Asphalt mixture fractional order viscoelastic model finite element numerical value implementation method

Abstract

The invention discloses an asphalt mixture fractional order viscoelastic model finite element numerical value implementation method, and belongs to the field of constitutive behaviors of asphalt mixtures. The method aims at solving the problem that an existing finite element method of a fractional order viscoelastic model cannot give consideration to calculation efficiency and precision. Comprising the following steps: acquiring a complex frequency domain modulus expression of a fractional order viscoelastic model based on Laplace transformation; through time domain / complex frequency domain equivalent substitution, constructing a one-dimensional asphalt mixture fractional order viscoelastic model stress strain lowest order differential expression; on the basis of a fractional derivative item in a non-classical method discrete stress-strain differential form, constructing an asphalt mixture fractional viscoelastic model stress updating expression in a discrete form; constructing a three-dimensional fractional order viscoelastic model stress updating expression and a Jacobian matrix updating expression based on a normal / non-normal Poisson's ratio hypothesis and an incremental iteration method; and writing a user subprogram UMAT for finite element numerical calculation. The method is used for realizing the finite element numerical value of the fractional order viscoelastic model.

Description

Technical Field

[0001] The invention relates to a finite element numerical realization method of a fractional-order viscoelastic model of asphalt mixture, and belongs to the field of constitutive behavior of asphalt mixture. Background Art

[0002] As a typical artificial material, asphalt mixture is one of the main materials for paving asphalt pavement. Asphalt pavement is subjected to environmental and vehicle effects during service, resulting in cracks and rutting. Mechanical analysis of asphalt pavement is the main means to carry out asphalt pavement structure design and ensure its service performance. For mechanical analysis, the material constitutive model is the main factor affecting the accuracy of mechanical analysis. When the internal strain of asphalt mixture is less than 150uε or the cumulative strain is less than 5000uε, the viscoelastic model can well characterize its stress-strain behavior.

[0003] At present, viscoelastic models are mainly divided into two categories: integer-order models and fractional-order models. Integer-order models are formed by series-parallel connection of spring elements and linear viscoelastic pots, mainly including Maxwell model, Kelvin model, standard linear solid model, Burgers model, generalized Maxwell model and generalized Kelvin model. Integer-order models usually have explicit creep or relaxation modulus expressions, which are equivalent to Prony series. In finite element analysis, with the help of the multiplication property of exponential function, the explicit expressions required for updating the stress and Jacobian matrix of integer-order models can be derived. However, for asphalt mixtures, due to the wide range of its relaxation spectrum, more springs or viscoelastic pot elements are required in integer-order models, so that the parameter identification of integer-order models requires solving ill-conditioned equations, which ultimately leads to the identified model parameters having no physical meaning. In addition, integer-order models are easily affected by noise during parameter identification, which leads to the occurrence of overfitting.

[0004] By replacing the linear viscoelastic elements in the integer-order viscoelastic model with parabolic viscoelastic or Abelian viscoelastic, a fractional-order viscoelastic model can be constructed. Studies have shown that parabolic viscoelastic and Abelian viscoelastic are mathematically equivalent. Compared with integer-order models, fractional-order viscoelastic models can use fewer parameters to characterize the viscoelastic behavior of asphalt mixtures in a wide frequency domain. Typical fractional-order viscoelastic models include FMM, FZM, FBM, MFZM, S2P1DM, 2S3PM, etc. Among the above models, the MFZM model has a moderate number of elements and a wide frequency domain representation range, and is recommended for mechanical analysis of asphalt pavements at different temperatures. However, due to the lack of simple creep or relaxation expressions in fractional-order models, it is usually difficult to use them directly for finite element analysis. Although some researchers use interactive conversion methods or analytical approximation methods to convert the complex modulus of fractional-order viscoelastic models into relaxation moduli, and then use the collocation method or the improved Dombi method to expand the relaxation modulus into an exponential function, so that finite element calculations can be carried out using exponential algorithms. However, in the calculation process of the above method, solving the indeterminate equation system or high-order equation requires special processing to ensure that the parameters are positive real numbers.

