0127] Example 3

[0128] Verification of the model formula of the present invention

[0129] Test materials and schemes

[0130] Test materials

[0131] The materials in this example are taken from the seasonal freezing region in the north.

[0132] According to the "Highway Geotechnical Test Regulations" (JTG 3430-2020), it is determined that for the clay materials in the seasonal freezing region in the north, its liquid limit is 48.2%, plastic limit is 21.4%, plasticity index is 26.8, maximum dry density is 1.86 g / cm3, optimum moisture content is 14.1%, specific gravity is 2.69, and the content of fine-grained components (particle size less than 0.075 mm) is 93.1%. It is classified as low liquid limit clay.

[0133] Considering the significant influence of vehicle dynamic loads on the roadbed and the survey data on the moisture level of the subgrade in the seasonal freezing region, the test soil samples in this invention are prepared under the optimum moisture content condition, and the dry density is uniformly controlled at 1.79 g / cm 3 .

[0134] The sample preparation process is as follows: First, the soil sample dried in an oven at 105°C is thoroughly mixed with water to achieve the preset moisture content and dry density. Subsequently, using the static compaction method, a cylindrical specimen with a diameter of 100 mm and a height of 200 mm is made in five layers; after each layer is compacted, the surface is roughened with a scraper to enhance the tightness of the interlayer bonding. After the preparation is completed, the specimen is wrapped and sealed with a plastic film and left standing for 24 hours for subsequent experiments.

[0135] Static triaxial strength performance test

[0136] The confining pressure range corresponding to common subgrade filling heights usually ranges between 30 and 100 kPa. Therefore, in this invention, 30 kPa, 60 kPa, and 90 kPa are selected as the test confining pressures to effectively characterize the subgrade performance under different filling heights. Considering the rapid overconsolidation stress history formed by cohesive soil under the influence of heavy compaction equipment during subgrade construction, its low permeability characteristics, and the action characteristics of environmental conditions and traffic loads during operation, the consolidation strain of cohesive soil often only increases under the instantaneous action of dynamic loads and then takes a long time to reach complete consolidation. Therefore, the undrained and unconsolidated loading is selected for the triaxial test, the loading rate is set at 0.8% / min, and the test is terminated when the axial strain of the specimen reaches 15%. For the determination of the peak strength, when the curve shows strain softening or stable characteristics, the peak deviator stress is taken as the peak strength; when the curve shows strain hardening characteristics, the deviator stress value corresponding to an axial strain of 15% is taken as the peak strength.

[0137] The freeze-thaw cycle test is carried out in a programmable constant temperature and humidity test chamber, and the temperature control range of this equipment is from -40°C to 60°C, and the temperature control accuracy reaches 0.1°C (as Figure 8 (a) shown). Referring to existing research, each freeze-thaw cycle includes freezing the specimen in an environment of -20°C for 12 hours, and then melting it at 20°C for 12 hours, and so on in a cycle. The test designed 0, 1, 3, 6, and 10 freeze-thaw cycles. After the specimen completes the specified number of cycles, it is taken out for subsequent loading tests, and the remaining specimens continue the freeze-thaw process. In addition, the specific scheme of the static triaxial test is shown in Table 1, and the test uses the Dynatriax100 / 14 static and dynamic triaxial test system of Changsha University of Science and Technology, as shown in Figure 8 (b).

[0138] Table 1 Static triaxial test scheme

[0139]

[0140] Dynamic triaxial stability performance test

[0141] Based on the peak strength data measured from the static triaxial tests, the present invention further conducted dynamic triaxial stability tests on plastic strain under different stress levels. The confining pressure and freeze-thaw conditions in the dynamic triaxial tests were kept consistent with those in the static triaxial tests to ensure the comparability of the results. The loading frequency was 1 Hz, the acting time was 0.2 s, the intermittent time was 0.8 s, and the load waveform was a half-sine wave. The test termination conditions were N load = 10000 times or the plastic strain reached 10%. Among them, the test was mainly divided into the following two steps:

[0142] Step A, determination of the dynamic critical failure point: That is, the dynamic critical failure point of the specimen was explored through the stress level (σ SL , defined as the ratio of the deviator stress to the peak strength of the static triaxial test). The specimen was initially loaded starting from 0.5σ SL . If the specimen became unstable and failed under 0.5σ SL , then the load was continued with a reduction of 0.1σ SL until the plastic strain curve was in the stable and critical state. Similarly, if the plastic strain curve was still in the stable and critical state under 0.5σ SL , then the load was continued with an increase of 0.1σ SL until the curve entered the failure state.

