Design method of asphalt pavement based on humidity environment

Abstract

The application discloses a design method of an asphalt pavement based on humidity environment, which comprises the following steps: (1) determining an asphalt pavement structure model; (2) determining material parameters of the asphalt pavement structure model; (3) establishing a finite element model of a temperature field of the asphalt pavement structure; (4) establishing a finite element model of a water and gas concentration field of the asphalt pavement structure; and (5) considering the establishment of a fatigue life model of the asphalt pavement structure under the humidity environment. Based on the finite element analysis method, the application establishes a pavement structure temperature field model and a non-uniform water and gas concentration field model, introduces the influence of water and gas factors into the pavement structure, compares the pavement structures under the conditions of not considering water and gas and the non-uniform water and gas concentration, further establishes a pavement structure fatigue cracking model, analyzes the change of the fatigue life of the asphalt pavement under the two water and gas concentration conditions, and provides a scientific basis for reducing the generation of crack diseases and constructing long-life and durable asphalt pavements.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of asphalt pavement design, and particularly relates to a design method for asphalt pavement under a humidity environment. BACKGROUND

[0002] As the main structure form of the surface layer of highways in China, asphalt pavement not only endures the action of vehicle load and temperature for a long time, but also experiences the humidity action caused by the humidity gradient between the soil subgrade and the air. In the real environment, the asphalt pavement is inevitably in an environment with different water vapor contents due to the existence of relative humidity. However, the current research on the humidity sensitivity of the performance of asphalt mixture is still insufficient, and the water damage problem caused by the penetration of water into the asphalt pavement is mainly analyzed from the perspective of liquid water. However, in fact, gaseous water is more likely to invade the adhesion interface between the asphalt and the aggregate or the interior of the asphalt mortar in the initial service period of the pavement, which reduces the adhesion and cohesion bond energies and causes the asphalt to peel off from the surface of the aggregate and thus causes initial water damage. On the other hand, temperature and humidity are interrelated environmental factors, and therefore it is necessary to study the performance of asphalt mixture under the condition of temperature and humidity coupling in order to more accurately predict the development of the fatigue cracking performance of the asphalt pavement in the real service environment and more accurately predict the service life of the asphalt pavement.

[0003] Asphalt pavement is a pavement structure with asphalt mixture as the surface layer. The base layer can be divided into a flexible base layer, a rigid base layer or a semi-rigid base layer. In the real service process, the asphalt pavement often shows fatigue failure under the repeated action of vehicle load. The current international asphalt pavement design methods can be roughly divided into three types: empirical method, mechanical-empirical method and performance-based design method. Although the existing asphalt pavement design methods consider the water stability of asphalt mixture, they do not consider the influence of environmental humidity on the fatigue cracking of asphalt mixture, and thus cannot consider the process of accelerated peeling of the asphalt film from the aggregate in the process of use of asphalt mixture in a high-humidity environment, resulting in accelerated damage of the asphalt pavement in a high-humidity environment.

[0004] Some scholars have obtained the mechanical response of the pavement structure under different water vapor conditions through numerical simulation method, and have estimated the fatigue life of the asphalt pavement by referring to the calculation method of the fatigue life of the pavement structure in the current specifications in China. However, the fact shows that the correlation between the prediction results obtained according to the mechanical response of the pavement structure and the actual fatigue life of the asphalt pavement is poor, which is mainly due to the fact that the fatigue failure of the asphalt pavement is often accompanied by the generation of cracks, and the expansion of the cracks will lead to the inconsistency between the simulation results and the stress condition of the actual pavement. Therefore, simple simulation cannot represent the asphalt pavement structure under the multi-field coupling action of the humidity environment.

[0005] In summary, in the design process of asphalt pavement, the design parameters of the asphalt pavement need to consider the action of humidity so as to accurately reflect the structural performance of the asphalt pavement in the all-weather service process. SUMMARY

[0006] The present application aims to overcome the deficiencies of the prior art, and provides a design method for asphalt pavement under humidity environment, which combines the actual climate conditions of engineering projects and establishes a pavement structure temperature field model based on finite element software analysis method; on this basis, a non-uniform water vapor concentration field model is established according to the accumulation type water vapor diffusion theory, and the water vapor concentration distribution of the pavement structure is obtained. The results of the temperature field and the water vapor concentration field are used as predefined fields, the influence of water vapor factors is introduced into the pavement structure, the pavement structure under the condition of not considering water vapor (the water vapor concentration in the pavement structure is zero) and the non-uniform water vapor concentration is compared, and the improved Paris formula is used to further establish the pavement structure fatigue cracking model, so as to analyze the change of asphalt pavement fatigue life under the two water vapor concentration conditions, and provide a scientific basis for reducing the generation of crack diseases and constructing long-life durable asphalt pavement.

[0007] To achieve the above purpose, the technical scheme adopted by the present application is:

[0008] A design method for asphalt pavement under humidity environment, which is based on the multi-field coupling effect under humidity environment, and simulates the fatigue cracking change of asphalt pavement in finite element software.

[0009] The design method for asphalt pavement specifically includes the following steps:

[0010] (1) Determine the asphalt pavement structure model: adopt the typical asphalt pavement structure form in Jiangxi Province, establish the pavement structure model in the finite element software, determine the model size, and use the partition tool of the finite element software to complete the division of the pavement structure layer;

[0011] (2) Determine the material parameters of the asphalt pavement structure model: the material parameters include thermal parameters, mechanical parameters, fracture parameters and fatigue cracking parameters, the thermal parameters include thermal conductivity, heat capacity and material density, the mechanical parameters include viscoelastic parameters, elastic modulus and Poisson's ratio, the fracture parameters include fracture energy and tensile strength, and the fatigue cracking parameters include improved Paris formula parameters and limit fracture energy release rate;

[0012] (3) Establishing the finite element model of asphalt pavement structure temperature field: The solar radiation, effective radiation of pavement, air temperature and convective heat exchange of pavement structure are simulated respectively, the thermal boundary conditions of pavement structure in the finite element software are defined, the grid division is carried out based on the asphalt pavement structure model determined in step (1), the thermal parameters of each structure layer are input in the finite element software, the finite element definition of air temperature and convective heat exchange, effective radiation of pavement are completed in the interaction module combined with the user subroutine FILM, and the finite element definition of solar radiation is completed in the load module combined with the user subroutine DFLUX, the atmospheric temperature in low temperature weather in Jiangxi Province (referring to the data of the National Meteorological Science Data Center, the average value of atmospheric temperature of the 10 days with the lowest temperature in January 2020 in Yichun City of Jiangxi Province is selected) is used as the temperature data of pavement structure, the pavement structure temperature is simulated in the finite element, and the pavement structure temperature field is obtained;

[0013] (4) Establishing the finite element model of asphalt pavement structure water vapor concentration field: First, the water vapor diffusion coefficient of asphalt pavement structure is determined and tested, and fitting is carried out to obtain water vapor diffusion coefficient data at different temperatures, the pavement structure temperature field established in step (3) is used as the predefined field, the grid division is the same as the temperature field, in the material parameter module, Fick's law is selected as the mass diffusion rule, the temperature and the corresponding water vapor diffusion coefficient data are input in the diffuse reflectance module of the finite element software, then the upper and lower boundary conditions of pavement structure water vapor concentration are defined and input, the pavement structure water vapor diffusion under the influence of temperature field is simulated in the finite element, and the pavement structure water vapor concentration field under the influence of temperature field is obtained;

[0014] (5) Establishing the fatigue life model of asphalt pavement structure considering humidity environment: Based on the asphalt pavement structure model determined in step (1), the wheel load is set as a half-sine dynamic load, and the wheel load function is defined, the material parameters determined in step (2) are set in the finite element software, the temperature distribution corresponding to the lowest temperature (5h) in the temperature field finite element model established in step (3) is used as the temperature predefined field, the water vapor concentration corresponding to the equilibrium state (2400d) in the water vapor concentration field finite element model established in step (4) is used as the water vapor concentration predefined field of pavement structure, the temperature and water vapor concentration are introduced into the pavement structure model, finally the task is created and submitted in the operation module, and the fatigue life model of asphalt pavement structure considering humidity environment is obtained. At the same time, the case that the water vapor concentration of pavement structure as a whole is zero is set as the control group, which is compared with the pavement fatigue cracking considering humidity environment, and the fatigue cracking of asphalt pavement structure is simulated.

[0015] The specific method for simulating fatigue cracking of the asphalt pavement structure is: fatigue cracking of the asphalt pavement is simulated based on the extended finite element XFEM method and the improved Paris formula, the crack propagation can be directly observed, when the crack enters the unstable failure stage under the action of the load, the crack continuously expands and finally penetrates through the entire surface layer, at this time, the number of load cycles is the fatigue life of the pavement structure, the fatigue life of the pavement under the humidity environment is compared with the control group, so as to adjust the design standard in the actual asphalt pavement design process, and the anti-cracking ability of the pavement is improved.

[0016] Preferably, in step (1), the pavement structure model has a size of 3m in height and 3.75m in width. The pavement structure includes, from top to bottom, a surface layer, a base layer (cement stabilized gravel), a bottom base layer (graded gravel), and a soil base. The surface layer includes an upper surface layer (AC-13C asphalt mixture), a middle surface layer (AC-20C asphalt mixture), and a lower surface layer (AC-25C asphalt mixture). The base layer includes a first base layer (5% cement stabilized gravel) and a second base layer (3% cement stabilized gravel).