[0005] In order to apply fractional viscoelastic models to finite element analysis, researchers have proposed various numerical calculation methods. At present, the Grünwald–Letnikov (GL) method is widely used. The GL method does not require explicit creep or relaxation expressions. It uses Grünwald coefficients to numerically discretize the fractional viscoelastic model in differential form at equal intervals on the time scale, so that the stress at the current moment is expressed as a function of the stress or strain at the previous moment. Since the GL method uses equal intervals for discretization on the time scale, a fixed increment step is required during analysis, and the time increment should not be too large. When conducting long-term creep analysis or mechanical analysis under variable temperature, it is difficult to balance computational efficiency and accuracy. In addition, the GL method needs to store unit historical stress and strain, resulting in a large amount of memory required for calculation. In addition to the GL method, some researchers use the Mittag Leffler (ML) function to characterize the relaxation modulus of the fractional viscoelastic model based on the Laplace transform, thereby constructing the stress update algorithm required for finite element calculation. The calculation method based on ML function can use automatic increment step and only needs to store the stress and strain at the current moment, so the memory required for calculation can be significantly reduced. However, the calculation process of expressing the relaxation modulus of asphalt mixture as ML function is cumbersome. In addition, the calculation of ML function is relatively complicated, usually using the approximate formula composed of series form or gamma function, which affects the solution accuracy and calculation efficiency of the model.

[0006] Therefore, it is urgent to propose a finite element numerical implementation method for the fractional viscoelastic model of asphalt mixture with high calculation accuracy and efficiency, low memory occupancy and simple calculation, so as to promote the application of fractional viscoelastic model in pavement mechanics analysis. Summary of the invention

[0007] In view of the problem that the existing finite element method of the fractional-order viscoelastic model cannot take into account both the calculation efficiency and accuracy, the present invention provides a finite element numerical implementation method of the fractional-order viscoelastic model of asphalt mixture.

[0008] A finite element numerical implementation method of a fractional-order viscoelastic model of asphalt mixture according to the present invention comprises:

[0009] Based on the stress-strain relationship and Laplace transform of each sub-element in the fractional-order viscoelastic model, the relationship between the total stress and total strain of the model in the complex frequency domain is established;

[0010] Introducing the equivalent substitution of time domain and complex frequency domain derivatives, constructing the lowest-order differential expression of the one-dimensional form of the total stress and total strain relationship of the model;

[0011] The non-classical method is used to discretize the fractional derivative terms in the lowest-order differential expression and construct the discrete form model stress update expression.

[0012] Combining the constant Poisson's ratio assumption or the non-extraordinary Poisson's ratio assumption with the incremental iteration method, the stress update expression and Jacobian matrix update expression of the three-dimensional model are constructed for finite element numerical calculation.

[0013] According to the finite element numerical realization method of the fractional-order viscoelastic model of asphalt mixture of the present invention, based on the stress update expression and the Jacobian matrix update expression of the three-dimensional model, the subroutine development interface in ABAQUS is utilized, and the user material subroutine UMAT of the fractional-order viscoelastic model is developed by FORTRAN; a finite element geometric model is created in ABAQUS, and after meshing, the material parameters, loads and boundary conditions of the fractional-order viscoelastic model of the asphalt mixture are input, and the user material subroutine UMAT is called to realize the finite element numerical calculation of the fractional-order viscoelastic model of the asphalt mixture.

[0014] According to the finite element numerical realization method of the fractional viscoelastic model of asphalt mixture of the present invention, the fractional viscoelastic model is selected as a modified fractional Zener model, and the sub-elements include spring element 1, spring element 2, fractional Abel viscoelastic element 1 and fractional Abel viscoelastic element 2;

[0015] The relationship between the total stress and total strain of the model in the complex frequency domain is established as:

[0016]

[0017] In the formula is the complex frequency domain modulus, E 1 is the modulus of spring element 1, η 1 is the viscosity of the fractional Abel viscosity element, η 2is the viscosity of the fractional Abel viscosity pot element 2, s is the independent variable in the complex frequency domain, α 1 is the fractional order of the fractional Abel viscosity element, α 2 is the fractional order of the fractional Abel viscosity element 2, 0<α 1 <α 2 <1,E 2 is the modulus value of spring element 2.

[0018] According to the finite element numerical realization method of the asphalt mixture fractional viscoelastic model of the present invention, an equivalent substitution of a derivative in the time domain or complex frequency domain with an initial value of zero is introduced:

[0019]

[0020] Where L is the Laplace transform, D t is the symbol of fractional derivative, α is the fractional order, z(t) is the function to be differentiated, and t is time;

[0021] The lowest order differential expression in one-dimensional form is:

[0022]

[0023] Where ε(t) is the total strain and σ(t) is the total stress.

