Finite element numerical implementation method of asphalt mixture fractional viscoelastic model

Abstract

The application relates to a finite element numerical implementation method of an asphalt mixture fractional viscoelastic model, and belongs to the field of constitutive behaviors of asphalt mixtures. The application aims at the problem that the existing finite element method of the fractional viscoelastic model cannot consider both the calculation efficiency and the accuracy. The application comprises the following steps: obtaining a complex frequency domain modulus expression of the fractional viscoelastic model based on Laplace transformation; constructing a one-dimensional form of a stress-strain minimum-order differential expression of the asphalt mixture fractional viscoelastic model through time domain / complex frequency domain equivalent substitution; constructing a discrete form of a stress updating expression of the asphalt mixture fractional viscoelastic model based on non-classical method discrete fractional derivative terms in the stress-strain differential form; constructing a three-dimensional form of stress updating expressions and Jacobian matrix updating expressions of the fractional viscoelastic model based on common / abnormal Poisson ratio assumptions and an incremental iteration method; and writing a user subroutine UMAT for finite element numerical calculation. The application is used for finite element numerical implementation of the fractional viscoelastic model.

Description

TECHNICAL FIELD

[0001] The present application relates to a finite element numerical implementation method of a fractional viscoelastic model of asphalt mixture, and belongs to the field of constitutive behavior of asphalt mixture. BACKGROUND

[0002] As a typical artificial material, asphalt mixture is one of the main materials for paving asphalt pavement. The asphalt pavement bears the action of the environment and vehicles during service, and cracks and rutting diseases occur. The mechanical analysis of asphalt pavement is a main means to carry out the design of asphalt pavement structure and ensure the service performance. For the mechanical analysis, the material constitutive model is a main factor affecting the accuracy of the mechanical analysis. When the internal strain of the asphalt mixture is less than 150uε or the cumulative strain is less than 5000uε, the viscoelastic model can well characterize the stress-strain behavior at this time.

[0003] At present, the viscoelastic model mainly includes two categories of integer order model and fractional order model. The integer order model is formed by series and parallel connection of spring elements and linear viscous pots, and mainly includes Maxwell model, Kelvin model, standard linear solid model, Burgers model, generalized Maxwell model and generalized Kelvin model. The integer order model usually has an explicit expression of creep or relaxation modulus, and has an equivalent relationship with Prony series. In finite element analysis, by means of the multiplication property of exponential function, the explicit expression required for updating the stress and Jacobian matrix of the integer order model can be derived. However, for the asphalt mixture, due to the wide range of its relaxation spectrum, more spring or viscous pot elements need to be used in the integer order model, so that the parameter identification of the integer order model needs to solve ill-conditioned equations, and finally the identified model parameters have no physical meaning. In addition, the integer order model is easily affected by noise in the parameter identification process, thereby causing overfitting phenomenon.

[0004] The linear dashpot element in the integer order viscoelastic model is replaced by a parabolic dashpot or an Abel dashpot to construct a fractional order viscoelastic model. Previous studies have shown that the parabolic dashpot and the Abel dashpot are mathematically equivalent. Compared with the integer order model, the fractional order viscoelastic model can represent the viscoelastic behavior of asphalt mixture in a wide frequency range with fewer parameters. 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 range, and is recommended for asphalt pavement mechanics analysis at different temperatures. However, due to the lack of a simple creep or relaxation expression in the fractional order model, it is usually difficult to directly use it in finite element analysis. Although some researchers use the interactive conversion method or the analytical approximation method to convert the complex modulus of the fractional order viscoelastic model into the relaxation modulus, and then use the configuration method or the improved Dombi method to expand the relaxation modulus into an exponential function, so as to use the exponential algorithm to carry out finite element calculation. However, the above methods need to solve indefinite equations or high-order equations in the calculation process, and special treatment is needed to ensure that the parameters are positive real numbers.

[0005] In order to use the fractional order viscoelastic model for 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 need to use an explicit creep or relaxation expression. It uses Grünwald coefficients to numerically discretize the fractional order viscoelastic model in time scale with equal interval, so that the stress at the current time can be expressed as a function of the stress or strain at the previous time. Since the GL method uses equal interval discretization in time scale, it needs to use fixed increment step in analysis, and the time increment should not be too large. When carrying out long-term creep analysis or mechanical analysis at variable temperature, it is difficult to balance the calculation efficiency and accuracy. In addition, the GL method needs to store the history stress and strain, which requires a large amount of memory in calculation. In addition to the GL method, some researchers use the Mittag Leffler (ML) function to represent the relaxation modulus of the fractional order viscoelastic model based on Laplace transform, so as to construct the stress updating 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 time, so it can significantly reduce the memory required for calculation. However, the calculation process of the relaxation modulus of asphalt mixture represented by the ML function is complicated. In addition, the calculation of the ML function is complex, which is usually calculated by using the series form or the approximate formula composed of gamma function, thereby affecting the solving accuracy and calculation efficiency of the model.

