Method for calculating asphalt pavement viscoelasticity under multi-axle moving load based on boltzmann superposition principle

Abstract

Based on the Boltzmann superposition principle of multiaxial moving load under asphalt pavement viscoelastic mechanics calculation method, the present application solves the problem that the existing asphalt pavement mechanics model assumes that the isotropic elastic body under fixed load cannot accurately evaluate the material performance by using elasticity to describe its performance. The calculation method comprises the following steps: 1, constructing the viscoelastic mechanics response model of the asphalt pavement structure under static load; 2, calculating the mechanics response of the asphalt pavement structure under moving load, adopting the Boltzman superposition principle and pointing out that the deformation of the asphalt pavement is a function of the load history, and when multiple loads act on the viscoelastic material, the final deformation performance of the asphalt pavement is formed by the superposition of the independent action of each load, and the uniaxial load form, the double shaft load form and the three shaft load form are constructed respectively. The calculated viscoelastic mechanics of the asphalt pavement under the moving load is more consistent with the actual stress condition of the asphalt pavement, and the complex modeling process in the finite element software is avoided.

Description

Technical Field

[0001] This invention relates to a method for calculating the viscoelastic mechanical response of asphalt pavement, specifically a method for calculating the viscoelastic mechanical response of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle. Background Technology

[0002] The mechanical response of asphalt pavement structures is one of the important methods for evaluating pavement structure performance. Currently, in the mechanical analysis of asphalt pavement structures, the pavement mechanical model is assumed to be an isotropic elastic body under fixed loads. This mechanical analysis method has a research history of nearly 80 years, accumulating rich theoretical and procedural results, and has been maturely applied in asphalt pavement structure design specifications. However, during the service life of asphalt pavements, the service life is far shorter than the design life. One important reason is the significant difference between the calculation results of the pavement mechanical model and the actual stress on the pavement structure, making it difficult to accurately assess the service life of the pavement structure. The mechanical model has the following shortcomings:

[0003] Asphalt mixtures are typical viscoelastic materials. Using elasticity to describe their performance cannot accurately assess the material properties (currently, the calculation methods for viscoelastic mechanics of asphalt pavement are relatively mature). The loads on the pavement are all moving loads. It is difficult to truly obtain the internal mechanical response of the pavement when using fixed loads for structural mechanical analysis of asphalt pavement. Therefore, it is impossible to accurately predict the service life of asphalt pavement structures. Summary of the Invention

[0004] The purpose of this invention is to address the problem that existing asphalt pavement mechanical models assume an isotropic elastic body under a fixed load, and the use of elasticity to describe its performance cannot accurately assess material properties. Therefore, this invention provides a viscoelastic mechanical calculation method for asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle.

[0005] The viscoelastic mechanical calculation method for asphalt pavement under multiaxial moving load based on the Boltzmann superposition principle of this invention is implemented according to the following steps:

[0006] I. Viscoelastic Mechanical Response of Asphalt Pavement Structure under Static Load

[0007] The mechanical response model under static load (fixed) is shown in Equation (1).

[0008]

[0009] in

[0010] In the formula σ r (s) represents the horizontal stress in the Laplace domain, σ θ (s) represents the auxiliary stress in the Laplace domain, σ z(s) represents the vertical stress in the Laplace domain, τ zr (s) represents the shear stress in the Laplace domain, u(s) represents the horizontal displacement, and w(s) represents the vertical displacement. The Hankel and Laplace transformations represent uniformly distributed loads, J0 represents the 0th-order Bessel function, J1 represents the 1st-order Bessel function, E and μ are the modulus and Poisson's ratio, respectively, r is the horizontal distance of the calculation point, z is the vertical distance of the calculation point, and A, B, C, and D are stress coefficients.

[0011] Applying the Laplace numerical inverse transform to Equation (1) yields the time-domain solution of the viscoelastic mechanical response of the asphalt pavement structure, as shown in Equation (2).

[0012]

[0013] In the formula: t is time, k is the order parameter, a and T are the model parameters respectively, i is the imaginary unit, F() is the Laplace domain calculation formula, and f(t) represents the time domain solution of the viscoelastic mechanical response of the asphalt pavement structure.

