# A Research on Fatigue Damage Constitutive Equation of Asphalt Mixture

Authors: Yazhen Sun; Chenze Fang; Dong Fan; Jinchang Wang; Xuezhong Yuan
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The laboratory investigations of fatigue damage constitutive equation of asphalt mixture were carried out by three-point bending fatigue tests. The three-point bending fatigue tests were performed at three levels of stress-strength ratio (SSR), temperature, and loading rate. The coupled multifactor (stress-strength ratio, temperature, and loading rate) fatigue life equation was established, which can well predict the fatigue life of the asphalt mixture. Both a damage model and a damage evolution equation have been established based on theE-N curve, which indicate that fatigue damage evolution is nonlinear and consists of three stages. The sensitivity analysis of damage model parameters indicates that each parameter has different effects on the three stages of damage evolution. Based on the researches above, the fatigue damage constitutive equations were finally built based on theσ-εcurves, which consist of two parts: the damage accumulation stage and the fatigue failure stage. The elasticity-power hardening model was used to describe the constitutive relation of damage accumulation stage. The elasticity-power hardening model and theSidoroffdamage model were used to describe the constitutive relation of damage failure stage. The constitutive equations can well characterize the fatigue damage performance of the asphalt mixtures under cyclic loading.


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## Body

## 1. Introduction

In recent years, the road construction industry has made great progress with the rapid development of economy, but the quality of the existing roads is uneven. The asphalt concrete pavement, as the surface course of the road structure, is exposed completely to the environment, receiving repeated actions of vehicle loads and environment influence by seasonal changes. The internal stresses and strains of the material change continuously, resulting in the degradation of strength under repetitive load cycles. The pavement under a cumulative load is prone to fatigue failure. The fatigue behaviors of pavement materials have become the research directions of an increasing number of researchers [1–4].In order to describe the damage evolution and stress-strain relationship of asphalt mixture, many damage models have been proposed [1, 2, 5], such asLolanddamage model,Sidoroff damage model,Mazarsdamage model, andExpAssodamage model. TheLolanddamage model indicates that the stress-strain relationship of material is linear after reaching the peak stress, which is an approximate version [6]. The constitutive equation corresponding toSidoroff damage model, obtained based on the energy equivalence assumption, indicates that the stress-strain relationship of material is linear before reaching the peak stress, which is an approximate version [7]. TheMazarsdamage model indicates that material is not damaged before reaching the peak stress, which is an approximate version [8]. TheExpAssodamage model consisted of four parameters indicating that fatigue damage evolution is nonlinear and consists of three stages [9]. However, the physical meanings of model parameters have not been analyzed.Although the above researches are a big step towards the goal of describing damage evolution and constitutive equation asphalt mixture, there are concerns associated with these researches. The above models only studied the constitutive equations of materials under a single load and did not study the constitutive equations of materials subjected to cyclic loading. However, asphalt pavement is subjected to vehicle cyclic loading, and the researches on the fatigue damage constitutive equation of asphalt mixture under cyclic loading are rarely found.To describe the stress-strain relationship of asphalt mixtures under cyclic loading more intuitively and accurately, it is necessary to obtain a simplified and specific constitutive equation. Due to the discontinuity and nonuniformity of the asphalt mixture at the micro level, the constitutive relationship of the asphalt mixture in a fatigue test is more complicated. The constitutive equation and damage evolution of each damage model also differ widely. It is not appropriate to describe the constitutive equations only by using elastic or plastic theory, nor is it to describe the fatigue damage constitutive equation by applying only one existing damage model directly [4, 10, 11].Laboratory fatigue tests have been carried out to solve the problems above. The influence of experimental factors on test results is studied to obtain the coupled multifactor fatigue life equation. Both a damage model and a damage evolution equation have been established based on theE-N curve. The sensitivity analysis of damage model parameters was used to study the physical meanings of model parameters. Based on the researches above, the fatigue damage constitutive equations were finally built based on theσ-εcurves, which consists of two parts: the damage accumulation stage and fatigue failure stage.


## 2. Establishment of Fatigue Equation

### 2.1. Test Design

#### 2.1.1. Test Materials

Bitumen penetration grade 70 was used as asphalt binder for preparations of the specimens, with its specifications listed in Table1. The aggregate gradation type is AC-13, as listed in Table 2.Table 1
Properties of asphalt rubber.
Material property Used standard Value Softening point T0606—2000 57 (°C) Viscosity T0625—2000 1.5-4.0 (Pa·s) Elastic recovery T0662—2000 30 (%)Table 2
Aggregate gradation.
Sieve size (mm) 16.0 13.2 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075 Passing percentage 100.0 91.1 80.2 54.0 33.2 22.5 16.0 12.1 8.7 5.5


#### 2.1.2. Preparation of Specimen

A rut board of 400 mm × 400 mm × 70 mm was performed by using a hydraulic sample forming machine, presented in Figure1. By cutting the rut board, the specimen beams of 250 mm × 30 mm × 35 mm were obtained, and it is presented in Figure 2.Figure 1
Rut board forming machine.Figure 2
Specimen beams.


#### 2.1.3. Test Conditions and Methods

The laboratory fatigue test is presented in Figure3. The flexural tensile strengths of specimens under different test conditions were measured. The specimens were tested using three stress-strength ratios: 0.3, 0.4, and 0.5; three temperatures: 5, 15, and 25°C; and three loading rates: 5, 10, and 20 mm/min.Figure 3
Laboratory fatigue test.