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

[0007] In order to solve the problem that the finite element method of the existing fractional viscoelastic model cannot balance the calculation efficiency and accuracy, the application provides a fractional viscoelastic model finite element numerical implementation method of asphalt mixture.

[0008] The fractional viscoelastic model finite element numerical implementation method of asphalt mixture provided by the application comprises the following steps.

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

[0010] The lowest order differential expression of the one-dimensional form of the total stress and total strain relationship formula of the model is constructed by introducing time domain and complex frequency domain derivation equivalent substitution.

[0011] The fractional derivative term in the lowest order differential expression is discretized by using a non-classical method to construct a discrete form model stress updating expression.

[0012] The three-dimensional form model stress updating expression and Jacobian matrix updating expression are constructed by combining the constant Poisson ratio assumption or the non-Poisson ratio assumption with the incremental iteration method, and are used for finite element numerical calculation.

[0013] According to the fractional viscoelastic model finite element numerical implementation method of asphalt mixture provided by the application, based on the three-dimensional form model stress updating expression and Jacobian matrix updating expression, the user material subroutine UMAT of the fractional viscoelastic model is developed by using the subroutine development interface in ABAQUS and using FORTRAN; the finite element geometric model is created in ABAQUS, the grid is divided, and then the material parameters, load and boundary conditions of the fractional viscoelastic model of asphalt mixture are input, and the user material subroutine UMAT is called to realize the fractional viscoelastic model finite element numerical calculation of asphalt mixture.

[0014] According to the fractional viscoelastic model finite element numerical implementation method of asphalt mixture provided by the application, the fractional viscoelastic model is selected as the modified fractional Zener model, and the sub-elements include spring element one, spring element two, fractional Abel dashpot element one and fractional Abel dashpot element two.

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

[0016]

[0017] In the formula, σ is the total stress, ε is the total strain, G0 is the spring constant, G1 is the fractional Abel dashpot constant, ω is the angular frequency, and j is the imaginary unit. is the modulus value of the spring element one, η1 is the viscosity of the fractional order Abel dashpot element one, η2 is the viscosity of the fractional order Abel dashpot element two, s is a complex frequency domain independent variable, α1 is the fractional order of the fractional order Abel dashpot element one, α2 is the fractional order of the fractional order Abel dashpot element two, 0 < α1 < α2 < 1, and E2 is the modulus value of the spring element two.

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

[0019]

[0020] In the formula, L is a Laplace transform, D t is a fractional order derivative symbol, α is a fractional order, z(t) is a function to be derived, and t is time.

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

[0022]

[0023] In the formula, ε(t) is total strain, and σ(t) is total stress.

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

[0025]

[0026] In the formula, t n represents the time corresponding to the nth increment step, J is the total number of Gaussian Jacobi quadrature nodes, A j , f j , s j are intermediate coefficients, Δt is the increment step size of adjacent time points, is the horizontal coordinate of the jth Gaussian Jacobi quadrature node, is the weight of the jth Gaussian Jacobi quadrature coefficient;

[0027] The stress updating expression of the discrete form modified fractional order Zener model is constructed as:

[0028]

[0029] In the formula, B1, B2, h1, B3, h2 are all intermediate variables, and Δε is a strain increment, wherein:

[0030]

[0031] In the formula, χ is the independent variable of the intermediate variable.

[0032] The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to the present invention

[0033] Using the assumption of constant Poisson ratio or very Poisson ratio, the coefficients in formula (5) are transformed in three dimensions;

[0034] If we use Poisson's ratio, we get:

[0035]

[0036] If the assumption of a very Poisson's ratio is adopted, then in formula (6) It was measured directly through triaxial testing;

[0037] In the formula, 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, and combining formulas (6) and (5), the update expressions for the normal stress, shear stress, and Jacobian matrix of the three-dimensional modified fractional Zener model are obtained.