[0017] Preferably, in step (2), the viscoelastic parameters are Prony series of relaxation modulus of the asphalt pavement surface layer. The Prony series of relaxation modulus expression is determined based on the indoor uniaxial tensile dynamic modulus test of the asphalt mixture under different temperatures, different water vapor concentrations, and different loading frequencies, the dynamic modulus and phase angle master curve are established, and then the dynamic modulus is converted into relaxation modulus, so that the Prony series of relaxation modulus of the asphalt mixture is determined.

[0018] In the indoor uniaxial tensile dynamic modulus test, the different temperatures include 5℃, 20℃, 35℃, and 50℃, the different water vapor concentrations include 0.35g / m 3 , 8.64g / m 3 , 13.82g / m 3 , and 17.27g / m 3 (corresponding to 2%, 50%, 80%, and 100% relative humidity levels at 20℃, respectively), and the different loading frequencies include 25Hz, 10Hz, 5Hz, 1Hz, 0.5Hz, and 0.1Hz. Finally, the master curve of the non-destructive viscoelastic property parameters under the reference temperature of 20℃ and the reference humidity of 0.35g / m 3 is obtained.

[0019] Preferably, the Prony series of relaxation modulus expression is:

[0020]

[0021] wherein E(t) is a relaxation modulus function, the unit is MPa; t is time; Eg G'(t) is the instantaneous relaxation modulus, unit is MPa; i = 1, 2, …, m, m is the total number of model elements in parallel; E i Ei is the relaxation modulus of the i-th spring; τ i τi is the relaxation time of the i-th spring in the generalized Maxwell model, unit is s; the relaxation modulus Prony series is the parameter E g and E i .

[0022] Preferably, the method of converting dynamic modulus to relaxation modulus comprises:

[0023] S1, respectively determine the Prony series expression of complex modulus, storage modulus and loss modulus, first Fourier transform the Prony series expression of the relaxation modulus of the generalized Maxwell model, let t = iw, w is the angular frequency, rad / s, the complex modulus expression can be obtained as:

[0024] Then according to the viscoelastic relationship, the expression of the storage modulus can be obtained as:

[0025] The expression of the loss modulus is:

[0026] S2, based on the dynamic modulus and phase angle data of the asphalt mixture, solve the storage modulus and instantaneous modulus E g ;

[0027] S3, use the configuration method to solve the parameters in the storage modulus expression E i , select the preset relaxation time point to set the configuration point, and solve the parameters E i corresponding to each relaxation time.

[0028] S4, substitute the solved parameters E i into the relaxation modulus expression to solve the relaxation modulus.

[0029] Preferably, in step (2), the fracture parameters are obtained by indoor semi-circular bending test of the asphalt mixture under different temperatures and different water vapor concentrations, the different temperatures include 5℃, 15℃, 25℃, and the different water vapor concentrations include 0.35g / m 3 , 8.64g / m 3 , 13.82g / m 3 , 17.27g / m 3 , the test loading rate is 2mm / min, and the pre-crack length is 0mm and 10mm respectively, the specimen with 0mm pre-crack length is used for testing the tensile strength of the mixture, and the specimen with 10mm pre-crack length is used for testing the fracture energy of the mixture.

[0030] The fracture parameters are fitted according to the GEP model to obtain the following fitting relationship:

[0031] Fracture energy of AC-13C asphalt mixture: y = 0.88x1 2 -67.19x1+4.65x2 2 -15.81x2+3204.96, R 2 = 0.9821;

[0032] Tensile strength of AC-13C asphalt mixture: y = 0.005x1 2 -0.371x1-0.058x2+8.799, R 2 = 0.9431;

[0033] Fracture energy of AC-20C asphalt mixture: y = -0.54x1 2 -16.85x1+3.93x2 2 -2.53x2+2592.53, R 2 = 0.9842;

[0034] Tensile strength of AC-20C asphalt mixture: y = -0.008x1 2 -0.032x1-0.006x2 2 +0.015x2+8.672, R 2 = 0.9560;

[0035] Fracture energy of AC-25C asphalt mixture: y = -1.64x1 2 +17.13x1+2.44x2 2 +7.57x2+2117.36, R 2 = 0.9798;

[0036] Tensile strength of AC-25C asphalt mixture: y = 0.007x1 2 -0.404x1-0.003x2 2 +0.001x2+7.545, R 2 = 0.9491.

[0037] Wherein, y is the model dependent variable, corresponding to tensile strength, fracture energy respectively; x1, x2 are model independent variables, x1 is temperature (℃), x2 is water vapor concentration (g / m 3 ).

[0038] The fitting relationship can be used to obtain more fracture parameters in the range of temperature and water vapor concentration, and the fracture parameters of the control group with zero water vapor concentration in step (5) can be calculated by the above fitting relationship.

[0039] Preferably, in step (2), the improved Paris formula is expressed by the fracture release rate G, specifically as follows:

[0040]

[0041] where da is the crack propagation rate, a is the crack propagation length, N is the load cycle number variable, ΔG is the change value of fracture release rate, and C' and m' are material-related parameters.

[0042] The fatigue crack propagation can be divided into three main stages: low-speed propagation stage, stable propagation stage, and unstable failure stage. Under the action of load, the fracture energy release rate G generated at the crack tip increases continuously, reaches the fracture energy release rate threshold G t , at which the crack propagation condition is met, and the fatigue crack propagation occurs; with the continuous action of load, the fracture energy release rate at the crack tip continues to increase until the ultimate fracture energy release rate G pl is reached, and the structure fails unstably. Studies have shown that the remaining fatigue life in the third stage is much smaller than that in the first two stages and can be ignored.

[0043] Based on the improved Paris formula, the low-speed stable propagation stage of crack propagation is described to determine the crack initiation condition, and the criterion is: where f is the fatigue crack initiation factor, f ≥ 1.0 when the crack initiates and enters the stable propagation stage; N0 is the load cycle number; and c1 and c2 are both material-related parameters.

[0044] When the crack propagation enters the stable propagation stage, the crack propagation rate is mainly controlled by the improved Paris formula, as shown in the following formula: where c3 and c4 are both material-related parameters.

[0045] c1, c2, c3, and c4 are the improved Paris formula parameters, which can be fitted by standard tests.

[0046] Preferably, in step (3),

[0047] The solar radiation is calculated according to the principle of Fourier series, and the calculation formula is:

[0048] where Q(t) is the solar radiation daily variation function, t is the time variable, Q0 is the daily maximum solar radiation, and Q0 = 0.131nQ t , n = 12 / c, Q t is the total solar radiation per day, J / m 2 , and c is the actual effective sunshine duration, h.

[0049] Fourier coefficient In this formula, t=0 is assumed to be 6:00 am;

[0050] The calculation formula for the effective radiation of the road surface is: e =εσ[(T r -T Z ) 4 -(T a -T Z ) 4 ]; where: q e is the effective radiation of the road surface, J / m 2 ; ε is the emissivity of asphalt pavement, which is taken as 0.81; σ is the Boltzmann constant, which is taken as 5.670×10 -8 W / (m 2 ·K 4 );T r is the road surface temperature, °C; T a is the daily average temperature, ℃; T z The absolute zero value is -273℃.

[0051] The relationship for simulating daily temperature changes is:

[0052] T=T a +T m {0.96sin[ω(t-t0)]+0.14sin[2ω(t-t0)]}; where: T a is the daily average temperature, ℃; T m is the daily temperature amplitude, ℃; t0 is the initial phase, usually set to 9h; ω is the angular frequency, ω=π / 12; T max is the maximum temperature of a single day, ℃; T min The lowest temperature of the day, ℃.

[0053] The heat exchange coefficient is calculated as: k c =3.7v+9.4; where: k c is the heat exchange coefficient, W / (m2·℃); ν is the daily average wind speed, m / s.

[0054] Preferably, in step (4), the water vapor diffusion coefficient is specifically an accumulation type water vapor diffusion coefficient. The accumulation type water vapor diffusion coefficient data at different temperatures can be obtained through indoor water vapor tests, and then the relationship between the accumulation type water vapor diffusion coefficient and temperature is fitted through the Arrhenius equation to expand the water vapor diffusion coefficient of the asphalt surface layer.

[0055] The fitted model is:

[0056] D=A1e -E / RT; wherein: D is the accumulative water vapor diffusion coefficient, A1 is a pre-factor, dimensionless; E is the reaction activation energy, J / mol, R is the universal gas constant, J / (K·mol); T is the K temperature, K.

[0057] The fitting result is: A1=3.753, E=39.197 J / mol, and R=8.314 J / (K·mol).

[0058] In step (4), the upper boundary condition of the water vapor concentration of the pavement structure is the water vapor concentration of the lower layer of the asphalt pavement, which is calculated according to the temperature field finite element model result; and the lower boundary condition of the water vapor concentration of the pavement structure is the water vapor concentration in the atmosphere, which is the water vapor concentration in the atmosphere in Jiangxi Province under low-temperature weather (referring to the data of the National Meteorological Science Data Center, and selecting the water vapor concentration in the 10 days with the lowest temperature in January 2020 in Yichun City, Jiangxi Province).

[0059] Preferably, in step (5), the wheel load function expression is:

[0060]

[0061] In the formula, p(t) is the wheel load function; t is the time variable; p max is the uniform load action value (usually 700 kpa); T is the load action time (s), which is related to the driving speed v and the radius r of the tire contact area, and is reasonably assumed according to the action characteristics of the driving load; when the load action point is more than 6r away from a certain point, it is considered that the point is no longer affected by the load, so the load action time T=12r / v, v is the vehicle speed (m / s), r is the wheel load action radius (m), and in the present model, r=(0.213 / 2)m.