### 2.2. Test Results

The fatigue test was carried out according to the test scheme, and the fatigue lives are listed in Table3, from which it can be seen that there is a negative correlation between the fatigue life and the stress-strength ratio and loading rate and a positive correlation between the fatigue life and the temperature.Table 3
The fatigue life.
Loading rate (V) / mm / min Stress-strength ratio (SSR) Value (Nf) 5°C 15°C 25°C 5 0.3 1,586 2,770 4,842 0.4 631 1,413 2,000 0.5 324 490 920 10 0.3 1,324 2,313 4,044 0.4 617 1,122 1,585 0.5 229 407 776 20 0.3 924 1,613 2,820 0.4 432 631 1,096 0.5 175 303 537


### 2.3. The Influence of Experimental Factors on Test Results

The fatigue failure of the beam specimen is mainly caused by the tensile stress at the bottom of the specimen, because the ultimate tensile stress of asphalt mixture is far weaker than its ultimate compressive stress. When the cyclic load below the ultimate bending resistance is applied to the beam specimen, the fatigue failure of the beam specimen occurs because of the stress concentration at the bottom, and the fatigue damage is mainly manifested by fatigue cracks. The growth process of a fatigue crack consisted of two stages: the crack initiation and crack propagation. Damage gradually accumulates with the increase of load number. Fatigue cracks occur at the bottom of the beam specimen when the damage is accumulated to a certain extent. Fatigue cracks will expand due to the increase of load number. The cumulative result of the expansion is the fracture of beam specimen [12–14].


#### 2.3.1. The Influence of Stress-Strength Ratio on Fatigue Test

The fatigue lives under the stress-strength ratio of 0.3, 0.4, and 0.5, respectively, are presented in Table3. It can be seen from Table 3 that there is a negative correlation between the fatigue life and the stress level. The larger the stress-strength ratio is and the greater the load is applied, the greater the accumulation of damage is after each cycle of loading and the shorter the fatigue life becomes [15].Linear fitting for the logarithmic fatigue life measured under the stress-strength ratio of 0.3, 0.4, and 0.5, respectively, was carried out, and the results are presented in Figure4. The curve equations of fatigue life are listed in Table 4.Table 4
Linear regression results.
Temperature Loading rate (V) / mm / min (T) / °C 5 10 20 5 lg ⁡ N f=4.21-3.45SSR, R2=0.98312 lg ⁡ N f=4.28-3.80SSR, R2=0.98901 lg ⁡ N f=4.06-3.61SSR, R2=0.99507 15 lg ⁡ N f=4.59-3.76SSR, R2=0.96751 lg ⁡ N f=4.51-3.77SSR, R2=0.98162 lg ⁡ N f=4.28-3.62SSR, R2=0.98989 25 lg ⁡ N f=4.75-3.60SSR, R2=0.96643 lg ⁡ N f=4.66-3.58SSR, R2=0.96643 lg ⁡ N f=4.51-3.60SSR, R2=0.96643Figure 4
The logarithmic fatigue lives at different stress-strength ratios.
(a)
5°C (b)
15°C (c)
25°CIt can be seen from Figure4 and Table 4 that the logarithmic fatigue life decreases with the increase of stress-strength ratio. A linear function can well describe the relation of logarithmic fatigue life to stress-strength ratio, which can be expressed as(1)lg⁡Nf=a-bSSRwhere lg⁡Nf is the logarithmic fatigue life, SSR is the stress-strength ratio, and a and b are constants.


#### 2.3.2. The Influence of Temperature on Fatigue Test

The fatigue lives at temperature of 5, 15, and 25°C, respectively, are presented in Figure5. As shown in the figure, there is a positive correlation between the fatigue life and the temperature. Asphalt mixture is a viscoelastic material. The healing of asphalt mixtures improves with the increase of temperature. In the meantime, the corresponding fatigue life increases [16–18].Figure 5
Fatigue lives at different temperatures.
(a)
5 mm/min (b)
10 mm/min (c)
20 mm/min


#### 2.3.3. The Influence of Loading Rate on Fatigue Test

The fatigue lives under the loading rate of 5 mm/min, 10 mm/min, and 20 mm/min, respectively, are presented in Figure6. As shown in the figure, there is a negative correlation between the fatigue life and the loading rate. The mechanical behavior of asphalt mixture under dynamic load is different from that under static load. In addition to the magnitude of the load, the mechanical behavior is also related to the loading rate and the deformation rate of the material. When a cyclic loading was applied to the beam specimen, the deformation rate increased with the increase of the loading rate. After each loading cycle, the healing of asphalt mixtures is not fully exploited, which leads to a rise in the rate of damage accumulation and a decline in fatigue life [12, 18–20].Figure 6
Fatigue lives under different loading rates.
(a)
5°C (b)
15°C (c)
25°C


### 2.4. Construction of Coupled Multifactor Fatigue Equation

Based on the analysis above, it is obvious that the fatigue life is related to stress-strength ratio, temperature, and loading rate. In order to accurately describe the relationship between fatigue life and experimental factors, linear regressions were made in the three experimental factors. A linear function can well describe the relation of logarithmic fatigue life to the three experimental factors, which can be expressed as(2)lg⁡Nf=4.26838+0.02355T-3.64776SSR-0.01619V,R2=0.99374where lg⁡Nf is the logarithmic fatigue life,T is the temperature, SSR is the stress-strength ratio, and V is the loading rate.Equation (2) is a fatigue equation that couples the stress-strength ratio, the temperature, and the loading rate.


### 2.5. The Analysis of the Fatigue Equation

The load magnitude applied to the asphalt mixture is reflected by the stress-strength ratio of fatigue tests and the axle load of asphalt pavements. The specimen subjected to a larger stress-strength ratio, which corresponds to a larger axle load that a pavement is subjected to, has a shorter fatigue life.The number of vehicle loads in a specific time is reflected by the loading rate of fatigue tests and the traffic volume of asphalt pavements. The specimen subjected to a larger loading rate, which corresponds to a larger traffic volume that a pavement is subjected to, has a shorter fatigue life.It can be seen from the qualitative analysis that the coupled multifactor fatigue equation can well characterize the influence of the vehicle load and traffic volume on the fatigue life of asphalt pavements.