[0039] The beneficial effects of this invention are as follows: First, based on the Laplace transform, the complex frequency domain modulus expression of the fractional-order viscoelastic model of asphalt mixtures is obtained. Then, through time-domain / complex frequency-domain equivalent substitution, a one-dimensional form of the lowest-order differential expression of stress and strain for the fractional-order viscoelastic model of asphalt mixtures is constructed. Based on this, the fractional derivative terms in the discrete stress-strain differential form are discretized using a non-classical method, thereby constructing a discrete form of the stress update expression for the fractional-order viscoelastic model of asphalt mixtures. Subsequently, based on the constant / non-perfect Poisson's ratio assumption and the incremental iteration method, the three-dimensional form of the stress update expression and Jacobian matrix expression for the fractional-order viscoelastic model of asphalt mixtures is derived. Finally, based on the subroutine development interface in ABAQUS, a User Material Subroutine (UMAT) is developed using FORTRAN to realize the finite element numerical calculation of the fractional-order viscoelastic model of asphalt mixtures. This invention supports variable incremental step size, has adaptive step size, and low memory usage, enabling the finite element numerical calculation of the fractional-order viscoelastic model.

[0040] One traditional numerical method for implementing fractional-order viscoelastic models using the finite element method is the Grünwald–Letnikov method. This method uses a fixed increment size and requires storing the stress and strain history of all elements at integration points, resulting in high computational cost and large memory usage, making it unsuitable for long-term creep analysis. The method of this invention uses a variable increment size and only needs to store the stress and strain history of the previous increment step during calculation, resulting in high computational efficiency and low memory usage.

[0041] One of the traditional finite element numerical methods for fractional-order viscoelastic models is the Mittag-Leffler function method. This method is suitable for simple fractional-order viscoelastic models where the time-domain creep compliance or relaxation modulus can be expressed using Mittag-Leffler functions. However, for complex fractional-order viscoelastic models, it is difficult to derive the time-domain expressions for their creep compliance or relaxation modulus. Furthermore, its calculation requires the use of approximate formulas for Mittag-Leffler functions, resulting in a large computational burden and affecting the accuracy. The method of this invention employs a differential form of constitutive equations, which eliminates the need to derive the time-domain expressions for the model's creep compliance or relaxation modulus, making it applicable to the finite element numerical calculation of complex fractional-order viscoelastic models.

[0042] The method of this invention is universal, with unified solution steps and calculation format for different fractional-order models, making it easier to implement using the finite element method. Attached Figure Description

[0043] Figure 1 This is a flowchart of the finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixture described in this invention;

[0044] Figure 2 This is a flowchart of the UMAT development process for a fractional-order viscoelastic model of asphalt mixtures;

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

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

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

[0048] Figure 6 It is the ε of the modified fractional Zener model 22 Creep curve. Detailed Implementation

[0049] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0050] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0051] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0052] Specific Implementation Method 1: Combination Figures 1 to 3 As shown, this invention provides a finite element numerical implementation method for a fractional-order viscoelastic model of asphalt mixtures, including,

[0053] Based on the stress-strain relationship of each sub-element and the Laplace transform 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] The total stress and total strain relationship of the model in the complex frequency domain is simplified, and equivalent substitutions of time domain and complex frequency domain differentiation are introduced to construct the lowest order differential expression of the total stress and total strain relationship of the model in one dimension.

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

[0056] By combining the constant Poisson's ratio assumption or the very constant Poisson's ratio assumption with the incremental iteration method, we derive and construct the stress update expression and Jacobian matrix update expression for the three-dimensional formal model, which are then used for finite element numerical calculations.

[0057] Furthermore, combined with Figure 1 As shown, based on the stress update expression and Jacobian matrix update expression of the three-dimensional formal model, the user material subroutine UMAT (User Material Subroutine) for the fractional-order viscoelastic model is developed using FORTRAN through the subroutine development interface in ABAQUS. 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 asphalt mixture are input. The user material subroutine UMAT is then called to realize the finite element numerical calculation of the fractional-order viscoelastic model of asphalt mixture.

[0058] As an example, combined Figure 3As shown, in this embodiment, the fractional-order viscoelastic model is selected as the modified fractional-order Zener model, and the sub-elements include spring element one, spring element two, fractional-order Abel sticky pot element one, and fractional-order Abel sticky pot element two.