[0062] Technical effects of the present application:

[0063] 1. The present application combines the actual climate conditions of the engineering project, establishes a pavement structure temperature field model based on the finite element software analysis method; on this basis, a non-uniform water vapor concentration field model is established according to the accumulative water vapor diffusion theory, and the water vapor concentration distribution of the pavement structure is obtained. The results of the temperature field and the water vapor concentration field are used as predefined fields, the influence of the water vapor factor is introduced into the pavement structure, the pavement structure under the conditions of not considering the water vapor state (the water vapor concentration in the pavement structure is zero) and the non-uniform water vapor concentration state is compared, the improved Paris formula is further used to establish a pavement structure fatigue cracking model, and the change of the fatigue life of the asphalt pavement under the two water vapor concentration states is analyzed, so as to provide a scientific basis for reducing the generation of crack diseases and constructing long-life durable asphalt pavement.

[0064] 2、Compared with the prior art, the mechanical response of the pavement structure under different water and gas conditions is obtained by using a numerical simulation method, and the fatigue life of the asphalt pavement is estimated by referring to the calculation method of the fatigue life of the pavement structure in the current specification in China. BRIEF DESCRIPTION OF DRAWINGS

[0065] Figure 1 It is a typical asphalt pavement structure finite element component diagram of the application;

[0066] Figure 2 It is a dynamic modulus master curve of AC-13C asphalt mixture of the application;

[0067] Figure 3 It is a phase angle master curve of AC-13C asphalt mixture of the application;

[0068] Figure 4 It is a dynamic modulus master curve of AC-20C asphalt mixture of the application;

[0069] Figure 5 It is a phase angle master curve of AC-20C asphalt mixture of the application;

[0070] Figure 6 It is a dynamic modulus master curve of AC-25C asphalt mixture of the application;

[0071] Figure 7 It is a phase angle master curve of AC-25C asphalt mixture of the application;

[0072] Figure 8 It is a generalized Maxwell model diagram for describing the stress relaxation behavior of asphalt mixture of the application;

[0073] Figure 9 It is the fracture energy of AC-13C asphalt mixture of the application under different test conditions;

[0074] Figure 10 It is the fracture energy of AC-20C asphalt mixture of the application under different test conditions;

[0075] Figure 11 It is the fracture energy of AC-25C asphalt mixture of the application under different test conditions;

[0076] Figure 12 It is the tensile strength of AC-13C asphalt mixture of the application under different test conditions;

[0077] Figure 13 Tensile strength of AC-20C asphalt mixture under different test conditions of the present application;

[0078] Figure 14 Tensile strength of AC-25C asphalt mixture under different test conditions of the present application;

[0079] Figure 15 Grid division schematic diagram of the temperature field finite element model of the present application;

[0080] Figure 16 24h temperature field simulation results of the road surface and the layer bottom of each asphalt surface layer of the present application;

[0081] Figure 17 Water vapor concentration changes of the layer bottom of each asphalt surface layer during humidity field simulation of the present application;

[0082] Figure 18 Name identification statements used for defining different asphalt mixtures when establishing an asphalt pavement cracking model in the finite element software of the present application;

[0083] Figure 19 Pavement structure temperature predefinition field set when establishing an asphalt pavement cracking model in the finite element software of the present application;

[0084] Figure 20 Pavement structure water vapor concentration predefinition field set when establishing an asphalt pavement cracking model in the finite element software of the present application;

[0085] Figure 21 ENRRTXFEM field output setting when establishing an asphalt pavement cracking model in the finite element software of the present application.

[0086] Figure 22 Crack propagation conditions directly observed when simulating the cracking conditions of an asphalt pavement in the finite element software of the present application [(a) simulation conditions of N=0 times, a=2cm, (b) simulation conditions of N=3.079×10 7 times, a=8cm, (c) simulation conditions of N=5.584×10 7 times, a=14cm, (d) simulation conditions of N=5.728×10 7 times, a=18cm];

[0087] Figure 23 Schematic diagram of the relationship between the crack propagation length and the number of load cycle actions when simulating the cracking conditions of an asphalt pavement in the finite element software of the present application. DETAILED DESCRIPTION

[0088] The above scheme is further described below in combination with specific implementations; it should be understood that these embodiments are used to illustrate the basic principles, main features and advantages of the present application, and the present application is not limited in scope by the following embodiments; the implementation conditions used in the embodiments can be further adjusted according to specific requirements, and the implementation conditions not specified are usually the conditions in conventional experiments.

[0089] In the following embodiments, the specific details of asphalt pavement design and construction are referred to the existing highway specifications, such as “Highway Asphalt Pavement Design Specification” JTG D50-2017, “Highway Asphalt Pavement Construction Technical Specification” (JTGF40), “Highway Engineering Asphalt and Asphalt Mixture Inspection Regulation” (JTG E20-2011) and the like.

[0090] Embodiment 1

[0091] The present embodiment provides a design method for asphalt pavement under humidity environment, which is based on the finite element software under the multi-field coupling of humidity environment, and the design method is specifically as follows.

[0092] I. Determining the asphalt pavement structure model

[0093] A typical asphalt pavement structure form in Jiangxi Province is adopted, and a pavement structure model is established in ABAQUS finite element software, the model size is 3m high and 3.75m wide. And according to the project construction drawing design book, the form of pavement structure is determined as “4cm AC-13C asphalt mixture upper layer + 6cm AC-20C asphalt mixture middle layer + 8cm AC-25C asphalt mixture lower layer + 20cm 5% cement stabilized gravel base + 18cm 3% cement stabilized gravel base + 20cm graded gravel bottom base + soil base” (see Figure 1 ).

[0094] The material information of the three kinds of asphalt mixtures of the surface layer is shown in Table 1.

[0095] Table 1

[0096]

[0097] II. Determining the material parameters of the asphalt pavement structure model

[0098] The material parameters of the asphalt pavement structure model include thermal parameters, mechanical parameters, fracture parameters and fatigue cracking parameters, the thermal parameters include thermal conductivity, heat capacity and material density, the mechanical parameters include viscoelastic parameters, elastic modulus and Poisson's ratio, the viscoelastic parameters are Prony series of relaxation modulus of the asphalt pavement surface layer, the fracture parameters include fracture energy and tensile strength, and the fatigue cracking parameters include improved Paris formula parameters of the asphalt pavement surface layer and ultimate fracture energy release rate.

[0099] 1. Thermal parameters

[0100] Thermal parameters include thermal conductivity, heat capacity and material density. The thermal parameters of each layer of pavement material are shown in Table 2.

[0101] Table 2

[0102]

[0103] 2. Viscoelastic parameters of asphalt pavement surface layer

[0104] The Prony series of relaxation modulus of asphalt mixture is a commonly used viscoelastic parameter in finite element method. The present invention establishes the dynamic modulus|E through indoor uniaxial tensile dynamic modulus test. * | and phase angle The Prony series of the relaxation modulus of the asphalt mixture can be determined by converting the dynamic modulus into the relaxation modulus. The method for establishing the master curve of the dynamic modulus and phase angle under humidity conditions can be referred to Reference 1 of the inventor's research group ("Study on the Effect of Humidity on the Viscoelastic Properties and Fatigue Cracking Performance of Asphalt Mixtures," Hou Qiang, Master's Thesis, Wuhan University of Technology, 2020).

[0105] (1) Indoor uniaxial tensile dynamic modulus test

[0106] With reference to the "Testing Procedures for Asphalt and Asphalt Mixtures in Highway Engineering" (JTG E20-2011), uniaxial tensile dynamic modulus tests were carried out indoors on asphalt mixture (AC-13, AC-20 and AC-25) specimens. The dynamic modulus tests were conducted at four temperatures (5°C, 20°C, 35°C and 50°C), six loading frequencies (25Hz, 10Hz, 5Hz, 1Hz, 0.5Hz and 0.1Hz), and a load of 0.35g / m 3 、8.64g / m 3 、13.82g / m 3 、17.27g / m 3 Four water vapor concentration levels (which are absolute humidity, corresponding to the four relative humidity levels of 2%, 50%, 80% and 100% at 20°C). This test uses standard asphalt mixture cylindrical specimens with a diameter of 100mm and a height of 150mm. The various properties of the asphalt mixture selected for the test were verified to meet the requirements through the Marshall test. After the asphalt mixture specimens were cured at four relative humidity levels (2%, 50%, 80% and 100%) for more than 6 months, indoor tests were carried out to measure the dynamic modulus |E of the asphalt mixture at different temperatures, different humidity levels and different loading frequencies. * | and phase angle φ data, the results are shown in Table 3-5.

[0107] Table 3 Dynamic test results of AC-13C asphalt mixture (dynamic modulus unit: MPa, phase angle unit: °)

[0108]

[0109]

[0110] Table 4 Dynamic test results of AC-20C asphalt mixture (dynamic modulus unit: MPa, phase angle unit: °)

[0111]

[0112] Table 5 Dynamic test results of AC-25C asphalt mixture (dynamic modulus unit: MPa, phase angle unit: °)

[0113]

[0114] As shown in Tables 3-5, the viscoelastic behavior of asphalt mixtures changes under the same temperature and frequency conditions after humidity curing at different water vapor concentrations. With increasing water vapor concentration, the dynamic modulus of the mixture decreases and the phase angle increases. This phenomenon is particularly pronounced under high-temperature, low-frequency test conditions. This is because the aggregate has a strong affinity for water. Under the continuous action of water vapor, water vapor intrudes into the interface between the asphalt and the aggregate, causing a decrease in adhesion and thus a change in the overall viscoelastic behavior. To obtain viscoelastic parameters of the asphalt mixture beyond the experimentally defined temperature, frequency, and humidity, the measured dynamic modulus and phase angle will be processed based on the master curve model and the temperature-frequency-humidity joint shift factor to provide a reference for the determination of subsequent numerical simulation parameters.