[0143] Step B, study on the evolution law of plastic strain: That is, after determining the dynamic critical failure point of the specimen, to explore the development law of plastic strain of subgrade soil under the stable and critical states, based on the critical dynamic stress value under the most unfavorable working conditions (σ3 = 30 kPa, N FT = 10), 80 kPa, 100 kPa, 120 kPa, and 140 kPa lower than this value were selected as the deviator stresses (σ d ) in the dynamic triaxial tests and dynamic triaxial tests with N load of 10000 times were carried out.

[0144] Analysis of test results and model verification

[0145] Verification of the fractional-order damage model

[0146] The applicability of the fractional-order damage model established in the present invention was verified by using the stable-type, critical-type, and failure-type plastic strain curves (classified according to σ FT ) under different σ3 (30 kPa, 60 kPa, 90 kPa) and different N d (0, 1, 3, 6, 10) conditions. At the same time, the model parameters were regressively fitted by the least squares method, as shown in Tables 2 and 3 specifically. Among them, I n and N load-criIt is obtained from indoor triaxial tests, while G1 and η1 in the Kelvin (generalized) element, α and η2 in the fractional-order damage element, and β and η3 in the unsteady viscoplastic element are obtained by regression fitting.

[0147] As can be seen from Table 2, as σ3 increases from 30 kPa to 90 kPa, the parameter G1 increases from 19.32 to 31.25, with an increase of about 61.75%, showing an obvious positive correlation, which indicates that the increase in σ3 effectively enhances G1 of the specimen. The parameter η1 hardly changes with the change of σ3, and the value is stable at about 467, reflecting the insensitivity of η1 to σ3. The parameter α generally does not show an obvious regularity with the increase of σ3, and its fluctuation range is between 0.13 and 0.66. The parameter η2 shows an upward trend with the increase of σ3 under the steady state (σ d = 100 kPa), and when σ3 increases from 30 kPa to 90 kPa, η2 increases from 0.73 to 10.27. The parameters β and η3 increase rapidly with the increase of σ3 under the failure state (σ d = 0.8σ SL ). It should be noted that at the same σ3, due to the small residual viscoelastic deformation generated by the Kelvin (generalized) element, the development of the plastic strain state is not sufficient to induce obvious changes in the parameters G1 and η1.

[0148] As can be seen from Table 3, the parameter G1 shows an obvious negative correlation with N FT . As N FT increases from 0 to 10, G1 gradually decreases from 25.17 to 14.34, with a decrease of about 43%. This is because the freeze-thaw cycle induces the weakening of the cementation between soil particles, resulting in the deterioration of G1. The parameter η1 changes slightly under different N FT , indicating that the instantaneous viscosity of the soil reflected by the Kelvin (generalized) element is not sensitive to N FT . The parameter α shows a certain degree of randomness during the accumulation process of N FT . At the same time, when the steady state enters the critical state, the parameter η2 gradually increases, but when the critical state enters the failure state, the parameter η2 begins to decrease. The parameters β and η3 decrease with the increase of N FT under the failure state.

[0149] Table 2The fitting results of model parameters under the condition of N FT = 0

[0150] Table 2The fitting results of model parameters under the condition of N FT = 0

[0151]

[0152] Table 3 Different N under the condition of σ3 = 60 kPa FT Fitting results of model parameters

[0153] Table 3 Fitting results of model parameters of different N FT under the condition of σ3 = 60 kPa

[0154]

[0155] The fitting calculation results of the fractional - order damage model under the working conditions shown in Table 2 and Table 3 are as follows Figure 9 Figure 10 shown. It can be found that the model established by the present invention is in good agreement with the test data. When the plastic strain state is in the shakedown and critical states, the strain curve mainly shows the stages of decelerating, stage Ⅰ - constant - velocity and stage Ⅱ - constant - velocity plastic strain. Compared with the Kelvin (generalized) model, the newly established model effectively improves the simulation accuracy of the non - linear behavior in the above - mentioned stages. When the plastic strain state is in the failure state, the strain curve shows an accelerating plastic strain stage. The fitting of the Kelvin (generalized) model has obvious deviations in the constant - velocity plastic strain stage, while the newly established model can better reflect the various development stages of plastic strain in the shakedown, critical and failure states.