[0024] According to the finite element numerical implementation method of the asphalt mixture fractional viscoelastic model of the present invention, the calculation method of the fractional derivative term in the lowest-order differential expression is:

[0025]

[0026] Where t n represents the time corresponding to the nth incremental step, J is the total number of Gauss Jacobian quadrature nodes, A j 、f j 、s j is the intermediate coefficient, Δt is the incremental step size of adjacent time points, is the abscissa of the jth Gauss Jacobian quadrature node, is the weight of the Gauss Jacobian j-th quadrature coefficient;

[0027] The stress update expression of the discrete modified fractional-order Zener model is constructed as follows:

[0028]

[0029] Where B 1 , B 2 、h 1 , B 3 、h 2 are all intermediate variables, Δε is the strain increment, among which:

[0030]

[0031] Where χ is the function independent variable of the intermediate variable.

[0032] According to the finite element numerical realization method of the asphalt mixture fractional viscoelastic model of the present invention,

[0033] Using the constant Poisson's ratio or the non-constant Poisson's ratio assumption, the coefficients in formula (5) are transformed into three dimensions;

[0034] If a constant Poisson's ratio is used, we get:

[0035]

[0036] If the extraordinary Poisson's ratio assumption is adopted, the Directly measured by triaxial test;

[0037] Where K is a subscript, indicating that the corresponding variable is related to volumetric strain or volumetric stress; G is a subscript, indicating that the corresponding variable is related to deviatoric strain or deviatoric stress; ν is the constant Poisson's ratio;

[0038] Based on the incremental iteration method in ABAQUS, the updated expressions of normal stress, shear stress and Jacobian matrix of the three-dimensional modified fractional-order Zener model are obtained by combining formula (6) and formula (5).

[0039] Beneficial effects of the present invention: The method of the present invention first obtains the complex frequency domain modulus expression of the asphalt mixture fractional viscoelastic model based on Laplace transform; then, through time domain / complex frequency domain equivalent substitution, the lowest order differential expression of stress and strain of the asphalt mixture fractional viscoelastic model in one-dimensional form is constructed; on this basis, the fractional derivative terms in the differential form of stress and strain are discretized based on the non-classical method (non-classical method), so as to construct the discrete form of the asphalt mixture fractional viscoelastic model stress update expression; thereafter, based on the constant / inconstant Poisson's ratio assumption and the incremental iteration method, the three-dimensional asphalt mixture fractional viscoelastic model stress update expression and Jacobian matrix expression are derived; finally, based on the subroutine development interface in ABAQUS, the user material subroutine (User Material Subroutine, UMAT) is developed using FORTRAN to realize the finite element numerical calculation of the asphalt mixture fractional viscoelastic model. The method of the present invention supports variable incremental step size, has the characteristics of step size adaptation and low memory occupancy, and can realize the finite element numerical calculation of the fractional viscoelastic model.

[0040] One of the traditional finite element numerical implementation methods of fractional viscoelastic models is the Grünwald–Letnikov method. It uses a fixed increment size and needs to store all stress and strain histories at the unit integration points during calculation, resulting in high computational cost and high memory usage, making it unsuitable for long-term creep analysis. The method of the present invention uses a variable increment size and only needs to store the stress and strain history in the previous increment during the calculation process, which has high computational efficiency and low memory usage.

[0041] The second method for realizing the finite element numerical implementation of the traditional fractional viscoelastic model is the Mittag-Leffler function method. It is applicable to simple fractional viscoelastic models whose time domain creep compliance or relaxation modulus can be expressed by the Mittag-Leffler function. For complex fractional viscoelastic models, it is difficult to derive the time domain expression of its creep compliance or relaxation modulus. In addition, it needs to use the approximate calculation formula of the Mittag-Leffler function when calculating, which has a large amount of calculation and the calculation accuracy is affected by the approximate calculation formula. The method of the present invention adopts the constitutive equation in differential form, which does not need to derive the time domain expression of the model creep compliance or relaxation modulus, and can be applied to the finite element numerical calculation of complex fractional viscoelastic models.