[0115] (2) Drawing of dynamic modulus and phase angle master curves of asphalt mixture

[0116] a. Based on the time-temperature-humidity equivalence principle, the free volume theory and the Doolittle equation are used to establish the joint displacement factor equation for temperature-humidity coupling displacement, specifically:

[0117]

[0118] Among them, TH is the joint shift factor, C1, C2, C3 are shift parameters, C1 = B / 2.303f0, C2 = f0 / α T , C3=f0 / α H , T is the actual temperature, H is the actual humidity, T0 is the reference temperature, H0 is the reference humidity, B is the material parameter, f0 is the free volume fraction of the material at temperature T0 and humidity H0, α T , α Hare the coefficient of thermal expansion and the coefficient of hygroscopic expansion, respectively.

[0119] b. The generalized sigmoidal model is used as the dynamic modulus master curve model, and based on this, the phase angle master curve model is derived according to the approximate Kramers-Kronig relationship. Combined with the joint shift factor equation for calculation, a dynamic modulus and phase angle master curve model that includes temperature, humidity, and loading frequency is constructed.

[0120] The generalized sigmoid model is:

[0121] The phase angle master curve model is:

[0122] Among them, |E * (f)| is the dynamic modulus function, is the phase angle function, f is the test loading frequency, ζ TH is the joint shift factor, δ, α, γ, β, and λ are the model parameters of the master curve, δ and δ+α are the extreme values ​​of the dynamic modulus master curve, and γ, β, and λ are the shape parameters of the dynamic modulus master curve.

[0123] c. According to the least squares fitting method, the reference temperature is 20℃ and the reference humidity is 0.35g / m 3 Master curves of dynamic modulus and phase angle under different conditions.

[0124] The reference temperature is selected as 20℃ and the reference humidity is 0.35g / m 3 The dynamic modulus and phase angle master curves, as well as the joint shift factor equation of temperature-humidity coupling shift, are fitted to minimize the objective function value. The model parameters and shift parameters are obtained with the help of the planning and solving function of Excel software, as shown in Tables 6 and 7.

[0125] Table 6 Model parameters for dynamic modulus and phase angle master curve fitting

[0126]

[0127]

[0128] Table 7 Shift parameters and fitting determination coefficients of the joint shift factor equation for coupled temperature and humidity shift

[0129]

[0130] According to the model parameters and shift parameters in Tables 6 and 7, the reference temperature is 20 °C and the reference humidity is 0.35 g / m 3 The main curve of dynamic modulus and phase angle is shown in Figures 2-7, which are the dynamic modulus master curves and phase angle master curves of the three asphalt mixtures. The fitting determination coefficients of the dynamic modulus master curves and phase angle master curves of the three asphalt mixtures are all above 0.9, indicating that the master curves can well characterize the viscoelastic properties of asphalt mixtures.

[0131] (3) Determine the Prony series of the relaxation modulus of asphalt mixture

[0132] a. First, determine the Prony series expression of the relaxation modulus of the generalized Maxwell model

[0133] The generalized Maxwell model is used to describe the stress relaxation behavior of asphalt mixture. The constitutive relationship of the model is as follows: Figure 8 (In the figure, E i is the relaxation strength of the i-th spring in the model, MPa; η i is the viscosity of the i-th viscosity pot, i = 1, 2, ..., n, where n is the total number of model elements connected in parallel; Ee is the equilibrium modulus, MPa). As shown in Figure 1, the stress-strain relationship of asphalt mixture under load can be expressed using the Boltzmann superposition principle. Therefore, the relaxation modulus is expressed using the generalized Maxwell model, and the Prony series expression for the relaxation modulus is determined as:

[0134]

[0135] Where E(t) is the relaxation modulus function, in MPa; t is time; E g is the instantaneous relaxation modulus, in MPa; i = 1, 2, ..., m, where m is the total number of model elements in parallel; E i is the relaxation modulus of the i-th spring; τ i is the relaxation time of the i-th spring in the generalized Maxwell model, in seconds; the relaxation modulus Prony series is the parameter E g and E i .

[0136] b. Based on the dynamic modulus and phase angle master curves of the three asphalt mixtures obtained above, the dynamic modulus is converted into the relaxation modulus by combining the viscoelastic parameter conversion relationship and the collocation method. The relaxation modulus parameter of the Prony series expression of the relaxation modulus of the generalized Maxwell model is determined and used as the viscoelastic parameter of the asphalt mixture in the finite element software.

[0137] The conversion relationship of viscoelastic parameters into dynamic modulus |E * |, complex modulus E * , storage modulus E', loss modulus E", phase angle The relationship between: E * =E'+iE",

[0138] The relaxation modulus of asphalt mixture can be converted from the complex modulus, storage modulus, and loss modulus expressions. The specific method for converting dynamic modulus to relaxation modulus includes:

[0139] S1, respectively determine the Prony series expression of complex modulus, storage modulus, and loss modulus. First, Fourier transform the Prony series expression of the generalized Maxwell model relaxation modulus, let t = iw, w is the angular frequency, rad / s, and the complex modulus expression is:

[0140] Then, according to the viscoelastic relationship, the storage modulus expression is:

[0141] The loss modulus expression is:

[0142] S2, based on the dynamic modulus and phase angle data of the asphalt mixture, solve the storage modulus and instantaneous modulus E g ;

[0143] S3, use the configuration method to solve each parameter in the storage modulus expression E i . Select a preset relaxation time point to set the configuration point, and set the relationship between the configuration point w i i and the relaxation time τ i = 2; use the planning solution function in EXCEL to solve the parameters corresponding to each relaxation time E i ;

[0144] S4, substitute each parameter E i solved into the relaxation modulus expression to solve the relaxation modulus.

[0145] The relaxation modulus Prony series of the asphalt pavement surface layer (asphalt mixture) obtained by the above method is shown in Table 8.

[0146] Table 8

[0147]

[0148] 3, Elastic modulus and Poisson's ratio

[0149] The linear elastic parameters are the elastic modulus and Poisson's ratio of the asphalt pavement base layer, subbase layer, and soil base, which are fixed parameters, as shown in Table 9.

[0150] Table 9

[0151]

[0152] 4, Fracture parameters

[0153] The fracture parameters include fracture energy and tensile strength. The cracking behavior of asphalt mixture can be mainly divided into linear viscoelastic fracture mechanics and viscoelastic-plastic fracture mechanics. When the temperature is 10℃ higher than the low temperature limit of the PG grade of the used asphalt, the cracking behavior of the mixture is considered to conform to the linear viscoelastic fracture mechanics theory, and the fracture energy G f is often used to characterize the crack resistance of the mixture; when the temperature is in the medium temperature range, the cracking behavior of the mixture shows viscoelastic-plastic fracture behavior, and the fracture energy G f cannot comprehensively characterize the crack resistance of the mixture, and the tensile strength also needs to be used for comprehensive evaluation. The fracture energy and the tensile strength are not only important indicators for evaluating the failure and damage of asphalt mixture, but also important parameters for simulating the cracking of the mixture. Among them, the fracture energy is used to define the damage evolution criterion of the model, and the tensile strength is used to define the damage initiation criterion of the model. The fracture energy and the tensile strength need to be tested and calculated.

[0154] (1) Semi-circular bending test for testing fracture energy and tensile strength

[0155] According to the semi-circular bending test procedure of AASHTO TP 105-2013 standard of American National Highway and Transportation Association, the semi-circular bending test of asphalt mixture (AC-13, AC-20 and AC-25) specimens is carried out in the laboratory. The test temperature conditions are medium temperature (5℃, 15℃ and 25℃), the relative humidity conditions are 2%, 50%, 80% and 100%, the loading rate is 2mm / min, and the pre-crack length is 0mm and 10mm. The specimen without pre-cutting is used for testing the tensile strength of the mixture, and the specimen with 10mm pre-cutting is used for testing the fracture energy of the mixture. For the case of 10mm cutting depth, a handheld cutting saw is used to cut the bottom of the semi-circular bending specimen at the centerline position, and the cutting width is 1mm.

[0156] In this test, the diameter of the semi-circular bending specimen of asphalt mixture is 150mm, and the thickness is 50mm. The properties of the selected asphalt mixture are verified by Marshall test to meet the requirements. After the asphalt mixture sample is humidified for four months at four relative humidity levels (2%, 50%, 80% and 100%), the semi-circular bending test is carried out to measure the fracture energy and the peak load size data of the three types of asphalt mixture under different conditions. The fracture energy and the tensile strength are calculated according to the following calculation formula, and the results are shown in Table 1. Figures 9-14 .

[0157] The fracture energy calculation formula is as follows:

[0158] In the formula, G f is the fracture energy, mJ / mm 2 ; W fFor fracture energy, J; A act = (r-c) x b, A act For effective fracture area, mm 2 ; r is the radius of the semi-circular bending specimen, mm; c is the pre-crack length, mm; b is the thickness of the semi-circular bending specimen, mm.