[0014] II. Calculation of the Structural Mechanical Response of Asphalt Pavement under Moving Loads

[0015] Boltzmann's superposition principle states that prior load history influences the deformation properties of viscoelastic materials. Specifically, the deformation of asphalt pavement is a function of load history. When multiple loads act on a viscoelastic material, the final deformation properties of the asphalt pavement are formed by the superposition of each independent load. In other words, each load step is independent and can be superimposed on others. The load trajectory is divided into several segments [r] i ,r i-1 Within each section, the properties of the asphalt mixture are assumed to be stable, and the mechanical response of the asphalt pavement structure under moving load is analyzed using the Boltzmann superposition principle.

[0016] A. Uniaxial load type:

[0017] Suppose the load is located at position r0 at time t0, and the viscoelastic response at point p is R(r0, z0, t0). As the load gradually moves across the road surface, at time t... n Time to travel to r n When the position is calculated, the viscoelastic mechanical response generated at the point p is shown in formula (5). The viscoelastic mechanical response of each term on the right side of the equation is calculated by substituting the corresponding horizontal distance, vertical distance and time into formula (2).

[0018]

[0019] In the formula: R(r0,z0,t) nLet be the mechanical response produced by the load at the calculation point p, i.e., at a distance (r0, z0) from point p, where the load lasts for t. n R(r) i ,z0,t n -t i-1 The mechanical response produced by the load at the calculation point p is the response of the load after t. i-1 After the movement time, the distance from point p is (r i The load at this location lasts for a duration of t (z0). n -t i-1 R(r) i-1 ,z0,t n -t i-1 () represents the mechanical response produced by the load at the calculation point p, after the load has passed through t i-1 After the movement time, at a distance p of (r i-1 Apply a reverse load (z0) to achieve (r) i-1 The load at z0 is zero, and the reverse load at this point lasts for t. n -t i-1 ;

[0020] The formula for calculating time t is as follows:

[0021]

[0022] In the formula t i —The moving load travels to r i Time (s) at position;

[0023] v — Vehicle speed (m / s);

[0024] Assuming time t0 is 0, the viscoelastic mechanical response of the asphalt pavement structure under uniaxial moving load is obtained as follows:

[0025]

[0026] B. Duplex load type:

[0027] Let the distance between the axes of the two axes be l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows:

[0028]

[0029] In the formula: l d Let be the shaft spacing (m) of the twin shafts. Formula (8) is the mechanical response generated by adding a load to the shaft based on Formula (7), i.e. The horizontal distance between the second axis and the calculation point p increased by l compared to the distance between the first axis and the second axis. d ;

[0030] C. Triple-axis load configuration:

[0031] Assume the distance between any two adjacent axes in a triplex system is l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows:

[0032]

[0033] Formula (9) is the mechanical response generated by adding a load on one shaft to Formula (8), that is... The horizontal distance between the third axis and the calculation point p increased by l compared to the distance between the second axis and the third axis. d .

[0034] This invention can be extended to a viscoelastic mechanical calculation method for asphalt pavement under multi-axis moving load: First, determine the pavement structure and viscoelastic parameters of the asphalt surface layer, then determine the calculation points, the speed of the moving load, and other parameters. Select the viscoelastic mechanical response calculation method for the moving load according to the axis type (single axis, double axis, triple axis), as shown in formulas (7), (8), and (9). Calculate the mechanical response R at different locations using the theoretical formula (1) of the viscoelastic system theory. Obtain the mechanical response of the asphalt pavement structure under multi-axis moving load according to the calculation method formulas (7), (8), and (9).

[0035] The viscoelastic mechanical calculation method for asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle of this invention has the following beneficial effects:

[0036] 1. This invention is more consistent with the actual stress conditions of asphalt pavement;

[0037] 2. This invention avoids the complex modeling process in finite element software;

[0038] 3. The theoretical results of this invention are simple in form, easy to program, and have high computational efficiency. Attached Figure Description

[0039] Figure 1 This is a flowchart of the calculation method for the viscoelasticity of asphalt pavement under multiaxial moving load based on the Boltzmann superposition principle of the present invention;

[0040] Figure 2 This is the viscoelastic mechanical model of the asphalt pavement structure in the embodiment;

[0041] Figure 3a The figure shows a comparison between the calculated results of the vertical viscoelastic displacement of the asphalt pavement over time and the calculation results of Abaqus in the example, where ■ represents the present invention and ● represents the Abaqus calculation;

[0042] Figure 3bThe figure shows a comparison between the calculation results of the horizontal stress of asphalt pavement over time and the calculation results of Abaqus in the example, where ■ represents the present invention and ● represents the Abaqus calculation.