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

[0060]

[0061] In the formula Let E1 be the modulus in the complex frequency domain, η1 be the viscosity of the fractional-order Abel sticky pot element 1, η2 be the viscosity of the fractional-order Abel sticky pot element 2, s be the independent variable in the complex frequency domain, α1 be the fractional-order Abel sticky pot element 1, α2 be the fractional-order Abel sticky pot element 2, 0 < α1 < α2 < 1, and E2 be the modulus of the spring element 2.

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

[0063]

[0064] In the formula, L is the Laplace transform, and D t α represents the fractional derivative, z(t) represents the fractional order, z(t) represents the function to be differentiated, and t represents time.

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

[0066]

[0067] In the formula, ε(t) is the total strain and σ(t) is the total stress.

[0068] For example, Represents the α2-α1 and α2 derivative operators with respect to time.

[0069] The fractional derivative term in the lowest-order differential expression is calculated as follows:

[0070]

[0071] In the formula t n Let J represent the time corresponding to the nth increment step, J be the total number of Gaussian Jacobian quadrature nodes, and A be the time corresponding to the nth increment step. j f j s j The intermediate coefficient is Δt, where Δt is the increment step size between adjacent time points, and Δt = t n -t n-1 ; Let x be the x-coordinate of the j-th quadrature node in the Gaussian-Jacobi quadrature. Let the weight of the j-th quadrature coefficient of the Gaussian Jacobian be . Related to the order of fractions;

[0072] The stress update expression for the discrete-form modified fractional Zener model is as follows:

[0073]

[0074] In the formula, B1, B2, h1, B3, and h2 are all intermediate variables, and Δε is the strain increment, where:

[0075]

[0076] In the formula, χ is the independent variable of the intermediate variable.

[0077] Furthermore, the coefficients in formula (5) are transformed in three dimensions by adopting the assumption of constant Poisson ratio or very constant Poisson ratio;

[0078] If we use Poisson's ratio, we get:

[0079]

[0080] If the assumption of a very Poisson's ratio is adopted, then in formula (6) It was measured directly through triaxial testing;

[0081] In the formula, 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, and combining formulas (6) and (5), the update expressions for the normal stress, shear stress, and Jacobian matrix of the three-dimensional modified fractional Zener model are obtained.

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

[0084] Furthermore, the normal stress update expression is:

[0085]

[0086] In the formula, oo represents the normal stress components in different directions, and oo takes values ​​of 11, 22, and 33. For the mean principal strain, Δε is the average principal strain increment. oo The strain increment is in the oo direction. For the mean principal stress, The deviatoric stress is in the oo direction. The increment of the average principal stress. The deviatoric stress increment is in the oo direction, and g1 and g2 are intermediate variables:

[0087]

[0088] In the formula The strain is biased in the oo direction. The strain increment is in the oo direction;

[0089]

[0090] The shear stress update expression is:

[0091]

[0092] In the formula σ kl The shear stress is in the kl direction, where kl takes values ​​of 12, 13, and 23; ε kl Let Δε be the shear strain in the kl direction. kl This represents the shear strain increment in the kl direction.

[0093] g3 and g4 are intermediate variables:

[0094]

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

[0096]

[0097] In the formula This represents the stress increment in the pq direction at the (n+1)th increment step, where pq takes values ​​of 11, 22, 33, 12, 13, and 23. For the (n+1)th increment step, the strain increment in the uv direction is given, where uv takes values ​​of 11, 22, 33, 12, 13, and 23; E K and E G As an intermediate variable:

[0098]

[0099] Furthermore, the state update variable in the UMAT subroutine is:

[0100]

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

[0102] σ 11 =σ0(t(U(t)-U(t-t0))+U(t-t0)).

[0103] In the formula, σ0 is the stress load amplitude, and U is the unit step function.

[0104] Table 1 Material Parameters of MFZM Model

[0105]