[0042] The method of the present invention is universal, has a unified solution step and calculation format for different fractional-order models, and is more convenient for finite element implementation. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a flow chart of the finite element numerical realization method of the asphalt mixture fractional viscoelastic model of the present invention;

[0044] Figure 2 It is the flow chart of UMAT development of fractional viscoelastic model of asphalt mixture;

[0045] Figure 3 It is a schematic diagram of the modified fractional-order Zener model;

[0046] Figure 4 This is a schematic diagram of uniaxial loading of the ABAQUS finite element model;

[0047] Figure 5 is the ε of the modified fractional Zener model kl Creep curve;

[0048] Figure 6 is the ε of the modified fractional Zener model 22 Creep curve. DETAILED DESCRIPTION

[0049] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0050] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0051] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0052] Specific implementation method 1. Combination Figures 1 to 3 As shown, the present invention provides a finite element numerical implementation method of a fractional-order viscoelastic model of asphalt mixture, comprising:

[0053] Based on the stress-strain relationship and Laplace transform of each sub-element in the fractional-order viscoelastic model, the relationship between the total stress and total strain of the model in the complex frequency domain is established;

[0054] Simplifying the relationship between the total stress and total strain of the model in the complex frequency domain, and introducing the equivalent substitution of the derivatives in the time domain and the complex frequency domain to construct the lowest-order differential expression of the one-dimensional form of the relationship between the total stress and total strain of the model;

[0055] The non-classical method is used to discretize the fractional derivative terms in the lowest-order differential expression and construct the discrete form model stress update expression.

[0056] Combining the constant Poisson's ratio assumption or the non-extraordinary Poisson's ratio assumption with the incremental iteration method, the stress update expression and Jacobian matrix update expression of the three-dimensional model are derived and constructed for finite element numerical calculation.

[0057] Further, combined with Figure 1 As shown in the figure, based on the stress update expression and Jacobian matrix update expression of the three-dimensional model, the subroutine development interface in ABAQUS is used, and the user material subroutine UMAT (User Material Subroutine) of the fractional-order viscoelastic model is developed in FORTRAN; a finite element geometric model is created in ABAQUS, and the material parameters, loads and boundary conditions of the asphalt mixture fractional-order viscoelastic model are input after meshing, and the user material subroutine UMAT is called to realize the finite element numerical calculation of the asphalt mixture fractional-order viscoelastic model.

[0058] As an example, combining Figure 3As shown, in this embodiment, the fractional-order viscoelastic model is selected as a modified fractional-order Zener model, and the sub-elements include spring element 1, spring element 2, fractional-order Abel viscoelastic element 1 and fractional-order Abel viscoelastic element 2;

[0059] The relationship between the total stress and total strain of the model in the complex frequency domain is established as:

[0060]

[0061] In the formula is the complex frequency domain modulus, E 1 is the modulus of spring element 1, η 1 is the viscosity of the fractional Abel viscosity element, η 2 is the viscosity of the fractional Abel viscosity pot element 2, s is the independent variable in the complex frequency domain, α 1 is the fractional order of the fractional Abel viscosity element, α 2 is the fractional order of the fractional Abel viscosity element 2, 0<α 1 <α 2 <1,E 2 is the modulus value of spring element 2.

[0062] In this implementation, an equivalent substitution is introduced in the time domain or complex frequency domain with an initial value of zero:

[0063]

[0064] Where L is the Laplace transform, D t is the symbol of fractional derivative, α is the fractional order, z(t) is the function to be differentiated, and t is time;

[0065] The lowest order differential expression in one-dimensional form is:

[0066]

[0067] Where ε(t) is the total strain and σ(t) is the total stress.

[0068] For example, Represents α with respect to time 2 -α 1 Order and α 2 Derivative operator.

[0069] The calculation method of the fractional derivative term in the lowest-order differential expression is:

[0070]

[0071] Where t n represents the time corresponding to the nth incremental step, J is the total number of Gauss Jacobian quadrature nodes, Aj 、f j 、s j is the intermediate coefficient, Δt is the incremental step size of adjacent time points, Δt=t n -t n-1 ; is the abscissa of the jth Gauss Jacobian quadrature node, is the weight of the Gauss Jacobi jth quadrature coefficient, It is related to the fractional order;

[0072] The stress update expression of the discrete modified fractional-order Zener model is constructed as follows:

[0073]

[0074] Where B 1 , B 2 、h 1 , B 3 、h 2 are all intermediate variables, Δε is the strain increment, among which:

[0075]

[0076] Where χ is the function independent variable of the intermediate variable.