## 2.1. Test Design

### 2.1.1. Test Materials

Bitumen penetration grade 70 was used as asphalt binder for preparations of the specimens, with its specifications listed in Table1. The aggregate gradation type is AC-13, as listed in Table 2.Table 1
Properties of asphalt rubber.
Material property Used standard Value Softening point T0606—2000 57 (°C) Viscosity T0625—2000 1.5-4.0 (Pa·s) Elastic recovery T0662—2000 30 (%)Table 2
Aggregate gradation.
Sieve size (mm) 16.0 13.2 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075 Passing percentage 100.0 91.1 80.2 54.0 33.2 22.5 16.0 12.1 8.7 5.5


### 2.1.2. Preparation of Specimen

A rut board of 400 mm × 400 mm × 70 mm was performed by using a hydraulic sample forming machine, presented in Figure1. By cutting the rut board, the specimen beams of 250 mm × 30 mm × 35 mm were obtained, and it is presented in Figure 2.Figure 1
Rut board forming machine.Figure 2
Specimen beams.


### 2.1.3. Test Conditions and Methods

The laboratory fatigue test is presented in Figure3. The flexural tensile strengths of specimens under different test conditions were measured. The specimens were tested using three stress-strength ratios: 0.3, 0.4, and 0.5; three temperatures: 5, 15, and 25°C; and three loading rates: 5, 10, and 20 mm/min.Figure 3
Laboratory fatigue test.


## 2.1.1. Test Materials

Bitumen penetration grade 70 was used as asphalt binder for preparations of the specimens, with its specifications listed in Table1. The aggregate gradation type is AC-13, as listed in Table 2.Table 1
Properties of asphalt rubber.
Material property Used standard Value Softening point T0606—2000 57 (°C) Viscosity T0625—2000 1.5-4.0 (Pa·s) Elastic recovery T0662—2000 30 (%)Table 2
Aggregate gradation.
Sieve size (mm) 16.0 13.2 9.5 4.75 2.36 1.18 0.6 0.3 0.15 0.075 Passing percentage 100.0 91.1 80.2 54.0 33.2 22.5 16.0 12.1 8.7 5.5


## 2.1.2. Preparation of Specimen

A rut board of 400 mm × 400 mm × 70 mm was performed by using a hydraulic sample forming machine, presented in Figure1. By cutting the rut board, the specimen beams of 250 mm × 30 mm × 35 mm were obtained, and it is presented in Figure 2.Figure 1
Rut board forming machine.Figure 2
Specimen beams.


## 2.1.3. Test Conditions and Methods

The laboratory fatigue test is presented in Figure3. The flexural tensile strengths of specimens under different test conditions were measured. The specimens were tested using three stress-strength ratios: 0.3, 0.4, and 0.5; three temperatures: 5, 15, and 25°C; and three loading rates: 5, 10, and 20 mm/min.Figure 3
Laboratory fatigue test.


## 2.2. Test Results

The fatigue test was carried out according to the test scheme, and the fatigue lives are listed in Table3, from which it can be seen that there is a negative correlation between the fatigue life and the stress-strength ratio and loading rate and a positive correlation between the fatigue life and the temperature.Table 3
The fatigue life.
Loading rate (V) / mm / min Stress-strength ratio (SSR) Value (Nf) 5°C 15°C 25°C 5 0.3 1,586 2,770 4,842 0.4 631 1,413 2,000 0.5 324 490 920 10 0.3 1,324 2,313 4,044 0.4 617 1,122 1,585 0.5 229 407 776 20 0.3 924 1,613 2,820 0.4 432 631 1,096 0.5 175 303 537


## 2.3. The Influence of Experimental Factors on Test Results

The fatigue failure of the beam specimen is mainly caused by the tensile stress at the bottom of the specimen, because the ultimate tensile stress of asphalt mixture is far weaker than its ultimate compressive stress. When the cyclic load below the ultimate bending resistance is applied to the beam specimen, the fatigue failure of the beam specimen occurs because of the stress concentration at the bottom, and the fatigue damage is mainly manifested by fatigue cracks. The growth process of a fatigue crack consisted of two stages: the crack initiation and crack propagation. Damage gradually accumulates with the increase of load number. Fatigue cracks occur at the bottom of the beam specimen when the damage is accumulated to a certain extent. Fatigue cracks will expand due to the increase of load number. The cumulative result of the expansion is the fracture of beam specimen [12–14].


### 2.3.1. The Influence of Stress-Strength Ratio on Fatigue Test

The fatigue lives under the stress-strength ratio of 0.3, 0.4, and 0.5, respectively, are presented in Table3. It can be seen from Table 3 that there is a negative correlation between the fatigue life and the stress level. The larger the stress-strength ratio is and the greater the load is applied, the greater the accumulation of damage is after each cycle of loading and the shorter the fatigue life becomes [15].Linear fitting for the logarithmic fatigue life measured under the stress-strength ratio of 0.3, 0.4, and 0.5, respectively, was carried out, and the results are presented in Figure4. The curve equations of fatigue life are listed in Table 4.Table 4
Linear regression results.
Temperature Loading rate (V) / mm / min (T) / °C 5 10 20 5 lg ⁡ N f=4.21-3.45SSR, R2=0.98312 lg ⁡ N f=4.28-3.80SSR, R2=0.98901 lg ⁡ N f=4.06-3.61SSR, R2=0.99507 15 lg ⁡ N f=4.59-3.76SSR, R2=0.96751 lg ⁡ N f=4.51-3.77SSR, R2=0.98162 lg ⁡ N f=4.28-3.62SSR, R2=0.98989 25 lg ⁡ N f=4.75-3.60SSR, R2=0.96643 lg ⁡ N f=4.66-3.58SSR, R2=0.96643 lg ⁡ N f=4.51-3.60SSR, R2=0.96643Figure 4
The logarithmic fatigue lives at different stress-strength ratios.
(a)
5°C (b)
15°C (c)
25°CIt can be seen from Figure4 and Table 4 that the logarithmic fatigue life decreases with the increase of stress-strength ratio. A linear function can well describe the relation of logarithmic fatigue life to stress-strength ratio, which can be expressed as(1)lg⁡Nf=a-bSSRwhere lg⁡Nf is the logarithmic fatigue life, SSR is the stress-strength ratio, and a and b are constants.