[0043] Figure 4a This is a schematic diagram of the mechanical response of the asphalt pavement structure at time t0 in the embodiment;

[0044] Figure 4b This is a schematic diagram of the mechanical response of the asphalt pavement structure at time t1 in the embodiment;

[0045] Figure 5 This is a diagram showing the division of moving load sections in the embodiment;

[0046] Figure 6 This is a schematic diagram of a common vehicle axle type in the embodiments;

[0047] Figure 7 This is a diagram illustrating the evolution of vertical deformation of the road surface under load from a single-axle vehicle traveling at a speed of 54 km / h, as shown in the example.

[0048] Figure 8 This is a diagram illustrating the evolution of vertical deformation of the road surface under load from a single-axle vehicle traveling at a speed of 72 km / h, as shown in the example.

[0049] Figure 9 This is a diagram illustrating the evolution of vertical deformation of the road surface under load from a single-axle vehicle traveling at a speed of 90 km / h, as shown in the example.

[0050] Figure 10 This is a diagram illustrating the evolution of vertical deformation of the road surface under load from a single-axle vehicle traveling at a speed of 108 km / h, as shown in the example.

[0051] Figure 11 The figure shows a comparison between the simulated mechanical response and the actual field measurement under dynamic load in the embodiment. In the figure, 1 represents the measured longitudinal strain, 2 represents the simulated longitudinal strain, 3 represents the measured transverse strain, and 4 represents the simulated transverse strain. Detailed Implementation

[0052] Specific Implementation Method 1: This implementation method, based on the Boltzmann superposition principle, uses the following steps to calculate the viscoelasticity of asphalt pavement under multiaxial moving loads:

[0053] I. Viscoelastic Mechanical Response of Asphalt Pavement Structure under Static Load

[0054] The mechanical response model under static load (fixed) is shown in Equation (1).

[0055]

[0056] in

[0057] In the formula σ r (s) represents the horizontal stress in the Laplace domain, σ θ (s) represents the auxiliary stress in the Laplace domain, σ z (s) represents the vertical stress in the Laplace domain, τ zr (s) represents the shear stress in the Laplace domain, u(s) represents the horizontal displacement, and w(s) represents the vertical displacement. The Hankel and Laplace transformations represent uniformly distributed loads, J0 represents the 0th-order Bessel function, J1 represents the 1st-order Bessel function, E and μ are the modulus and Poisson's ratio, respectively, r is the horizontal distance of the calculation point, z is the vertical distance of the calculation point, and A, B, C, and D are stress coefficients.

[0058] Applying the Laplace numerical inverse transform to Equation (1) yields the time-domain solution of the viscoelastic mechanical response of the asphalt pavement structure, as shown in Equation (2).

[0059]

[0060] In the formula: t is time, k is the order parameter, a and T are the model parameters respectively, i is the imaginary unit, F() is the Laplace domain calculation formula, and f(t) represents the time domain solution of the viscoelastic mechanical response of the asphalt pavement structure.

[0061] II. Calculation of the Structural Mechanical Response of Asphalt Pavement under Moving Loads

[0062] Boltzmann's superposition principle states that prior load history influences the deformation properties of viscoelastic materials. Specifically, the deformation of asphalt pavement is a function of load history. When multiple loads act on a viscoelastic material, the final deformation properties of the asphalt pavement are formed by the superposition of each independent load. In other words, each load step is independent and can be superimposed on others. The load trajectory is divided into several segments [r] i ,r i-1 Within each section, the properties of the asphalt mixture are assumed to be stable, and the mechanical response of the asphalt pavement structure under moving load is analyzed using the Boltzmann superposition principle.

[0063] A. Uniaxial load type:

[0064] Suppose the load is located at position r0 at time t0, and the viscoelastic response at point p is R(r0, z0, t0). As the load gradually moves across the road surface, at time t... n Time to travel to r nWhen the position is calculated, the viscoelastic mechanical response generated at the point p is shown in formula (5). The viscoelastic mechanical response of each term on the right side of the equation is calculated by substituting the corresponding horizontal distance, vertical distance and time into formula (2).