[0106] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A finite element numerical implementation method for a fractional-order viscoelastic model of asphalt mixtures, characterized in that, include, Based on the stress-strain relationship of each sub-element and the Laplace transform 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. By introducing equivalent substitutions for time-domain and complex frequency-domain differentiation, the lowest-order differential expression of the one-dimensional form of the relationship between total stress and total strain in the model is constructed. A non-classical method is used to discretize the fractional derivative terms in the lowest-order differential expression, and a discrete-form model stress update expression is constructed. By combining the constant Poisson's ratio assumption or the very Poisson's ratio assumption with the incremental iteration method, stress update expressions and Jacobian matrix update expressions for three-dimensional formal models are constructed for finite element numerical calculations. The fractional-order viscoelastic model is selected as the modified fractional-order Zener model, and the sub-elements include spring element one, spring element two, fractional-order Abel sticky pot element one, and fractional-order Abel sticky pot element two. The relationship between the total stress and total strain in the complex frequency domain model is established as follows: In the formula Let E1 be the modulus in the complex frequency domain, η1 be the viscosity of the fractional-order Abel sticky pot element 1, η2 be the viscosity of the fractional-order Abel sticky pot element 2, s be the independent variable in the complex frequency domain, α1 be the fractional-order Abel sticky pot element 1, α2 be the fractional-order Abel sticky pot element 2, 0 < α1 < α2 < 1, and E2 be the modulus of the spring element 2. Introducing equivalent substitutions for time-domain or complex frequency-domain differentiation with initial values ​​of zero: In the formula, L is the Laplace transform, and D t α represents the fractional derivative, z(t) represents the fractional order, z(t) represents the function to be differentiated, and t represents time. The lowest-order differential expression in one-dimensional form is: In the formula, ε(t) is the total strain and σ(t) is the total stress; The fractional derivative term in the lowest-order differential expression is calculated as follows: In the formula t n Let J represent the time corresponding to the nth increment step, J be the total number of Gaussian Jacobian quadrature nodes, and A be the time corresponding to the nth increment step. j f j s j The intermediate coefficient is Δt, where Δt is the increment step size between adjacent time points. Let x be the x-coordinate of the j-th quadrature node in the Gaussian-Jacobi quadrature. Let be the weight of the j-th quadrature coefficient of the Gaussian Jacobian.

2. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 1, characterized in that, Based on the stress update expression and Jacobian matrix update expression of the three-dimensional formal model, the user material subroutine UMAT for the fractional-order viscoelastic model is developed using FORTRAN through the subroutine development interface in ABAQUS. 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 asphalt mixture are input. The user material subroutine UMAT is then called to realize the finite element numerical calculation of the fractional-order viscoelastic model of asphalt mixture.

3. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 1, characterized in that, The stress update expression for the discrete-form modified fractional Zener model is as follows: In the formula, B1, B2, h1, B3, and h2 are all intermediate variables, and Δε is the strain increment, where: In the formula, χ is the independent variable of the intermediate variable.

4. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 3, characterized in that, Using the assumption of constant Poisson ratio or very Poisson ratio, the coefficients in formula (5) are transformed in three dimensions; If we use Poisson's ratio, we get: If the assumption of a very Poisson's ratio is adopted, then in formula (6) and It was measured directly through triaxial testing; In the formula, 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, combined with formulas (6) and (5), the update expressions for the normal stress, shear stress and Jacobian matrix of the three-dimensional modified fractional Zener model are obtained.

5. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 4, characterized in that, The normal stress update expression is: In the formula, oo represents the normal stress components in different directions, and oo takes values ​​of 11, 22, and 33. For the mean principal strain, Δε is the average principal strain increment. oo The strain increment is in the oo direction. For the mean principal stress, The deviatoric stress is in the oo direction. The increment of the average principal stress. The deviatoric stress increment is in the oo direction, and g1 and g2 are intermediate variables: In the formula The strain is biased in the oo direction. The strain increment is in the oo direction; 6. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 5, characterized in that, The shear stress update expression is: s kl (t n )=g3(t n ,2e kl ,2No kl )+g4(t n ,s kl )(8), In the formula σ kl The shear stress is in the kl direction, where kl takes values ​​of 12, 13, and 23; ε kl Let Δε be the shear strain in the kl direction. kl This represents the shear strain increment in the kl direction. g3 and g4 are intermediate variables:

7. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 6, characterized in that, The update expression for the Jacobian matrix is: In the formula This represents the stress increment in the pq direction at the (n+1)th increment step, where pq takes values ​​of 11, 22, 33, 12, 13, and 23. For the (n+1)th increment step, the strain increment in the uv direction is given, where uv takes values ​​of 11, 22, 33, 12, 13, and 23; E K and E G As an intermediate variable:

8. The finite element numerical implementation method for the fractional-order viscoelastic model of asphalt mixtures according to claim 7, characterized in that, The state update variable in the UMAT subroutine is: Input material parameters in the user material subroutine UMAT, create a finite element geometric model using ABAQUS, select three-dimensional solid elements to discretize the finite element geometric model, apply loads and boundary conditions to the discretized finite element geometric model, set incremental steps and submit the calculation to obtain the mechanical response results of the finite element model.

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