### 2.3.2. The Influence of Temperature on Fatigue Test

The fatigue lives at temperature of 5, 15, and 25°C, respectively, are presented in Figure5. As shown in the figure, there is a positive correlation between the fatigue life and the temperature. Asphalt mixture is a viscoelastic material. The healing of asphalt mixtures improves with the increase of temperature. In the meantime, the corresponding fatigue life increases [16–18].Figure 5
Fatigue lives at different temperatures.
(a)
5 mm/min (b)
10 mm/min (c)
20 mm/min


### 2.3.3. The Influence of Loading Rate on Fatigue Test

The fatigue lives under the loading rate of 5 mm/min, 10 mm/min, and 20 mm/min, respectively, are presented in Figure6. As shown in the figure, there is a negative correlation between the fatigue life and the loading rate. The mechanical behavior of asphalt mixture under dynamic load is different from that under static load. In addition to the magnitude of the load, the mechanical behavior is also related to the loading rate and the deformation rate of the material. When a cyclic loading was applied to the beam specimen, the deformation rate increased with the increase of the loading rate. After each loading cycle, the healing of asphalt mixtures is not fully exploited, which leads to a rise in the rate of damage accumulation and a decline in fatigue life [12, 18–20].Figure 6
Fatigue lives under different loading rates.
(a)
5°C (b)
15°C (c)
25°C


## 2.3.1. The Influence of Stress-Strength Ratio on Fatigue Test

The fatigue lives under the stress-strength ratio of 0.3, 0.4, and 0.5, respectively, are presented in Table3. It can be seen from Table 3 that there is a negative correlation between the fatigue life and the stress level. The larger the stress-strength ratio is and the greater the load is applied, the greater the accumulation of damage is after each cycle of loading and the shorter the fatigue life becomes [15].Linear fitting for the logarithmic fatigue life measured under the stress-strength ratio of 0.3, 0.4, and 0.5, respectively, was carried out, and the results are presented in Figure4. The curve equations of fatigue life are listed in Table 4.Table 4
Linear regression results.
Temperature Loading rate (V) / mm / min (T) / °C 5 10 20 5 lg ⁡ N f=4.21-3.45SSR, R2=0.98312 lg ⁡ N f=4.28-3.80SSR, R2=0.98901 lg ⁡ N f=4.06-3.61SSR, R2=0.99507 15 lg ⁡ N f=4.59-3.76SSR, R2=0.96751 lg ⁡ N f=4.51-3.77SSR, R2=0.98162 lg ⁡ N f=4.28-3.62SSR, R2=0.98989 25 lg ⁡ N f=4.75-3.60SSR, R2=0.96643 lg ⁡ N f=4.66-3.58SSR, R2=0.96643 lg ⁡ N f=4.51-3.60SSR, R2=0.96643Figure 4
The logarithmic fatigue lives at different stress-strength ratios.
(a)
5°C (b)
15°C (c)
25°CIt can be seen from Figure4 and Table 4 that the logarithmic fatigue life decreases with the increase of stress-strength ratio. A linear function can well describe the relation of logarithmic fatigue life to stress-strength ratio, which can be expressed as(1)lg⁡Nf=a-bSSRwhere lg⁡Nf is the logarithmic fatigue life, SSR is the stress-strength ratio, and a and b are constants.


## 2.3.2. The Influence of Temperature on Fatigue Test

The fatigue lives at temperature of 5, 15, and 25°C, respectively, are presented in Figure5. As shown in the figure, there is a positive correlation between the fatigue life and the temperature. Asphalt mixture is a viscoelastic material. The healing of asphalt mixtures improves with the increase of temperature. In the meantime, the corresponding fatigue life increases [16–18].Figure 5
Fatigue lives at different temperatures.
(a)
5 mm/min (b)
10 mm/min (c)
20 mm/min


## 2.3.3. The Influence of Loading Rate on Fatigue Test

The fatigue lives under the loading rate of 5 mm/min, 10 mm/min, and 20 mm/min, respectively, are presented in Figure6. As shown in the figure, there is a negative correlation between the fatigue life and the loading rate. The mechanical behavior of asphalt mixture under dynamic load is different from that under static load. In addition to the magnitude of the load, the mechanical behavior is also related to the loading rate and the deformation rate of the material. When a cyclic loading was applied to the beam specimen, the deformation rate increased with the increase of the loading rate. After each loading cycle, the healing of asphalt mixtures is not fully exploited, which leads to a rise in the rate of damage accumulation and a decline in fatigue life [12, 18–20].Figure 6
Fatigue lives under different loading rates.
(a)
5°C (b)
15°C (c)
25°C


## 2.4. Construction of Coupled Multifactor Fatigue Equation

Based on the analysis above, it is obvious that the fatigue life is related to stress-strength ratio, temperature, and loading rate. In order to accurately describe the relationship between fatigue life and experimental factors, linear regressions were made in the three experimental factors. A linear function can well describe the relation of logarithmic fatigue life to the three experimental factors, which can be expressed as(2)lg⁡Nf=4.26838+0.02355T-3.64776SSR-0.01619V,R2=0.99374where lg⁡Nf is the logarithmic fatigue life,T is the temperature, SSR is the stress-strength ratio, and V is the loading rate.Equation (2) is a fatigue equation that couples the stress-strength ratio, the temperature, and the loading rate.


## 2.5. The Analysis of the Fatigue Equation

The load magnitude applied to the asphalt mixture is reflected by the stress-strength ratio of fatigue tests and the axle load of asphalt pavements. The specimen subjected to a larger stress-strength ratio, which corresponds to a larger axle load that a pavement is subjected to, has a shorter fatigue life.The number of vehicle loads in a specific time is reflected by the loading rate of fatigue tests and the traffic volume of asphalt pavements. The specimen subjected to a larger loading rate, which corresponds to a larger traffic volume that a pavement is subjected to, has a shorter fatigue life.It can be seen from the qualitative analysis that the coupled multifactor fatigue equation can well characterize the influence of the vehicle load and traffic volume on the fatigue life of asphalt pavements.