[0156] Example 4

[0157] Based on Examples 1 - 3, the present invention provides a method for determining the plastic strain state of subgrade soil under freeze - thaw damage conditions. The method uses model formula 30 and includes the following steps

[0158] (1) Obtain the peak strength, total cohesion c, total internal friction angle critical loading times N load-cri and freeze - thaw damage degree I n of the subgrade soil through triaxial tests on the subgrade soil in the laboratory; at the same time, obtain the confining pressure σ3 through the subgrade fill height and the deviator stress σ through different vehicle loads d ; other model parameters are obtained by regression fitting

[0159] (2) Substitute the parameters obtained in step (1) into the model formula to obtain the relationship curve between the loading times and plastic strain under fixed deviator stress and confining pressure

[0160] (3) Based on the relationship curve between the loading times and plastic strain under fixed deviator stress and confining pressure obtained, in the stage Ⅰ - constant - velocity plastic strain of the shakedown state curve

[0161]

[0162] In the formula: R PD is the plastic strain rate, and ε PD-i+n is the plastic strain value at the (i + n)-th time, and ε PD-i is the plastic strain value at the i-th time. n is the difference in the number of loading times between two plastic strain values;

[0163] When the total number of loading times is 10,000 times, i + n = 10,000, n / (i + n) = 0.4, i = 6,000 times, and n = 4,000 times; if the plastic strain reaches 10% before the total number of loading times reaches 10,000 times, here, for example, i + n = 5,000 times is taken as the final number of times, then n / (i + n) = 0.4, i = 3,000 times, and n = 2,000 times.

[0164] When R PD ≈ 0, the plastic strain is in the stage of Ⅰ - constant-rate plastic strain;

[0165] When R PD > 0 and the value of R PD is close to a constant value, the plastic strain is in the stage of Ⅱ - constant-rate plastic strain;

[0166] When R PD > 0 and the value of R PD increases non-linearly, the plastic strain is in the stage of accelerating plastic strain;

[0167] On this basis, looking at the entire plastic strain curve, if there is only the stage of Ⅰ - constant-rate plastic strain, the subgrade is in a stable state;

[0168] If there are both the stage of Ⅰ - constant-rate plastic strain and the stage of Ⅱ - constant-rate plastic strain, or there is only the stage of Ⅱ - constant-rate plastic strain, the subgrade is in a critical state;

[0169] If there is a stage of accelerating plastic strain, or when the plastic strain is greater than or equal to 10%, the subgrade is in a failure state.

[0170] Taking the clay in the seasonal frozen soil area in the north as an example, after it has experienced 10 freeze-thaw cycles, its peak strength is 529.2 kPa, the cohesion is 76.8 kPa, the internal friction angle is 38.6°, and the subgrade fill height is 3 m (corresponding to a confining pressure of about 60 kPa).

[0171] Using the model formula 30, the relationship curve between the number of loading times and the plastic strain under a fixed deviator stress and confining pressure is calculated, as shown in Figure 11 shown.

[0172] Its plastic strain is in a stable state under the 0.2646 MPa traffic load (deviator stress), so there is no need to treat the subgrade soil, but attention should be paid to the settlement deformation of the subgrade during the daily maintenance of the road.

[0173] The plastic strain is in a critical state under the traffic load (deviator stress) between 0.2646 MPa and 0.3175 MPa, that is, the subgrade will not reach the failure standard in the short term. Therefore, the corresponding measures are that the subgrade soil must be strengthened or replaced within the predicted road operation time, or the traffic load should be restricted to ensure road safety.

[0174] The plastic strain is in a failure state under the traffic load (deviator stress) of 0.3175 MPa and above, that is, the subgrade will reach the failure standard in the short term. Therefore, the subgrade soil should be strengthened or replaced in time, or the traffic load (i.e., height and weight limits) should be restricted to ensure road safety.