[0065]

[0066] In the formula: R(r0,z0,t) n Let be the mechanical response produced by the load at the calculation point p, i.e., at a distance (r0, z0) from point p, where the load lasts for t. n R(r) i ,z0,t n -t i-1 The mechanical response produced by the load at the calculation point p is the response of the load after t. i-1 After the movement time, the distance from point p is (r i The load at this location lasts for a duration of t (z0). n -t i-1 R(r) i-1 ,z0,t n -t i-1 () represents the mechanical response produced by the load at the calculation point p, after the load has passed through t i-1 After the movement time, at a distance p of (r i-1 Apply a reverse load (z0) to achieve (r) i-1 The load at z0 is zero, and the reverse load at this point lasts for t. n -t i-1 ;

[0067] The formula for calculating time t is as follows:

[0068]

[0069] In the formula t i —The moving load travels to r i Time (s) at position;

[0070] v — Vehicle speed (m / s);

[0071] Assuming time t0 is 0, the viscoelastic mechanical response of the asphalt pavement structure under uniaxial moving load is obtained as follows:

[0072]

[0073] B. Duplex load type:

[0074] Let the distance between the axes of the two axes be l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows:

[0075]

[0076] In the formula: l d Let be the shaft spacing (m) of the twin shafts. Formula (8) is the mechanical response generated by adding a load to the shaft based on Formula (7), i.e. The horizontal distance between the second axis and the calculation point p increased by l compared to the distance between the first axis and the second axis. d ;

[0077] C. Triple-axis load configuration:

[0078] Assume the distance between any two adjacent axes in a triplex system is l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows:

[0079]

[0080] Formula (9) is the mechanical response generated by adding a load on one shaft to Formula (8), that is... The horizontal distance between the third axis and the calculation point p increased by l compared to the distance between the second axis and the third axis. d .

[0081] This embodiment can use the road structure combination and material parameters shown in Table 1. The material parameters of the asphalt mixture are selected from the Prony series model, and the model parameters are shown in Table 2. The road structure mechanical model is established using ABAQUS to verify the calculation program of the viscoelastic mechanical response of the asphalt pavement. The calculation results are shown in Figure 3. It can be seen from the figure that the viscoelastic mechanical calculation of the asphalt pavement under static load is accurate.

[0082] Table 1 Road surface structure combination and material parameters

[0083]

[0084] Table 2 Prony series parameters

[0085]

[0086] Specific Implementation Method Two: This implementation method differs from Specific Implementation Method One in that the uniformly distributed load in step one... The expression is as follows:

[0087]

[0088] Where ξ is the integral variable of the Hankel integral transform, s is the integral variable of the Laplace integral transform, δ represents the radius of the load circle, and p(s) represents the Laplace transform expression of the load over time.

[0089] Specific Implementation Method 3: This implementation method differs from Specific Implementation Method 1 or 2 in that the asphalt mixture in the asphalt pavement structure in step 1 adopts the Prony series model.

[0090] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that the formula for calculating the modulus E in step one is as follows:

[0091]

[0092] In the formula: E i —Corresponding relaxation time τ i The relaxation modulus at that time;

[0093] τ i —Relaxation time, t—time.

[0094] Specific Implementation Method 5: This implementation method differs from Specific Implementation Methods 1 to 4 in that in step 1, formula (2) takes T = 2*t and a*T = 15.

[0095] Specific Implementation Method Six: This implementation method differs from Specific Implementation Methods One to Five in that the width of the section in step two is 1 to 20 cm.

[0096] Specific Implementation Method Seven: This implementation method differs from Specific Implementation Method Six in that the width of the section in step two is 20cm.

[0097] Specific Implementation Method Eight: This implementation method differs from Specific Implementation Methods One through Seven in that the axial spacing l in step two is... d The unit is meters.

[0098] Example: This example demonstrates a method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle. The method is implemented using the following steps:

[0099] I. The typical structural form and material parameters of asphalt pavement in a certain province of China are shown in Table 3. The material parameters of the upper, middle and lower layers are linear viscoelastic materials. The viscoelastic material properties are selected in the Prony series form shown in equation (3). The material parameters of different layers are shown in Table 4.