## 3. Construction of Damage Evolution Equation

### 3.1. Construction of Damage Evolution Equation

The stress-strain relation of damaged specimens during fatigue test is described on the basis of strain equivalence and damage mechanics, i.e.,(3)E=E01-D(4)D=1-EE0where E0 is the initial modulus,E is the modulus during fatigue loading, and D is the damage factor.TheE-N curve is shown in Figure 7. As shown in the figure, the modulus decreases with the increase of the number of loading cycles. TheE/E0-N/Nf curve was obtained by dividing the abscissa and the ordinate ofE-N curve by E0 and Nf, respectively. TheE/E0-N/Nf curve is shown in Figure 8. It can be seen that the process can be divided into three stages. In the first stage, the modulus decreases sharply and fatigue cracks emerge. In the second stage, the modulus decreases stably and fatigue cracks expand. In the third stage, the modulus decreases rapidly and the beam fractures finally. The cumulative result of the expansion was the fracture of the beam specimen. The following equation was used to fit theE/E0-N/Nf curve:(5)EE0=1-∑i=12Ai1-e-N/Nf/Kiwhere Ai and Ki are model parameters,N is the number of loading cycles, and Nf is the fatigue life.Figure 7
E-Ncurve.Figure 8
E/ E 0 -N/ N f curve.The model parameters are obtained by fitting the experimental data. As listed in Table5, the squared correlation coefficients (R2) are over 0.99. As can be seen from Table 5, the fitting parameters are related to the experimental factors, such as the temperature, the loading rate, and the stress-strength ratio. A quadratic polynomial is used to fit the experimental data. The regression results are as follows.Table 5
Fitting parameters.
Temperature(T) / °C Loading rate(V) / mm/min stress-strength ratio (SSR) A 1 K 1 A 2 K 2 5 5 0.3 0.495 0.034 0.538 0.997 0.4 0.465 0.072 0.633 1.089 0.5 0.350 0.094 1.332 1.837 10 0.3 0.471 0.041 0.511 0.837 0.4 0.442 0.087 0.601 1.304 0.5 0.333 0.113 1.271 3.137 20 0.3 0.434 0.054 1.184 2.588 0.4 0.308 0.068 1.233 2.360 0.5 0.192 0.122 1.134 1.878 15 5 0.3 0.422 0.034 0.535 0.502 0.4 0.350 0.053 0.648 0.729 0.5 0.302 0.096 1.253 2.546 10 0.3 0.419 0.044 0.520 0.710 0.4 0.332 0.063 0.615 0.875 0.5 0.287 0.115 1.190 3.055 20 0.3 0.308 0.043 0.605 0.888 0.4 0.320 0.080 1.221 2.200 0.5 0.109 0.112 1.276 1.677 25 5 0.3 0.554 0.013 0.269 0.117 0.4 0.499 0.017 0.307 0.228 0.5 0.485 0.025 0.401 0.598 10 0.3 0.527 0.016 0.283 0.161 0.4 0.486 0.022 0.327 0.298 0.5 0.477 0.032 0.421 0.546 20 0.3 0.435 0.032 0.393 0.365 0.4 0.329 0.057 0.482 0.518 0.5 0.315 0.081 0.635 0.953A 1 = 0.685−0.0282T+0.00106 T2+0.00474 × 1012V−5.364 × 10−4  V2−0.0157 SSR−0.825 SSR2,R2 = 0.874.K 1 = −0.00976+0.00286T−1.673 × 10−4  T2+0.00292 × 1011V−5.528 × 10−5  V2−0.0201 SSR+0.35628 SSR2,R2 = 0.82995.A 2 = 1.330+0.0355T−0.00210 T2−0.0347V+0.00205V2−4.924 SSR+8.983 SSR2,R2 = 0.757.K 2 = 2.628+0.0413T−0.00364 T2+0.0735V−0.00152 V2−13.579 SSR+23.267 SSR2, R2 = 0.614.The damage model function is expressed as(6)D=1-EiE0=∑i=12Ai1-e-N/Nf/KiDamage evolution equation is obtained by calculating the derivative of (6), and the damage evolution equation is expressed as(7)dDdN=∑i=12Aie-N/Nf/Ki1KiNfTheD-N/Nf curve is shown in Figure 9. As presented in the figure, the damage increases nonlinearly during the fatigue tests, and the process can be divided into three stages, which is similar to the process of degradation of modulus. In the first stage, the damage lasts a short time but increases rapidly, and, in the end of this stage, it reaches a large value. In the second stage, the damage lasts a long time and increases stably. In the third stage, the damage increases sharply and the beam fractures finally [9].Figure 9
D-N/ N f curve.


### 3.2. Sensitivity Analysis of Damage Model Parameters

As shown in Table5, the value ranges of A1, K1, A2, and K2 are 0.110-0.554, 0.013-0.122, 0.269-1.134, and 0.118-1.878, respectively. Three data sets are taken within the range of each parameter and are listed in Table 6. The influences of the four parameters (A1, K1, A2, K2) on the three stages of damage evolution are analyzed.Table 6
The parameter values.
Parameter A 1 K 1 A 2 K 2 Value 0.1,0.3,0.5 0.04,0.08,0.12 0.2,0.4,0.6 0.1,0.6,1.1


#### 3.2.1. Analysis of ParameterA1

Set the values for parameterA1 as 0.1, 0.3, and 0.5, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 10. As shown in the figure, the three stages of damage evolution are greatly influenced by parameter A1. The larger parameter A1 is, the more serious the degree of the damage of material becomes. Parameter A1 is a main parameter that influences the degree of the damage of material.Figure 10
D-N/ N f curves (A1=0.1, 0.3, and 0.5).