[0159] The tensile strength calculation formula is as follows:

[0160] In the formula: R is the tensile strength, MPa; P is the peak load size, N; L is the support spacing, mm, which is 0.8D in this test; t is the specimen thickness, mm; h is the specimen height (D / 2), mm; D is the diameter of the semi-circular bending specimen, mm.

[0161] (2) Fracture parameter fitting

[0162] Based on the fracture parameter data obtained from the above test, the GEP model is used to fit the fracture parameters of different types of asphalt mixtures, and the relative square root error RRSE and the determination coefficient R 2 The model is evaluated, and the model fitting results are shown in Table 10, where y is the dependent variable of the model, corresponding to the tensile strength and fracture energy respectively; x1, x2 are the independent variables of the model, where x1 is the temperature (℃), and x2 is the water vapor concentration (g / m 3 ).

[0163] Among them, the applicability function based on the relative square root error is as follows:

[0164] In the formula: RRSE is the relative square root error; is the data predicted by the model; y i is the data obtained by the test, is the average value of the data obtained by the test.

[0165] The determination coefficient is also a commonly used index to evaluate the fitting and prediction effect of the model, and the calculation method is as follows:

[0166] The parameter meanings are the same as the above RRSE calculation formula.

[0167] Table 10 GEP model fitting results of fracture parameters

[0168]

[0169] 5. Fatigue cracking parameters of asphalt pavement surface

[0170] The fatigue cracking parameters of the asphalt pavement surface include the improved Paris formula parameters and the ultimate fracture energy release rate.

[0171] (1) Improvement of Paris formula

[0172] The classic Paris formula is mainly applicable to linear elastic materials and cannot accurately evaluate the viscoelastic material such as asphalt mixture. On this basis, document 2 (Shen Q. Research on fatigue crack propagation of semi-rigid base asphalt pavement based on XFEM [D]. Changsha: Hunan University, 2018.) improves the Paris formula by using the fracture release rate G instead of the stress intensity factor, as shown in the following formula:

[0173]

[0174] da / dN is the crack propagation rate, a is the crack propagation length, N is the number of load cycles, ΔG is the difference between the maximum fracture release rate and the minimum fracture release rate, C' and m' are material-related parameters, which can be fitted by standard semi-circular bending fatigue test.

[0175] (2) Limiting fracture energy release rate of asphalt mixture

[0176] The propagation of fatigue cracks can be divided into three main stages: low-speed propagation stage, stable propagation stage and unstable destruction stage. Under the action of load, the fracture energy release rate G generated at the crack tip increases continuously, reaches the fracture energy release rate threshold G t , at this time, the crack propagation condition is met, and the crack occurs fatigue propagation; with the continuous action of load, the fracture energy release rate at the crack tip continues to increase, until the limiting fracture energy release rate G pl , the structure is unstable and destroyed. Studies have shown that the remaining fatigue life in the third stage is much smaller than that in the first two stages and can be ignored.

[0177] (3) Improvement of Paris formula parameters

[0178] The present application describes the low-speed stable propagation stage of fatigue crack propagation based on the phenomenological model, which is used to judge the crack initiation condition, and the judgment criterion is as follows:

[0179]

[0180] In the formula: f is the fatigue crack initiation factor, when f≥1.0, the crack occurs and enters the stable propagation stage; N0 is the load cycle number at which the fatigue crack propagation starts; c1 and c2 are both fatigue crack propagation initiation parameters related to the material.

[0181] When the crack propagation enters the stable propagation stage, the crack propagation rate is mainly controlled by the improved Paris formula, as shown in the following formula:

[0182]

[0183] wherein: c3 and c4 are both material-related parameters, which can be obtained by fitting standard semi-circular bending fatigue tests.

[0184] It can be known from document 3 (Liu Y. Dynamic response and fracture performance of asphalt mixture based on semi-circular bending test [D]. Harbin: Harbin Institute of Technology, 2009.) and document 4 (Liu Y, Zhang Xiaoning, Zou Guilian. Fatigue crack propagation analysis of asphalt mixture based on semi-circular specimen [J]. Science Technology and Engineering, 2009, 19 (9): 5712-5716.) that there is a certain difference in the fatigue cracking parameters of the asphalt mixture, but they are very close overall. In order to simplify the simulation process, it is assumed in the application that the Paris formula parameters of the asphalt surface layer material under different environmental conditions remain unchanged, the test method for determining the parameters and the initial value of the parameters are selected with reference to document 2 (Shen Q. Research on fatigue propagation of reflective cracking of semi-rigid base asphalt pavement based on XFEM [D]. Changsha: Hunan University, 2018.), the initial value in the document is fitted, and the limit fracture energy release rate is 1800 N / m, and the improved Paris formula parameters are: c1 is 0.5, c2 is -1.7, c3 is 2.64*10 -8 , and c4 is 1.16.

[0185] III. Establishing a finite element model of the temperature field of the asphalt pavement structure

[0186] The following assumptions are made when simulating the temperature field model of the road structure:

[0187] (1) Except for the surface layer asphalt concrete which is a viscoelastic material and is defined by viscoelastic parameters, the materials of the other layers are all regarded as linear elastic materials and are defined by linear elastic parameters.

[0188] (2) The road structure layers are in complete contact, the temperature and heat flow change continuously, and the displacement change is continuous.

[0189] 1. Heat transfer

[0190] The pavement structure is in the external environment, and heat transfer occurs continuously with the atmosphere with the change of seasons and day and night, resulting in a change in the temperature of the pavement structure. The main forms of heat transfer include solar radiation, heat conduction and heat convection, which can be realized by setting the form of the thermal boundary condition in the finite element software. The solar radiation of the pavement structure, the effective radiation of the pavement, the air temperature and the convective heat exchange are simulated respectively, and the definition of the thermal boundary condition of the pavement structure in the finite element software is completed.

[0191] (1) Solar radiation

[0192] In the finite element simulation, the solar radiation refers to the heat radiation directly reaching the ground surface from the sun. The solar radiation is calculated according to the principle of Fourier series, and the calculation formula is as follows:

[0193]

[0194] wherein Q(t) is a daily variation function of the solar radiation; t is a time variable; Q0 is a daily maximum solar radiation, and is usually taken as Q0=0.131nQ t , n=12 / c, Q t is a total solar radiation per day, J / m 2 ; and c is an actual effective sunshine duration, h.

[0195] Fourier coefficients In the formula, it is set that t=0 at 6 o'clock in the morning.

[0196] In the actual situation, the solar radiation is not completely absorbed by the asphalt pavement, and part of the solar radiation is reflected by the asphalt pavement. The solar radiation absorption rate of the asphalt pavement is generally taken as 0.9, and then the solar radiation absorbed by the pavement is Qs=0.9Q.

[0197] (2) Effective radiation of pavement

[0198] The effective radiation of pavement is the difference between the long-wave radiation emitted by the pavement and the atmospheric inverse radiation absorbed by the pavement, which represents the net loss of long-wave radiation of the pavement, and is mainly related to the pavement temperature, air temperature and atmospheric humidity. The following formula is used to realize the boundary condition definition of the effective radiation of pavement:

[0199] q e =εσ[(T r -T Z ) 4 -(T a -T Z ) 4 ];

[0200] In the formula, q e is the effective radiation of pavement, J / m 2 ; ε is the emissivity of asphalt pavement, taken as 0.81; σ is the Boltzmann constant, taken as 5.670×10 -8 W / (m 2 ·K 4 ); T r is the road surface temperature, ℃; T a is the average temperature per day, ℃; and T z is the absolute zero value, taken as -273 ℃.

[0201] (3) Air temperature and convective heat exchange

[0202] In addition to the effect of solar radiation, atmospheric temperature and heat exchange will also affect the temperature of the pavement structure, which can be simulated by the following formula:

[0203] T=T a +T m {0.96sin[ω(t-t0)]+0.14sin[2ω(t-t0)]}

[0204] T a =(T max +T min ) / 2

[0205] T m =(T max -T min ) / 2;

[0206] Where: T a is the daily average temperature, ℃; T m is the daily temperature amplitude, ℃; t0 is the initial phase, usually set to 9h; ω is the angular frequency, ω=π / 12; T max is the maximum temperature of a single day, ℃; T min The lowest temperature of the day, ℃.

[0207] The heat exchange coefficient between the pavement structure and the atmosphere is affected by wind speed, and the expression is as follows:

[0208] k c =3.7v+9.4;

[0209] Where: k c is the heat exchange coefficient, W / (m 2 ·℃); ν is the daily average wind speed, m / s. According to statistics from the National Meteorological Science Data Center, the annual average wind speed in Jiangxi Province is about 2.7 m / s.

[0210] 2. Temperature data selection of pavement structure

[0211] Studies have shown that cracking is the primary symptom of asphalt pavement damage in the medium- and low-temperature range. To investigate the impact of moisture on the crack resistance of asphalt pavements, this study selected actual climate data, including temperature, humidity, and wind speed, from a section of the Yisui Expressway in Yichun City, Jiangxi Province, to conduct relevant research. Referring to actual meteorological data and to mitigate the effects of inclement weather, such as rainy days, this study selected the average atmospheric temperature of the ten lowest days in January 2020 in Yichun City, Jiangxi Province, as the temperature data for the pavement structure, which is listed in Table 11.