[0077] Furthermore, the coefficients in formula (5) are transformed into three dimensions using the constant Poisson's ratio or the non-constant Poisson's ratio assumption;

[0078] If a constant Poisson's ratio is used, we get:

[0079]

[0080] If the extraordinary Poisson's ratio assumption is adopted, the Directly measured by triaxial test;

[0081] Where K is a subscript, indicating that the corresponding variable is related to volumetric strain or volumetric stress; G is a subscript, indicating that the corresponding variable is related to deviatoric strain or deviatoric stress; ν is the constant Poisson's ratio;

[0082] Based on the incremental iteration method in ABAQUS, the updated expressions of normal stress, shear stress and Jacobian matrix of the three-dimensional modified fractional-order Zener model are obtained by combining formula (6) and formula (5).

[0083] As an example, the Poisson's ratio v=0.3.

[0084] Going further, the normal stress update expression is:

[0085]

[0086] Where oo represents the normal stress component in different directions, and the values ​​of oo are 11, 22, and 33; is the average principal strain, is the average principal strain increment, Δε oo is the strain increment in the oo direction, is the mean principal stress, is the deviator stress in the oo direction, is the average principal stress increment, is the deviatoric stress increment in the oo direction, g 1 and g 2 For the intermediate variable:

[0087]

[0088] In the formula is the deviatoric strain in the oo direction, is the deviatoric strain increment in oo direction;

[0089]

[0090] The updated expression of shear stress is:

[0091]

[0092] Where σ kl is the shear stress in the kl direction, and the values ​​of kl are 12, 13, and 23; ε kl is the shear strain in the kl direction, Δε kl is the shear strain increment in the kl direction,

[0093] g 3 and g 4 For the intermediate variable:

[0094]

[0095] The update expression of the Jacobian matrix is:

[0096]

[0097] In the formula is the stress increment in the pq direction in the n+1th incremental step, and the values ​​of pq are 11, 22, 33, 12, 13, and 23; is the strain increment in the uv direction of the n+1th incremental step, and the uv values ​​are 11, 22, 33, 12, 13, 23; E K and E G For the intermediate variable:

[0098]

[0099] Going further, the state update variables in the UMAT subroutine are:

[0100]

[0101] Input material parameters according to Table 1 into the user material subroutine UMAT, create a finite element geometry model using ABAQUS, use the three-dimensional 8-node reduced integration element C3D8R to discretize the finite element geometry model, fix the normal displacement of the back node of the three-dimensional 8-node reduced integration element in the x direction, and apply the stress load σ in the x direction 11 ; Set the time increment size as Figure 5 and Figure 6 As shown in the figure, that is, Δt = 0.1s, Δt = 0.01s and 0.01≤Δt≤2s, the creep curve of the finite element model in the x direction is:

[0102] σ 11 =σ 0 (t(U(t)-U(tt 0 ))+U(tt 0 )).

[0103] Where σ 0 is the stress load amplitude, and U is the unit step function.

[0104] Table 1 MFZM model material parameters

[0105]

[0106] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the present invention. It should therefore be understood that many modifications may be made to the exemplary embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in a manner different from that described in the original claims. It should also be understood that the features described in conjunction with a single embodiment may be used in other described embodiments.

Claims

1. Finite element numerical implementation method of fractional viscoelastic model of asphalt mixture, characterized by: include, Based on the stress-strain relationship and Laplace transform of each sub-element in the fractional-order viscoelastic model, the total stress and total strain relationship of the model in the complex frequency domain is established; Introducing the equivalent substitution of time domain and complex frequency domain derivatives, constructing the lowest-order differential expression of the one-dimensional form of the total stress and total strain relationship of the model; The non-classical method is used to discretize the fractional derivative terms in the lowest-order differential expression and construct the discrete form model stress update expression; Combining the constant Poisson's ratio assumption or the non-extraordinary Poisson's ratio assumption with the incremental iteration method, the stress update expression and Jacobian matrix update expression of the three-dimensional model are constructed for finite element numerical calculation.

2. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 1 is characterized in that: Based on the stress update expression and Jacobian matrix update expression of the three-dimensional model, the subroutine development interface in ABAQUS was used, and the user material subroutine UMAT of the fractional-order viscoelastic model was developed in FORTRAN. A finite element geometric model was created in ABAQUS, and the material parameters, loads and boundary conditions of the asphalt mixture fractional-order viscoelastic model were input after meshing. The user material subroutine UMAT was called to realize the finite element numerical calculation of the asphalt mixture fractional-order viscoelastic model.

3. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 2 is characterized in that: Selecting the fractional-order viscoelastic model as a modified fractional-order Zener model, wherein the sub-elements include a spring element 1, a spring element 2, a fractional-order Abel viscoelastic element 1, and a fractional-order Abel viscoelastic element 2; The relationship between the total stress and total strain of the model in the complex frequency domain is established as: In the formula is the complex frequency domain modulus, E1 is the modulus value of spring element one, η1 is the viscosity of fractional-order Abel viscosity pot element one, η2 is the viscosity of fractional-order Abel viscosity pot element two, s is the complex frequency domain independent variable, α1 is the fractional order of fractional-order Abel viscosity pot element one, α2 is the fractional order of fractional-order Abel viscosity pot element two, 0<α1<α2<1, E2 is the modulus value of spring element two.

4. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 3 is characterized in that: Introduce an equivalent substitution in the time domain or complex frequency domain with an initial value of zero: Where L is the Laplace transform, D t is the symbol of fractional derivative, α is the fractional order, z(t) is the function to be differentiated, and t is time; The lowest order differential expression in one-dimensional form is: Where ε(t) is the total strain and σ(t) is the total stress.

5. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 4 is characterized in that: The calculation method of the fractional derivative term in the lowest-order differential expression is: Where t n represents the time corresponding to the nth incremental step, J is the total number of Gauss Jacobian quadrature nodes, A j 、f j 、s j is the intermediate coefficient, Δt is the incremental step size of adjacent time points, is the abscissa of the jth Gauss Jacobian quadrature node, is the weight of the Gauss Jacobian j-th quadrature coefficient; The stress update expression of the discrete modified fractional-order Zener model is constructed as follows: Where B1, B2, h1, B3, h2 are all intermediate variables, Δε is the strain increment, where: Where χ is the function independent variable of the intermediate variable.

6. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 5 is characterized in that: Using the constant Poisson's ratio or the non-constant Poisson's ratio assumption, the coefficients in formula (5) are transformed into three dimensions; If a constant Poisson's ratio is used, we get: If the extraordinary Poisson's ratio assumption is adopted, the and Directly measured by triaxial test; Where K is a subscript, indicating that the corresponding variable is related to volumetric strain or volumetric stress; G is a subscript, indicating that the corresponding variable is related to deviatoric strain or deviatoric stress; ν is the constant Poisson's ratio. Based on the incremental iteration method in ABAQUS, the updated expressions of the normal stress, shear stress and Jacobian matrix of the three-dimensional modified fractional-order Zener model are obtained by combining formula (6) and formula (5).

7. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 6 is characterized in that: The normal stress update expression is: Where oo represents the normal stress component in different directions, and the values ​​of oo are 11, 22, and 33; is the average principal strain, is the average principal strain increment, Δε oo is the strain increment in the oo direction, is the mean principal stress, is the deviator stress in the oo direction, is the average principal stress increment, is the deviatoric stress increment in the oo direction, g1 and g2 are intermediate variables: In the formula is the deviatoric strain in the oo direction, is the deviatoric strain increment in oo direction; 8. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 7 is characterized in that: The updated expression of shear stress is: s kl (t n )=g3(t n ,2e kl ,2No kl )+g4(t n ,s kl )(8), Where σ kl is the shear stress in the kl direction, and the values ​​of kl are 12, 13, and 23; ε kl is the shear strain in the kl direction, Δε kl is the shear strain increment in the kl direction, g3 and g4 are intermediate variables:

9. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 8 is characterized in that: The update expression of the Jacobian matrix is: In the formula is the stress increment in the pq direction in the n+1th incremental step, and the values ​​of pq are 11, 22, 33, 12, 13, and 23; is the strain increment in the uv direction of the n+1th incremental step, and the uv values ​​are 11, 22, 33, 12, 13, 23; E K and E G For the intermediate variable:

10. The finite element numerical implementation method of the asphalt mixture fractional viscoelastic model according to claim 9, wherein The characteristic is that the state update variable in the UMAT subroutine is: Input material parameters in the user material subroutine UMAT, create a finite element geometry model using ABAQUS, select three-dimensional solid elements to discretize the finite element geometry model, and apply loads and boundary conditions to the discretized finite element geometry model; After setting the incremental step, submit the calculation to obtain the mechanical response results of the finite element model.

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