#### 3.2.2. Sensitivity Analysis of ParameterK1

Set the values for parameterK1 as 0.04, 0.08, and 0.12, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 11. As shown in the figure, as the parameter K1 changes, the second and third stages of damage evolution are less influenced, but the first stage is greatly influenced. The larger parameter K1 is, the longer the duration of the first stage lasts. Parameter K1 is a parameter that influences the rate of the damage evolution in the first stage.Figure 11
D-N/ N f curves (K1=0.04, 0.08, and 0.12).


#### 3.2.3. Sensitivity Analysis of ParameterA2

Set the values for parameterA2 as 0.2, 0.4, and 0.6, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 12. As shown in the figure, as parameter A2 changes, the first stage of damage evolution is less influenced, but the second and third stages are greatly influenced. The larger parameter A2 is, the faster the rate of damage evolution is, and the more serious the degree of the damage of material becomes in the second and third stages. Parameter A2 is a parameter that influences the rate and the degree of the damage evolution in the second and third stages.Figure 12
D-N/ N f curves (A2=0.2, 0.4, and 0.6).


#### 3.2.4. Sensitivity Analysis of ParameterK2

Set the values for parameterK2 as 0.1, 0.6, and 1.1, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 13. As shown in the figure, as parameter K2 changes, the first stage of damage evolution is less influenced, but the second and third stages are greatly influenced. The larger parameter K2 is, the slower the rate of damage evolution is, and the less serious the degree of the damage of material becomes in the second and third stages. Parameter K2 is a parameter that influences the rate and degree of the damage evolution in the second and third stages.Figure 13
D-N/Nf curves (K2=0.1, 0.6, and 1.1).


#### 3.2.5. Result of the Sensitivity Analysis

ParameterA1 is a main parameter that influences the degree of the damage of material. Parameter K1 is a parameter that influences the speed of the damage evolution in the first stage. Parameters A2 and K2 are parameters that influence the speed and degree of the damage evolution in the second and third stages. A conclusion can be drawn that each parameter has different effects on the three stages of damage evolution.


## 3.1. Construction of Damage Evolution Equation

The stress-strain relation of damaged specimens during fatigue test is described on the basis of strain equivalence and damage mechanics, i.e.,(3)E=E01-D(4)D=1-EE0where E0 is the initial modulus,E is the modulus during fatigue loading, and D is the damage factor.TheE-N curve is shown in Figure 7. As shown in the figure, the modulus decreases with the increase of the number of loading cycles. TheE/E0-N/Nf curve was obtained by dividing the abscissa and the ordinate ofE-N curve by E0 and Nf, respectively. TheE/E0-N/Nf curve is shown in Figure 8. It can be seen that the process can be divided into three stages. In the first stage, the modulus decreases sharply and fatigue cracks emerge. In the second stage, the modulus decreases stably and fatigue cracks expand. In the third stage, the modulus decreases rapidly and the beam fractures finally. The cumulative result of the expansion was the fracture of the beam specimen. The following equation was used to fit theE/E0-N/Nf curve:(5)EE0=1-∑i=12Ai1-e-N/Nf/Kiwhere Ai and Ki are model parameters,N is the number of loading cycles, and Nf is the fatigue life.Figure 7
E-Ncurve.Figure 8
E/ E 0 -N/ N f curve.The model parameters are obtained by fitting the experimental data. As listed in Table5, the squared correlation coefficients (R2) are over 0.99. As can be seen from Table 5, the fitting parameters are related to the experimental factors, such as the temperature, the loading rate, and the stress-strength ratio. A quadratic polynomial is used to fit the experimental data. The regression results are as follows.Table 5
Fitting parameters.
Temperature(T) / °C Loading rate(V) / mm/min stress-strength ratio (SSR) A 1 K 1 A 2 K 2 5 5 0.3 0.495 0.034 0.538 0.997 0.4 0.465 0.072 0.633 1.089 0.5 0.350 0.094 1.332 1.837 10 0.3 0.471 0.041 0.511 0.837 0.4 0.442 0.087 0.601 1.304 0.5 0.333 0.113 1.271 3.137 20 0.3 0.434 0.054 1.184 2.588 0.4 0.308 0.068 1.233 2.360 0.5 0.192 0.122 1.134 1.878 15 5 0.3 0.422 0.034 0.535 0.502 0.4 0.350 0.053 0.648 0.729 0.5 0.302 0.096 1.253 2.546 10 0.3 0.419 0.044 0.520 0.710 0.4 0.332 0.063 0.615 0.875 0.5 0.287 0.115 1.190 3.055 20 0.3 0.308 0.043 0.605 0.888 0.4 0.320 0.080 1.221 2.200 0.5 0.109 0.112 1.276 1.677 25 5 0.3 0.554 0.013 0.269 0.117 0.4 0.499 0.017 0.307 0.228 0.5 0.485 0.025 0.401 0.598 10 0.3 0.527 0.016 0.283 0.161 0.4 0.486 0.022 0.327 0.298 0.5 0.477 0.032 0.421 0.546 20 0.3 0.435 0.032 0.393 0.365 0.4 0.329 0.057 0.482 0.518 0.5 0.315 0.081 0.635 0.953A 1 = 0.685−0.0282T+0.00106 T2+0.00474 × 1012V−5.364 × 10−4  V2−0.0157 SSR−0.825 SSR2,R2 = 0.874.K 1 = −0.00976+0.00286T−1.673 × 10−4  T2+0.00292 × 1011V−5.528 × 10−5  V2−0.0201 SSR+0.35628 SSR2,R2 = 0.82995.A 2 = 1.330+0.0355T−0.00210 T2−0.0347V+0.00205V2−4.924 SSR+8.983 SSR2,R2 = 0.757.K 2 = 2.628+0.0413T−0.00364 T2+0.0735V−0.00152 V2−13.579 SSR+23.267 SSR2, R2 = 0.614.The damage model function is expressed as(6)D=1-EiE0=∑i=12Ai1-e-N/Nf/KiDamage evolution equation is obtained by calculating the derivative of (6), and the damage evolution equation is expressed as(7)dDdN=∑i=12Aie-N/Nf/Ki1KiNfTheD-N/Nf curve is shown in Figure 9. As presented in the figure, the damage increases nonlinearly during the fatigue tests, and the process can be divided into three stages, which is similar to the process of degradation of modulus. In the first stage, the damage lasts a short time but increases rapidly, and, in the end of this stage, it reaches a large value. In the second stage, the damage lasts a long time and increases stably. In the third stage, the damage increases sharply and the beam fractures finally [9].Figure 9
D-N/ N f curve.