[0175] The above are only the preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention are included in the protection scope of the present invention.

Claims

1. A fractional-order damage model of plastic strain for subgrade soil considering freeze-thaw damage, characterized in that The model is as follows: where: ε all-2 is the total plastic strain under the failure state; ε RD is the viscoelastic strain; is the viscoplastic strain; is the accelerated viscoplastic strain; β and η3 are the acceleration index and the initial value of the viscosity coefficient of the unsteady viscoplastic element; N load-cri is the critical number of loading cycles; σ max is the maximum dynamic stress value; t1 is the load application time within each cycle period; T is the load cycle; N load is the number of loading times; α and η2 are the fractional orders and viscosity coefficients of the fractional damage element, and 0 < α < 1; c and are the total cohesion and total internal friction angle of the soil mass respectively. σ3 is the confining pressure; G1 and η1 are the viscoelastic shear modulus and viscosity coefficient of the Kelvin (generalized) element; I n is the degree of freeze-thaw damage; e is a mathematical constant with a value of 2.71; Γ(α + 1) is the Gamma function of α.

2. Use of the fractional-order damage model for plastic strain of subgrade soil considering freeze-thaw damage according to claim 1, characterized in that It is used to determine the plastic strain state of subgrade soil under freeze-thaw damage conditions and to monitor and predict the subgrade settlement deformation under freeze-thaw damage conditions.

3. A method for determining the plastic strain state of subgrade soil under freeze-thaw damage conditions, characterized in that, Using the model described in claim 1, it includes the following steps: (1) The peak strength, total cohesion c, total internal friction angle φ, critical loading times N of the subgrade soil are obtained by conducting triaxial tests on the subgrade soil indoors load-cri and the degree of freeze-thaw damage I n ; meanwhile, the confining pressure σ3 is obtained from the fill height of the subgrade and the deviator stress σ is obtained from different vehicle loads d ; other model parameters are obtained by regression fitting. (2) Substitute the parameters obtained in step (1) into the model formula to obtain the relationship curve between the number of loadings and plastic strain under fixed deviator stress and confining pressure; (3) Based on the obtained relationship curve between the number of loadings and plastic strain under fixed deviator stress and confining pressure, when: When R PD ≈ 0, the plastic strain is in the stage of stage I - constant velocity plastic strain; When R PD > 0 and R PD is close to a constant value, the plastic strain is in the stage of II - constant - velocity plastic strain; When R PD > 0 and R PD has a non-linear growth in value, the plastic strain is in the stage of accelerating plastic strain; On this basis, looking at the entire plastic strain curve, if there is only the Ⅰ - constant velocity plastic strain stage, the subgrade is in a stable state; If there are both the Ⅰ - constant velocity plastic strain stage and the Ⅱ - constant velocity plastic strain stage, or there is only the Ⅱ - constant velocity plastic strain stage, then the subgrade is in a critical state; If there is an accelerating plastic strain stage, or when the plastic strain is greater than or equal to 10%, the subgrade is in a failure state.

4. The method for determining the plastic strain state of subgrade soil under freeze-thaw damage conditions according to claim 3, wherein Where: R PD is the plastic strain rate, ε PD-i+n is the plastic strain value at the (i + n)-th time, ε PD-i is the plastic strain value at the i-th time, and n is the difference in the number of loading times between two plastic strain values.

5. The method for determining the plastic strain state of subgrade soil under freeze-thaw damage conditions according to claim 3, wherein When it is determined that the subgrade is in a critical state, the relationship curve between the number of loadings and plastic strain under fixed deviator stress and confining pressure can be used to predict how many times the cyclic load reaches the failure standard when the subgrade will reach it. However, if it will not in the short term, it is necessary to optimize the subgrade soil within the expected operation time or restrict the vehicle load.

6. The method for determining the plastic strain state of subgrade soil under freeze-thaw damage conditions according to claim 3, characterized in that When it is determined that the subgrade is in a failure state, it is necessary to reinforce, replace, etc. the subgrade, or restrict the vehicle load.

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