[0100] II. Viscoelastic Mechanical Response of Asphalt Pavement Structure under Static Load

[0101] The mechanical response model under static load (fixed) is shown in Equation (1), and the schematic diagram of the mechanical model under static load is shown in [Figure / Diagram]. Figure 2 As shown;

[0102]

[0103] in

[0104] In the formula σ r (s) represents the horizontal stress in the Laplace domain, σ θ (s) represents the auxiliary stress in the Laplace domain, σ z (s) represents the vertical stress in the Laplace domain, τ zr (s) represents the shear stress in the Laplace domain, u(s) represents the horizontal displacement, and w(s) represents the vertical displacement. The Hankel and Laplace transformations represent uniformly distributed loads. ξ is the integral variable of the Hankel integral transform, s is the integral variable of the Laplace integral transform, J0 represents the 0th order Bessel function, J1 represents the 1st order Bessel function, E and μ are the modulus and Poisson's ratio, respectively, r is the horizontal distance of the calculation point, z is the vertical distance of the calculation point, and A, B, C, and D are stress coefficients.

[0105] Applying the Laplace numerical inverse transform to Equation (1) yields the time-domain solution of the viscoelastic mechanical response of the asphalt pavement structure, as shown in Equation (2).

[0106]

[0107] In the formula: t is time, k is the sequence parameter, a and T are model parameters. Based on experience, T = 2*t, a*T = 15, i is the imaginary unit, F() is the Laplace domain calculation formula, F() refers to the Laplace domain calculation formula of the 6 mechanical responses in formula (1), and f(t) is the time domain solution of the viscoelastic mechanical response of the asphalt pavement structure.

[0108] III. Calculation of the Structural Mechanical Response of Asphalt Pavement under Moving Loads

[0109] Boltzmann's superposition principle states that prior load history influences the deformation properties of viscoelastic materials. Specifically, the deformation of asphalt pavement is a function of load history. When multiple loads act on a viscoelastic material, the final deformation properties of the asphalt pavement are formed by the superposition of each independent load. In other words, each load step is independent and can be superimposed on others. The load trajectory is divided into several segments [r] i ,r i-1 Within each section, the properties of the asphalt mixture are assumed to be stable, and the mechanical response of the asphalt pavement structure under moving load is analyzed using the Boltzmann superposition principle.

[0110] A. Single-axle load type, commonly used for the front axle of automobiles and the rear axle of light vehicles, such as... Figure 6 As shown in a:

[0111] like Figure 4aThe viscoelastic response at point p is calculated as follows: Assume the load is at position r0 at time t0. The viscoelastic response at point p is R(r0, z0, t0). When the load moves to... Figure 4b When calculating the position, the viscoelastic response at point p at this moment consists of three parts: R(r1,z0,t1-t0) describes the viscoelastic response generated when the load moves to position r1, and t1-t0 describes the duration of the load at this position; R(r0,z0,t1) is the viscoelastic response generated by the load at time t0, that is, the viscoelastic response generated by the load (shown by the dashed line) at position r0 for a duration of t1; R(r0,z0,t1-t0) is the viscoelastic response generated by the reverse load applied at position r0. R(r0,z0,t1) and R(r0,z0,t1-t0) describe the case where the load at position r0 is 0. Therefore, the viscoelastic response generated at point p when the load travels to position r1 can be expressed as:

[0112] R p =R(r0,z,t1)+R(r1,z,t1-t0)-R(r0,z,t1-t0) (4)

[0113] As the load gradually moves across the road surface from r0 to r n At t n Time to travel to r n Location (e.g.) Figure 5 When (as shown), the formula for calculating the viscoelastic response at point p is extended from (4) as follows:

[0114]

[0115] In the formula: R(r0,z0,t) n Let be the mechanical response produced by the load at calculation point p, which is at a distance (r0, z0) from p, and the duration of the load at this point is t. n R(r) i ,z0,t n -t i-1 () represents the mechanical response produced by the load at the calculation point p, after the load has passed through t i-1 After the movement time, the distance from point p is (r i The load at this location lasts for a duration of t (z0). n -t i-1 R(r) i-1 ,z0,t n -t i-1 () represents the mechanical response produced by the load at the calculation point p, after the load has passed through t i-1 After the movement time, at a distance p of (r i-1 Apply a reverse load (z0) to achieve (r) i-1The load at z0 is zero, and the reverse load at this point lasts for t. n -t i-1 ;