## 3.2. Sensitivity Analysis of Damage Model Parameters

As shown in Table5, the value ranges of A1, K1, A2, and K2 are 0.110-0.554, 0.013-0.122, 0.269-1.134, and 0.118-1.878, respectively. Three data sets are taken within the range of each parameter and are listed in Table 6. The influences of the four parameters (A1, K1, A2, K2) on the three stages of damage evolution are analyzed.Table 6
The parameter values.
Parameter A 1 K 1 A 2 K 2 Value 0.1,0.3,0.5 0.04,0.08,0.12 0.2,0.4,0.6 0.1,0.6,1.1


### 3.2.1. Analysis of ParameterA1

Set the values for parameterA1 as 0.1, 0.3, and 0.5, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 10. As shown in the figure, the three stages of damage evolution are greatly influenced by parameter A1. The larger parameter A1 is, the more serious the degree of the damage of material becomes. Parameter A1 is a main parameter that influences the degree of the damage of material.Figure 10
D-N/ N f curves (A1=0.1, 0.3, and 0.5).


### 3.2.2. Sensitivity Analysis of ParameterK1

Set the values for parameterK1 as 0.04, 0.08, and 0.12, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 11. As shown in the figure, as the parameter K1 changes, the second and third stages of damage evolution are less influenced, but the first stage is greatly influenced. The larger parameter K1 is, the longer the duration of the first stage lasts. Parameter K1 is a parameter that influences the rate of the damage evolution in the first stage.Figure 11
D-N/ N f curves (K1=0.04, 0.08, and 0.12).


### 3.2.3. Sensitivity Analysis of ParameterA2

Set the values for parameterA2 as 0.2, 0.4, and 0.6, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 12. As shown in the figure, as parameter A2 changes, the first stage of damage evolution is less influenced, but the second and third stages are greatly influenced. The larger parameter A2 is, the faster the rate of damage evolution is, and the more serious the degree of the damage of material becomes in the second and third stages. Parameter A2 is a parameter that influences the rate and the degree of the damage evolution in the second and third stages.Figure 12
D-N/ N f curves (A2=0.2, 0.4, and 0.6).


### 3.2.4. Sensitivity Analysis of ParameterK2

Set the values for parameterK2 as 0.1, 0.6, and 1.1, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 13. As shown in the figure, as parameter K2 changes, the first stage of damage evolution is less influenced, but the second and third stages are greatly influenced. The larger parameter K2 is, the slower the rate of damage evolution is, and the less serious the degree of the damage of material becomes in the second and third stages. Parameter K2 is a parameter that influences the rate and degree of the damage evolution in the second and third stages.Figure 13
D-N/Nf curves (K2=0.1, 0.6, and 1.1).


### 3.2.5. Result of the Sensitivity Analysis

ParameterA1 is a main parameter that influences the degree of the damage of material. Parameter K1 is a parameter that influences the speed of the damage evolution in the first stage. Parameters A2 and K2 are parameters that influence the speed and degree of the damage evolution in the second and third stages. A conclusion can be drawn that each parameter has different effects on the three stages of damage evolution.


## 3.2.1. Analysis of ParameterA1

Set the values for parameterA1 as 0.1, 0.3, and 0.5, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 10. As shown in the figure, the three stages of damage evolution are greatly influenced by parameter A1. The larger parameter A1 is, the more serious the degree of the damage of material becomes. Parameter A1 is a main parameter that influences the degree of the damage of material.Figure 10
D-N/ N f curves (A1=0.1, 0.3, and 0.5).


## 3.2.2. Sensitivity Analysis of ParameterK1

Set the values for parameterK1 as 0.04, 0.08, and 0.12, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 11. As shown in the figure, as the parameter K1 changes, the second and third stages of damage evolution are less influenced, but the first stage is greatly influenced. The larger parameter K1 is, the longer the duration of the first stage lasts. Parameter K1 is a parameter that influences the rate of the damage evolution in the first stage.Figure 11
D-N/ N f curves (K1=0.04, 0.08, and 0.12).


## 3.2.3. Sensitivity Analysis of ParameterA2

Set the values for parameterA2 as 0.2, 0.4, and 0.6, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 12. As shown in the figure, as parameter A2 changes, the first stage of damage evolution is less influenced, but the second and third stages are greatly influenced. The larger parameter A2 is, the faster the rate of damage evolution is, and the more serious the degree of the damage of material becomes in the second and third stages. Parameter A2 is a parameter that influences the rate and the degree of the damage evolution in the second and third stages.Figure 12
D-N/ N f curves (A2=0.2, 0.4, and 0.6).


## 3.2.4. Sensitivity Analysis of ParameterK2

Set the values for parameterK2 as 0.1, 0.6, and 1.1, respectively, and other parameters keep invariant; theD-N/Nf curves are plotted in Figure 13. As shown in the figure, as parameter K2 changes, the first stage of damage evolution is less influenced, but the second and third stages are greatly influenced. The larger parameter K2 is, the slower the rate of damage evolution is, and the less serious the degree of the damage of material becomes in the second and third stages. Parameter K2 is a parameter that influences the rate and degree of the damage evolution in the second and third stages.Figure 13
D-N/Nf curves (K2=0.1, 0.6, and 1.1).