[0212] Table 11 Average temperature at each hour on the coldest day in January in Yichun, Jiangxi Province

[0213]

[0214] 3. Grid division of the temperature field finite element model

[0215] In the process of setting the pavement temperature field and water vapor concentration field by the finite element software, the grid of the model to be established in connection needs to be completely the same, so as to realize the calling of the unit material parameters. In order to import the parameters of temperature and water vapor concentration into the pavement cracking model, the grid division of the pavement temperature field and water vapor concentration field model needs to be consistent with the pavement cracking model. The finite element software limits the size of the model grid by the way of meshing, and the present application adopts the way of combining global meshing and edge meshing to divide the model grid. The grid is set in different forms of density, the grid of the surface layer is distributed more, the grid of the subbase is distributed less, the grid with large density is set in the crack propagation area and the load acting area, and the grid with less density is set in other areas, so as to balance the calculation efficiency and the calculation accuracy of the simulation. The model after the grid division is shown in Figure 15 .

[0216] 4. Temperature field simulation

[0217] The thermal parameters of each structural layer are input in the finite element software, the finite element definition of the air temperature and the convective heat exchange, the effective radiation of the pavement is completed in the interaction module combined with the user subroutine FILM, and the finite element definition of the solar radiation is completed in the load module combined with the user subroutine DFLUX. According to the data of the National Meteorological Science Center, the total amount of solar daily radiation of the pavement structure is determined as 10.61 MJ / m 2 , the average wind speed is 2.7 m / s, and the setting of the temperature boundary, the unit type, the boundary constraint condition and the like of the model is completed. The atmospheric temperature in the low temperature weather in Jiangxi Province is combined as the temperature data of the pavement structure (Table 11), the pavement structure temperature is simulated in the finite element, the simulation results of the road surface and the layer bottom of each asphalt surface layer are drawn into a graph as shown in Figure 16 , and the pavement structure temperature field model is obtained.

[0218] As shown in Figure 16 , the degree of influence of the pavement structure temperature on the external environment decreases with the increase of the depth of the pavement structure layer. The result file of the temperature field simulation is exported, which is used as the predefined field of the water vapor diffusion simulation of the pavement structure.

[0219] Four, establishment of the water vapor concentration field finite element model of the asphalt pavement structure

[0220] 1. Accumulation type water vapor diffusion coefficient

[0221] During the whole process of pavement construction, asphalt mixture is always in a high temperature state above 100℃, which leads to the mixture containing almost no moisture, and at this time the water vapor concentration inside the newly paved asphalt pavement can be considered to be zero. The surface layer of the newly paved asphalt pavement is completely exposed to the external environment, and the internal water vapor concentration will change dynamically under the influence of the environment. The overall pavement structure is built on the subgrade, and studies have shown that the humidity of the subgrade soil is close to the saturated humidity, always maintaining a relative humidity of 98% to 100%; at the same time, the external environment also maintains a certain humidity, which forms a water vapor concentration difference between the subgrade and the external environment above the asphalt pavement, driving water vapor to continuously diffuse from the place with high concentration to the place with low concentration, and finally reaching a dynamic balance state of water vapor diffusion between the subgrade soil, the pavement structure and the external environment. According to the change rule of the water vapor concentration inside the newly paved asphalt pavement, the water vapor diffusion in the pavement is divided into two types: accumulative water vapor diffusion and penetrating water vapor diffusion. Studies have shown that when the water vapor reaches an accumulative balance in the surface layer of the asphalt pavement, the water vapor distribution inside the pavement structure no longer changes greatly, and the accumulative balance state of the water vapor can be considered to be the true water vapor distribution of the pavement structure. Therefore, in the present application, the accumulative water vapor diffusion coefficient of the surface layer of the asphalt pavement is used to measure the strength of the accumulative balance of water vapor inside the asphalt pavement, and the unit is m 2 / s or mm 2 / s.

[0222] According to reference 5 (Xi Lei. Influence of Relative Humidity on the Mechanical Properties of Asphalt Mixture under Pressure[D]. Wuhan: Wuhan University of Technology, 2021), reference 6 (Chongzhi Tou. Influence Law and Simulation Method of Water Vapor Movement on Asphalt-Aggregate Adhesion[D]. Wuhan: Wuhan University of Technology, 2021.), reference 7 (Huang Tingting. Research on the Accumulative Water Vapor Movement Mechanism of Asphalt Mixture[D]. Wuhan: Wuhan University of Technology, 2018.) and reference 8 (Liu Ziyao. Research on the Influence Factors of Penetrating Water Vapor Diffusion of Asphalt Mixture[D]. Wuhan: Wuhan University of Technology, 2018.), the research team of the present inventors found that the time required for the humidity of AC-13C, AC-20C and AC-25C asphalt mixtures to reach a balanced state is roughly the same, and the accumulative water vapor diffusion coefficients of the three types of asphalt mixtures are in the same order of magnitude, so it can be assumed that the accumulative water vapor diffusion coefficients of the three types of asphalt mixtures are the same. Based on the fact that the accumulative water vapor diffusion coefficient of AC-20C limestone asphalt mixture has been measured by the research team of the present inventors (reference 7), the accumulative water vapor diffusion coefficients of the three types of asphalt mixtures at different temperatures can be obtained as shown in Table 12.

[0223] Table 12

[0224]

[0225] The present inventors' research group has only carried out the test of the accumulated water vapor diffusion coefficient in a partial temperature range, and in the actual pavement structure, the temperature range in the pavement structure is larger, in order to obtain the accumulated water vapor diffusion coefficient under more temperature conditions, the present application carries out fitting on the relationship between the accumulated water vapor diffusion coefficient and the temperature through the Arrhenius equation, and the fitting model is as follows:

[0226] D=A1e -E / RT , wherein D is the accumulated water vapor diffusion coefficient, A1 is a pre-factor, and is dimensionless; E is a reaction activation energy, J / mol, R is a general gas constant, J / (K·mol); and T is a K temperature, K.

[0227] The water vapor fitting result is shown in Table 13, and the model fitting coefficients are obtained: the fitting result is A1=3.753, E=39.197 J / mol, and R=8.314 J / (K·mol). From the fitting result, the goodness of fit R 2 of the accumulated water vapor diffusion coefficient prediction model can reach more than 0.997, which proves that the water vapor diffusion coefficient and the temperature relationship obtained by using the Arrhenius equation fitting has high accuracy.

[0228] Table 13

[0229]

[0230] The present application can expand the water vapor diffusion coefficient of the asphalt surface layer through the above fitting model to obtain the accumulated water vapor diffusion coefficient in a larger temperature range, and the material parameters for providing the pavement structure water vapor concentration field model are shown in the following Table 14.

[0231] Table 14

[0232]

[0233] 2, parameter setting

[0234] The pavement structure temperature field simulation result established in the foregoing is taken as a predefined field, in the material parameter module, Fick's law is selected as a mass diffusion rule, and the temperature and the corresponding water vapor diffusion coefficient are input in the diffusivity module of the finite element software. The solubility refers to the solubility of the diffusing gas in the diffusion medium, in order to reduce the influence of gas dissolution on water vapor diffusion, the solubility is uniformly set to 10 -7 in the model.

[0235] 3, determining the upper and lower boundary conditions of the water vapor concentration of the pavement structure

[0236] The relative humidity and the water vapor concentration can be converted by the ideal gas state equation, and the conversion relationship is shown in the following formula:

[0237]

[0238] wherein: R is a proportionality constant, J / (K·mol); T is temperature in Kelvin, K; P T is the water saturated vapor pressure at the corresponding temperature, Pa; m H2O is the relative molecular mass of water, 18.015 g / mol; C is the water vapor concentration, g / m 3

[0239] The pavement structure is built on the soil base, and some scholars calculate the relative humidity of the soil base based on the soil suction, which is always maintained at about 100%. In the pavement structure form selected in the present application, the base layer is in direct contact with the soil base, and the void ratio of the base layer is large, so its diffusion coefficient is much larger than that of the asphalt surface layer. To simplify the simulation calculation, it is assumed that the relative humidity of the base structure is maintained at 100%, and the lower boundary of the water vapor concentration of the pavement structure is the bottom surface of the asphalt lower surface layer. According to the calculation results of the temperature field, the temperature of the structure layer is stabilized at 7.34℃, and the water vapor concentration is calculated to be 7.93 g / m 3 The upper surface of the asphalt surface layer is in direct contact with the air, and the water vapor concentration in the air will affect the water vapor distribution in the pavement structure. In this paper, the average relative humidity values of the lowest 10 days of January 2020 in Yichun City are obtained from the data of the National Meteorological Science Data Center, as shown in Table 15 below, and the water vapor concentration values are converted by the conversion relationship between relative humidity and water vapor concentration as the upper boundary condition of the water vapor concentration of the pavement structure. Through calculation, the water vapor concentration in the air is 6.13 g / m 3

[0240] Table 15 Average relative humidity values of the lowest 10 days in January in Yichun City, Jiangxi Province

[0241]

[0242]

[0243] 4. Establishing the water vapor concentration field of the pavement structure

[0244] The upper and lower boundary conditions of the water vapor concentration of the pavement structure are input in the finite element, the non-uniform water vapor diffusion of the pavement structure is simulated, and the water vapor concentration field of the pavement structure under the influence of the temperature field is obtained.

[0245] The model is analyzed by using the mass diffusion analysis step, and since the temperature corresponding to each hour of the model is different, multiple analysis steps need to be set according to the running time of the model. The water vapor diffusion of the pavement structure needs a long time to reach a relatively stable state, and the present application simulates the water vapor diffusion phenomenon of the pavement structure (upper surface layer and lower surface layer) under this environment, as shown in Figure 17 ​​The simulation results show that when water vapor diffuses for about 2400 days, the water vapor concentration in the pavement structure reaches a relatively stable state. Influenced by the upper and lower water vapor concentration boundary conditions, the water vapor concentration in the asphalt surface layer is maintained at 6-7 g / m 3 Around, which shows that the pavement structure is in a relatively stable water vapor concentration environment, and it is very important to study the influence of water vapor on the anti-cracking performance of asphalt pavement. Subsequently, the simulation results of the temperature field and the water vapor concentration field will be used as predefined fields to study the influence of water vapor on the anti-cracking performance of asphalt pavement.