[0116] The formula for calculating time t is as follows:

[0117]

[0118] In the formula t i —The moving load travels to r i Time (s) at position;

[0119] v — Vehicle speed (m / s);

[0120] Assuming time t0 is 0, substituting equation (6) into equation (5), we obtain the viscoelastic mechanical response of the asphalt pavement structure under uniaxial moving load as follows:

[0121]

[0122] B. Twin-axis load type, such as Figure 6 As shown in b:

[0123] Let the distance between the axes of the two axes be l. d Compared to a uniaxial moving load, a biaxial load is equivalent to a load at a distance l from a uniaxial moving load. d If the same load is applied at a distance, then only the viscoelastic mechanical response generated by the second axis needs to be added to Equation (7). The viscoelastic mechanical response of the asphalt pavement structure under the biaxial moving load is as follows:

[0124]

[0125] In the formula: l d Let be the shaft spacing (m) of the twin shafts. Formula (8) is the mechanical response generated by adding a load to the shaft based on Formula (7), i.e. The horizontal distance between the second axis and the calculation point p increased by l compared to the distance between the first axis and the second axis. d Therefore, the horizontal distance in the formula increased by l. d .

[0126] C. Triple-axle load configuration, such as Figure 6 As shown in c:

[0127] Assume the distance between any two adjacent axes in a triplex system is l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows:

[0128]

[0129] Formula (9) is the mechanical response generated by adding a load on a shaft to Formula (8), that is... The horizontal distance between the third axis and the calculation point p increased by l compared to the distance between the second axis and the third axis. d Therefore, the horizontal distance in the formula increases by l. d .

[0130] This embodiment first calculates the lengths [r] of different segments. i ,r i-1 The pavement mechanical response and computational efficiency are as follows: Figure 7-10 As shown in Tables 5 and 6, from Figure 7-10 As shown in Table 5, at the same driving speed, the interval length has a very small impact on the mechanical response, with the error controlled within 0.5%. Table 5 also shows that driving speed has little impact on the program's computational efficiency, while the interval width significantly affects it. The computational efficiency is approximately 10 times that of a 1cm interval when the interval width is 20cm. Considering the load circle's radius is 10.65cm and diameter is 21.3cm, when the interval width is 20cm, the entire load is precisely distributed within that width. Therefore, considering all factors, the interval width is selected as 20cm. The tensile strain at the bottom of the asphalt surface layer is obtained through simulation calculations and on-site monitoring. Figure 11 As shown. Figure 11 The load in the model is a uniaxial load with a load size of 0.7 MPa and a load circle radius of 10 cm. The mechanical response in the three directions is obtained by calculating the transverse strain and longitudinal strain using formula (1), and then calculated using formula (10). Figure 11 The numerical calculation results shown are close to the measured results, indicating that the proposed calculation method can effectively analyze the mechanical response inside the structure.

[0131]

[0132] Table 3. Structural Combinations and Material Parameters for Six-Layer Asphalt Pavement

[0133]

[0134] Table 4. Prony series model parameters for materials at different layers.

[0135]

[0136] Table 5. Efficiency calculation (s) for different section widths and driving speeds

[0137]

[0138]

[0139] Table 6. Calculation results of vertical displacement (0.01 mm) for different section widths and driving speeds.

[0140]