[0246] Five, establishment of asphalt pavement structure fatigue life model considering humidity environment

[0247] 1, pavement structure model

[0248] The size of the pavement structure model is consistent with the temperature field and water vapor concentration field model as described in the first part, with a height of 3 m and a width of 3.75 m. According to the road structure size specified in the Figure 1 The structure layers are divided using the zoning tool according to the road structure size specified in the

[0249] 2, wheel load definition

[0250] The current asphalt pavement design specification in China specifies that the standard design axle load is a single axle double wheel load BZZ-100, with an axle load of 100 kN and a tire ground pressure of 0.7 MPa. In this model study, a double wheel group axle is used, with two equivalent circles of diameter 0.213 m representing the wheel load on each side. In the ABAQUS finite element software, different vehicle loads can be defined by setting the action area, action time and load size of the load.

[0251] The wheel load is set as a half-sine dynamic load to simulate the pavement structure, and the wheel load function expression is: Where p(t) is the wheel load function, t is the time variable, p maxis the uniform load value (usually 700 kpa); T is the load action time (s), which is related to the driving speed v and the radius r of the tire contact area, and is reasonably assumed according to the characteristics of the driving load, when the load action point is more than 6r away from a point, it is considered that the point is no longer affected by the load, so the load action time T = 12r / v, v is the speed (m / s), r is the wheel load action radius (m), in this model r = (0.213 / 2) m.

[0252] 3. Setting of material parameters of the asphalt pavement structure model

[0253] According to the material parameters (thermal parameters, viscoelastic parameters, elastic modulus, Poisson's ratio, fracture parameters and fatigue cracking parameters) of the asphalt pavement structure model determined in the first part, the corresponding material parameters are set.

[0254] Among them, the finite element realizes the definition of viscoelastic parameters and fracture parameters through UTRS and USDFLD subprograms. In order to be closer to the actual situation, the viscoelastic parameters (Prony series of relaxation modulus, see Table 8) are used to define the material parameters of the asphalt pavement surface layer, and the linear elastic material is used to define the base layer, subbase layer and other structure layers, and the elastic modulus and Poisson's ratio parameters (see Table 9) are selected. In addition, the fracture energy and tensile strength (see Table 10) are used to set the damage evolution criterion and damage initiation criterion of the model respectively. The load is set as the standard driving load. Figures 9-14

[0255] In the pavement structure, it is necessary to define three kinds of asphalt mixtures at the same time, at this time, the name recognition statement is needed to identify the type of mixture, and then the change of material parameters is carried out, and the name recognition statement of the USDFLD subprogram part is shown in Table 8. Figure 18

[0256] The finite element software can define the fatigue cracking criterion of the model by editing the keyword in the contact attribute module, so as to realize the fatigue analysis of the asphalt pavement. In order to realize the cyclic loading of the load, the model uses the cyclic analysis step based on the Fourier series method for analysis, in order to ensure that the number of cycles is sufficient to realize the fatigue failure of the asphalt pavement, the maximum number of cycles of the analysis step is set to 10 7 times. The fatigue cracking criterion of the model is controlled by the parameters of the improved Paris formula and the ultimate fracture energy release rate of the asphalt mixture.

[0257] 4. Setting of predefined field

[0258] Before simulating the crack propagation of the pavement structure, the temperature field and the water concentration field of the pavement structure also need to be set, so as to realize the calling of the material parameters of the asphalt surface layer.

[0259] (1) Temperature pre-defined field ​​

[0260] Research shows that under low temperature conditions, asphalt pavement is mainly damaged by cracking. Figure 16 The simulation results of the medium temperature field show that the temperature of the entire pavement structure remains below 19°C. The overall temperature of the pavement structure is lowest when the temperature field increment corresponds to 5 hours. Therefore, this paper intends to use the temperature distribution corresponding to the 5-hour temperature model as the predefined temperature field.

[0261] (2) Predefined field of water vapor concentration

[0262] from Figure 17 The simulation results of the water vapor concentration field show that the pavement structure reaches a state of water vapor accumulation equilibrium after 2400 days of water vapor diffusion. This paper primarily studies the impact of water vapor concentration, so the predefined temperature field remains consistent under different operating conditions. The water vapor concentration field is needed to characterize the long-term water vapor distribution of the pavement structure. Therefore, the water vapor concentration at equilibrium (2400 days) is selected as the predefined water vapor concentration field for the pavement structure.

[0263] By setting the above predefined fields, the temperature and water vapor concentration factors can be introduced into the pavement structure model, such as Figure 19 、 20 As shown, this is the predefined field of pavement structure temperature and the predefined field of water vapor concentration.

[0264] 5. Set up a control group

[0265] To quantify the impact of moisture on the crack resistance of asphalt pavements, this study used a control group to analyze the cracking behavior of the pavement structure at different moisture concentrations. The control group assumed a zero moisture concentration for the entire pavement structure. The model's fracture parameters (fracture energy and tensile strength) were calculated using the GEP model fitting results for the fracture parameters in Table 10 and defined directly in the material parameter module. The dynamic modulus data tested at this concentration was used to define the model's viscoelastic parameters.

[0266] 6. Job submission and post-processing

[0267] After completing the model settings, you need to import the user subroutine file in the manager to call the material parameters. After the task is completed, create a task in the job module and submit it.

[0268] In order to read the information of crack extension in post-processing, it is necessary to set the field output. However, the output of energy release rate cannot be defined in the CAE interface. The output of ENRRTXFEM can be realized by editing keywords, such as Figure 21 shown.

[0269] 7. Simulation of fatigue life of asphalt pavement under humidity environment

[0270] This paper simulates fatigue cracking of asphalt pavement based on the XFEM method and the improved Paris formula. The pavement structure model adopts a two-dimensional model, which can directly observe the expansion of cracks. During the simulation, the length a of the crack expansion and the corresponding number of load cycles N change as follows: Figure 22 (where (a) is the simulation case with N=0 times and a=2 cm, and (b) is the simulation case with N=3.079×10 7 For the simulation case of times and a=8 cm, (c) is N=5.584×10 7 For the simulation case of times and a=14 cm, (d) is N=5.728×10 7 times, a = 18 cm simulation case).

[0271] Depend on Figure 22 It can be seen that under the cyclic action of load, the cracks expand freely inside the unit and eventually penetrate the entire asphalt surface layer, which proves that the above method can realize the calculation of the fatigue life of asphalt pavement.

[0272] Assuming that the initial crack has extended to 2cm below the layer, the pavement structure under two environmental conditions, considering non-uniform water vapor concentration (humidity) and the control group, is simulated. The results of the change in crack extension length and load cycle number are shown in Table 16 and Figure 23 shown.

[0273] Table 16

[0274]

[0275]

[0276] From Table 16 and Figure 23 As can be seen from the figure, as the number of load cycles increases, when the fracture energy release rate at the crack tip reaches the threshold for fatigue cracking, the crack enters a stable expansion phase, and the crack length steadily increases. As the number of load cycles further increases, the fracture energy release rate at the crack tip reaches its upper limit, and the crack enters an unstable failure phase. At this point, the crack grows rapidly, penetrating the entire asphalt surface layer. The number of load cycles at this time is the fatigue life of the pavement structure. The lower and middle layers account for over 98% of the fatigue life of the entire surface layer, while the upper layer accounts for less than 2%. This indicates that the fatigue life of the asphalt surface layer is significantly affected by the middle and lower layers. When the crack extends to the upper layer, the crack growth rate is fastest. The fatigue life of the pavement structure under the control (zero water vapor concentration) is 6.544×10 7 times, while the fatigue life of the pavement structure considering the non-uniform water vapor concentration field is 5.728×10 7Secondly, the fatigue life is reduced by 14.246%. This shows that with the increase of water vapor concentration, the anti-fatigue cracking ability of asphalt pavement decreases, and fatigue cracks are more likely to occur. In areas with high water vapor concentration, the design standard of pavement fatigue life should be improved to improve the overall anti-cracking ability of pavement.

Claims

1. A method for designing asphalt pavement under humidity conditions, characterized by: The design method is based on the multi-field coupling effect of the humidity environment, and simulates the fatigue cracking changes of the asphalt pavement in the finite element software; The specific steps include: (1) Determine the asphalt pavement structure model: Use a typical asphalt pavement structure form to establish a pavement structure model in finite element software, determine the model size, and use finite element software to complete the division of the pavement structure layer; (2) Determine the material parameters of the asphalt pavement structure model: the material parameters include thermal parameters, mechanical parameters, fracture parameters and fatigue cracking parameters. The thermal parameters include thermal conductivity, heat capacity and material density. The mechanical parameters include viscoelastic parameters, elastic modulus and Poisson's ratio. The fracture parameters include fracture energy and tensile strength. The fatigue cracking parameters include improved Paris formula parameters and ultimate fracture energy release rate. (3) Establish a finite element model of the temperature field of the asphalt pavement structure: simulate the solar radiation, effective pavement radiation, air temperature and convective heat exchange of the pavement structure respectively, complete the definition of the thermal boundary conditions of the pavement structure in the finite element software, mesh the asphalt pavement structure model determined in step (1), input the thermal parameters of each structural layer in the finite element software, and complete the finite element definition of solar radiation, effective pavement radiation, air temperature and convective heat exchange respectively. Combined with the atmospheric temperature in low temperature weather in Jiangxi Province as the temperature data of the pavement structure, simulate the pavement structure temperature in the finite element to obtain the pavement structure temperature field; (4) Establish a finite element model of the water vapor concentration field of the asphalt pavement structure: First, determine and test the water vapor diffusion coefficient of the asphalt pavement structure and perform fitting to obtain water vapor diffusion coefficient data at different temperatures. The pavement structure temperature field established in step (3) is used as the predefined field. Fick's law is selected as the mass diffusion law. The temperature and corresponding water vapor diffusion coefficient data are input into the diffuseness module of the finite element software. Then, the upper and lower boundary conditions of the water vapor concentration of the pavement structure are defined and input. The water vapor diffusion of the pavement structure is simulated in the finite element to obtain the water vapor concentration field of the pavement structure under the influence of the temperature field. (5) Consider the establishment of the fatigue life model of the asphalt pavement structure under a humid environment: Based on the asphalt pavement structure model determined in step (1), the wheel load is set as a semi-sinusoidal dynamic load, and the wheel load function is defined. The material parameters determined in step (2) are set in the finite element software, and the temperature distribution corresponding to the lowest temperature in the temperature field finite element model established in step (3) is used as the temperature predefined field. The water vapor concentration when reaching the equilibrium state in the water vapor concentration field finite element model established in step (4) is used as the water vapor concentration predefined field of the pavement structure. The temperature and water vapor concentration are introduced into the pavement structure model. Finally, a task is created and submitted in the job module to obtain the fatigue life model of the asphalt pavement structure under a humid environment.

2. The method for designing asphalt pavement under humidity conditions according to claim 1, characterized in that: In step (1), the model size is: 3m high and 3.75m wide. The pavement structure includes a surface layer, a base layer, a subbase layer and a soil base from top to bottom. The surface layer adopts asphalt mixture, and includes an upper layer, a middle layer and a lower layer in sequence, and adopts asphalt mixtures AC-13C, AC-20C and AC-25C in sequence.

3. The method for designing an asphalt pavement under a humidity environment according to claim 1, characterized in that: In step (2), the viscoelastic parameter is the relaxation modulus Prony series of the asphalt pavement surface layer. The dynamic modulus and phase angle master curve can be established by conducting indoor uniaxial tensile dynamic modulus tests of asphalt mixtures at different temperatures, different water vapor concentrations, and different loading frequencies. At the same time, the relaxation modulus Prony series expression is determined based on the generalized Maxwell model. Then, the dynamic modulus is converted into the relaxation modulus to determine the relaxation modulus Prony series of the asphalt mixture.

4. The method for designing asphalt pavement under humidity conditions according to claim 1, characterized in that: In step (2), the fracture parameters are obtained through indoor semicircular bending tests of asphalt mixtures at different temperatures and different water vapor concentrations. The fracture parameters are fitted according to the GEP model to obtain the following fitting relationship: Fracture energy of AC-13C asphalt mixture: y=0.88x1 2 -67.19x1+4.65x2 2 -15.81x2+3204.96, R 2 =0.9821; Tensile strength of AC-13C asphalt mixture: y=0.005x1 2 -0.371x1-0.058x2+8.799, R 2 =0.9431; Fracture energy of AC-20C asphalt mixture: y=-0.54x1 2 -16.85x1+3.93x2 2 -2.53x2+2592.53, R 2 =0.9842; Tensile strength of AC-20C asphalt mixture: y=-0.008x1 2 -0.032x1-0.006x2 2 +0.015x2+8.672, R 2 =0.9560; Fracture energy of AC-25C asphalt mixture: y=-1.64x1 2 +17.13 x1+2.44x2 2 +7.57x2+2117.36, R 2 =0.9798; Tensile strength of AC-25C asphalt mixture: y=0.007x1 2 -0.404x1-0.003x2 2 +0.001x2+7.545, R 2 =0.9491; Among them, y is the model dependent variable, corresponding to tensile strength and fracture energy respectively; x1 and x2 are the model independent variables; x1 is temperature, ℃; x2 is water vapor concentration; g / m 3 .

5. The method for designing asphalt pavement under humidity conditions according to claim 1, characterized in that: In step (2), the improved Paris formula is expressed by the fracture release rate G, specifically: ； is the crack growth rate, a is the crack growth length, N is the number of load cycles, ΔG is the change in fracture release rate, C´ and m´ are material-related parameters; The improved Paris formula parameters include parameters of the low-speed stable expansion stage of crack expansion and parameters of the stable expansion stage of crack expansion; Based on the improved Paris formula, the slow and stable expansion stage of crack propagation is described to judge the crack initiation. The judgment criteria are: ; Where: ƒ is the fatigue crack initiation factor. When ƒ≥1.0, the crack starts to crack and enters the stable expansion stage. N0 is the number of load cycles. c1 and c2 are both material-related parameters. When the crack propagation enters the stable propagation stage, the crack propagation rate is mainly controlled by the improved Paris formula, as shown in the following formula: ; Where: c3 and c4 are parameters related to the material; c1, c2, c3 and c4 are the parameters of the improved Paris formula, which can be obtained by fitting the standard test.

6. The method for designing an asphalt pavement under a humidity environment according to claim 1, characterized in that: In step (3), The solar radiation is calculated according to the Fourier series principle, and the calculation formula is: Where Q(t) is the diurnal variation function of solar radiation; t is the time variable; Q0 is the maximum solar radiation of the day, usually Q0=0.131nQ t , n=12 / c, Q t is the total solar radiation per day, J / m 2 ; c is the actual effective sunshine duration, h; Fourier coefficient , in this formula, t=0 is set at 6:00 am; The calculation formula for the effective radiation of the road surface is: ; Where: q e is the effective radiation of the road surface, J / m 2 ; ε is the emissivity of asphalt pavement, which is taken as 0.81; σ is the Boltzmann constant, which is taken as 5.670×10 -8 W / (m 2 ·K 4 ) ; T r is the road surface temperature, °C; T a is the daily average temperature, ℃; T z For absolute zero, take -273°C; The relationship for simulating daily temperature changes is: Where: T a is the daily average temperature, ℃; T m is the daily temperature amplitude, °C; t0 is the initial phase, usually set to 9h; ω is the angular frequency, ω=π / 12; T max is the maximum temperature of a single day, ℃; T min is the lowest temperature of a single day, ℃; The heat exchange coefficient is calculated as: Where: k c is the heat exchange coefficient, W / (m2·℃); ν is the daily average wind speed, m / s.

7. The method for designing asphalt pavement under humidity conditions according to claim 1, characterized in that: In step (4), the water vapor diffusion coefficient is specifically an accumulation type water vapor diffusion coefficient. The accumulation type water vapor diffusion coefficient data at different temperatures can be obtained through indoor water vapor tests. Then, the relationship between the accumulation type water vapor diffusion coefficient and temperature is fitted using the Arrhenius equation, and the water vapor diffusion coefficient of the asphalt surface layer can be expanded. The fitted model is: Where: D is the cumulative water vapor diffusion coefficient, A1 is the dimensionless pre-exponential factor, E is the activation energy of the reaction, J / mol, R is the universal gas constant, J / (K·mol), and T is the temperature in Kelvin, K. The fitting results are: A1=3.753, E=39.197 J / mol, R=8.314 J / (K·mol); The upper boundary condition of the water vapor concentration in the pavement structure is the water vapor concentration in the layer below the asphalt pavement, which is calculated based on the temperature obtained from the results of the temperature field finite element model; the lower boundary condition of the water vapor concentration in the pavement structure is the water vapor concentration in the atmosphere, which is the atmospheric water vapor concentration under low temperature weather in the area.

8. The method for designing an asphalt pavement under a humidity environment according to claim 1, characterized in that: In step (5), the wheel load function expression is: ； Where p(t) is the wheel load function; t is the time variable; p max is the value of the uniform step load; T is the load action time, s; T=12r / v; v is the vehicle speed, m / s; r is the wheel load action radius, m.

9. The method for designing asphalt pavement under humidity conditions according to claim 1, characterized in that: Step (5) further includes: setting the condition where the water vapor concentration of the entire pavement structure is zero as a control group, comparing it with the pavement fatigue cracking condition under the humidity environment, and simulating fatigue cracking of the asphalt pavement structure; The specific method for simulating fatigue cracking of asphalt pavement structure is as follows: based on the extended finite element (XFEM) method and the improved Paris formula, the fatigue cracking of asphalt pavement is simulated, and the expansion of cracks can be directly observed. When the cracks enter the unstable failure stage under the action of load, the cracks continue to expand and eventually penetrate the entire surface layer. The number of load cycles at this time is the fatigue life of the pavement structure. The fatigue life of the pavement under the humidity environment is compared with the control group, so as to adjust the design standards in the actual asphalt pavement design process to improve the pavement's resistance to cracking.

10. The method for designing asphalt pavement under humidity conditions according to claim 3, characterized in that: The relaxation modulus Prony series expression is: ； Where E(t) is the relaxation modulus function, in MPa; t is time; E g is the instantaneous relaxation modulus, in MPa; i = 1, 2, …, m, where m is the total number of model elements in parallel; E i is the relaxation modulus of the i-th spring; τ i is the relaxation time of the i-th spring in the generalized Maxwell model, in seconds; the relaxation modulus Prony series is the parameter E g and E i .

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