Claims

1. A method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle, characterized in that... The method for calculating the viscoelasticity of asphalt pavement under multi-axis moving load is implemented according to the following steps: I. Viscoelastic Mechanical Response of Asphalt Pavement Structure under Static Load The mechanical response model under static load is shown in equation (1). in In the formula σ r (s) represents the horizontal stress in the Laplace domain, σ θ (s) represents the auxiliary stress in the Laplace domain, σ z (s) represents the vertical stress in the Laplace domain, τ zr (s) represents the shear stress in the Laplace domain, u(s) represents the horizontal displacement, and w(s) represents the vertical displacement. The Hankel and Laplace transformations represent uniformly distributed loads, J0 represents the 0th-order Bessel function, J1 represents the 1st-order Bessel function, E and μ are the modulus and Poisson's ratio, respectively, r is the horizontal distance of the calculation point, z is the vertical distance of the calculation point, and A, B, C, and D are stress coefficients. Applying the Laplace numerical inverse transform to Equation (1) yields the time-domain solution of the viscoelastic mechanical response of the asphalt pavement structure, as shown in Equation (2). In the formula: t is time, k is the order parameter, a and T are the model parameters respectively, i is the imaginary unit, F() is the Laplace domain calculation formula, and f(t) represents the time domain solution of the viscoelastic mechanical response of the asphalt pavement structure. II. Calculation of the Structural Mechanical Response of Asphalt Pavement under Moving Loads Boltzmann's superposition principle states that prior load history influences the deformation properties of viscoelastic materials. Specifically, the deformation of asphalt pavement is a function of load history. When multiple loads act simultaneously on a viscoelastic material, the final deformation properties of the asphalt pavement are formed by the superposition of the independent actions of each load. This involves dividing the load's trajectory into several segments [r]. i ,r i-1 Within each section, the properties of the asphalt mixture are assumed to be stable, and the mechanical response of the asphalt pavement structure under moving load is analyzed using the Boltzmann superposition principle. A. Uniaxial load type: Suppose the load is located at position r0 at time t0, and the viscoelastic response at point p is R(r0, z0, t0). As the load gradually moves across the road surface, at time t... n Time to travel to r n When the position is calculated, the viscoelastic mechanical response generated at the point p is shown in formula (5). The viscoelastic mechanical response of each term on the right side of the equation is calculated by substituting the corresponding horizontal distance, vertical distance and time into formula (2). In the formula: R(r0,z0,t) n Let be the mechanical response produced by the load at the calculation point p, i.e., at a distance (r0, z0) from point p, where the load lasts for t. n R(r) i ,z0,t n -t i-1 The mechanical response produced by the load at the calculation point p is the response of the load after t. i-1 After the movement time, the distance from point p is (r i The load at this location lasts for a duration of t (z0). n -t i-1 R(r) i-1 ,z0,t n -t i-1 () represents the mechanical response produced by the load at the calculation point p, after the load has passed through t i-1 After the movement time, at a distance p of (r i-1 Apply a reverse load (z0) to achieve (r) i-1 The load at z0 is zero, and the reverse load at this point lasts for t. n -t i-1 ; The formula for calculating time t is as follows: In the formula t i —The moving load travels to r i Time (s) at position; v — Vehicle speed (m / s); Assuming time t0 is 0, the viscoelastic mechanical response of the asphalt pavement structure under uniaxial moving load is obtained as follows: B. Duplex load type: Let the distance between the axes of the two axes be l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows: In the formula: l d For the shaft spacing of the twin shafts, formula (8) is the mechanical response generated by adding the load of one shaft to formula (7), i.e. The horizontal distance between the second axis and the calculation point p increased by l compared to the distance between the first axis and the second axis. d ; C. Triple-axis load configuration: Assume the distance between any two adjacent axes in a triplex system is l. d The viscoelastic mechanical response of the asphalt pavement structure under a twin-shaft moving load is as follows: Formula (9) is the mechanical response generated by adding a load on one shaft to Formula (8), that is... The horizontal distance between the third axis and the calculation point p increased by l compared to the distance between the second axis and the third axis. d .

2. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 1, characterized in that... Uniformly distributed load in step one The expression is as follows: Where ξ is the integral variable of the Hankel integral transform, s is the integral variable of the Laplace integral transform, δ represents the radius of the load circle, and p(s) represents the Laplace transform expression of the load over time.

3. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 1, characterized in that... In step one, the asphalt mixture in the asphalt pavement structure is modeled using the Prony series model.

4. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 1, characterized in that... The formula for calculating the modulus E in step one is as follows: In the formula: E i —Corresponding relaxation time τ i The relaxation modulus at that time; τ i —Relaxation time, t—time.

5. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 1, characterized in that... In formula (2) of step one, take T = 2*t and a*T = 15.

6. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 1, characterized in that... The width of the section in step two is 1 to 20 cm.

7. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 6, characterized in that... The width of the section in step two is 20cm.

8. The method for calculating the viscoelasticity of asphalt pavement under multiaxial moving loads based on the Boltzmann superposition principle according to claim 1, characterized in that... In step two, the axial spacing l d The unit is meters.

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