## 3.2.5. Result of the Sensitivity Analysis

ParameterA1 is a main parameter that influences the degree of the damage of material. Parameter K1 is a parameter that influences the speed of the damage evolution in the first stage. Parameters A2 and K2 are parameters that influence the speed and degree of the damage evolution in the second and third stages. A conclusion can be drawn that each parameter has different effects on the three stages of damage evolution.


## 4. Construction of Fatigue Damage Constitutive Equations

The stress-strain curves were obtained by calculating the fatigue test data. The stress-strain curves of the last five cyclic loadings were chosen in order to clearly draw the stress-strain curves, which are presented in Figure14. The residual strain increased with the increase of the number of cyclic loadings. Plastic deformation occurred when the damage was accumulated, and the plastic deformation would influence the damage accumulation. Plastic deformation and damage accumulation are interrelated [9, 21, 22].Figure 14
Stress-strain curves.Constitutive equation under cyclic loading consists of two parts: one for the damage accumulation stage and the other for the fatigue failure stage. The constitutive equations for the damage accumulation stage before the last cyclic loading are in the same form, and the stress-strain curves are presented in Figure15. As shown in the figure, the stress-strain curves of damage accumulation stage consist of four parts. The form of the constitutive equations of the last cyclic loading, which is called the damage failure stage, is different from the form for the damage accumulation stage. The stress-strain curves are presented in Figure 16. As shown in the figure, the stress-strain curves of damage accumulation stage consist of three parts.Figure 15
Stress-strain curves of damage accumulation.Figure 16
Stress-strain curves of fatigue destruction.


### 4.1. Construction of Constitutive Equations of the Damage Accumulation Stage

The elasticity-power hardening model can be used to describe the stress-strain relationship of the material [21] and is expressed as(8)σ=Eε,0≤σ≤σ0kεn,σ>σ0whereσis the stress,ε is the strain,Eis the elastic modulus, and k and n are material constants obtained by fitting the test data.The elasticity-power hardening model, a widely used mechanical model, can not only well describe the plastic deformation of the material, but also can well describe the elastic deformation of the material [21].Figure15 shows that constitutive relation of the loading stage consisted of two parts. The material in partAB is elastic, and the material in partBC is elastic-plastic. The asphalt mixture in the partAB has the characteristics of linear deformation; the asphalt mixture in partBC has the characteristic of nonlinear deformation. Therefore, the elasticity-power hardening model was used to describe the constitutive relation of damage accumulation stage in the loading stage. Constitutive relation of the unloading stage consisted of two parts. The asphalt mixture in the partCD has the characteristics of linear deformation. The asphalt mixture in the partDE has the characteristics of nonlinear deformation. The elasticity-power hardening model indicates that linear function can well describe the elastic deformation of the material, and the power function can well describe the elastic-plastic deformation of the material. Therefore, the elasticity-power hardening model was used to describe the constitutive relation of damage accumulation stage in the unloading stage. In this paper, the constitutive equations for the elastic and the elastic-plastic state are expressed by the linear function and the power function, respectively [9, 21, 23].The constitutive equations for the damage accumulation state are expressed as follows.Loading stage is(9)σ=E01-Dε∗=E01-∑i=12Ai1-e-N/Nf/Kiε∗,εA≤ε<εB1-Da1ε∗-a2a3=1-∑i=12Ai1-e-N/Nf/Kia1ε∗-a2a3,εB≤ε<εCUnloading stage is(10)σ=E01-Dε∗∗=E01-∑i=12Ai1-e-N/Nf/Kiε∗∗,εC≤ε<εD1-Da4ε∗∗∗+1a5=1-∑i=12Ai1-e-N/Nf/Kia4ε∗∗∗+1a5,εD≤ε≤εEwhere εA, εB, εC, εD, and εE are the stains at pointsA,B,C,D, and E, respectively, ε∗= ε-εA, ε∗∗=ε-εB, ε∗∗∗=ε-εC, a1, a2, a3, a4, and a5 are fitting parameters.The properties of asphalt mixture are stable after 30 cycles of loading. The constitutive equation of the 30th loading cycle of the fatigue test at the temperature of 25°C, the stress-strength ratio of 0.4, and the loading rate of 10 mm/min was obtained by fitting the data. The fitting results are listed in Table7, and the squared correlation coefficient (R2) is larger than 0.98. The fitting and the test curve are shown in Figure 17. As shown in the figure, the fatigue behavior of the asphalt mixture is well represented by the proposed constitutive equation. Because the characteristics of the constitutive relations before the last cyclic loading are the same, the proposed constitutive equations of the damage accumulation stage are typical.Table 7
Fitting results for damage accumulation.
Fitting parameters a 1 a 2 a 3 a 4 a 5 Value 146.20 -3.01×10−4 0.79 0.12 1658.19Figure 17
Comparison ofσ-εcurve for damage accumulation based on the test and fitting curve.


### 4.2. Construction of Constitutive Equations of the Damage Failure Stage

TheSidoroff damage model can be used to describe the stress-strain relationship of the material [5], and it is expressed as(11)σ=Eε,0≤ε≤εfEεfεfε3,ε>εfwhere εf is the strain at the peak stress.Figure16 shows that the constitutive relation of damage failure stage consists of three parts. The material in partHI is elastic, and the material in partIJ is elastic-plastic. The constitutive relation of the partHJ, similar to the constitutive relation of the partAC, is described with the elasticity-power hardening model. The material in partJK is elastic-plastic, and the asphalt mixture in the partJK has the characteristics of nonlinear deformation. The Sidoroff damage model assumes that damage occurs and cracks are generated when the material reaches its peak stress. The Sidoroff damage model can well describe the nonlinear deformation of the material, so it was used to describe the constitutive relation of the unloading part in the damage failure stage [7, 21, 23].The constitutive equations of the damage failure stage are expressed as follows:(12)σ=E01-Dε′=E01-∑i=12Ai1-e-N/Nf/Kiε′,εH≤ε<εI1-Db1ε′-b2b3=1-∑i=12Ai1-e-N/Nf/Kib1ε′-b2b3,εI≤ε<εJb4b5ε′-b63,εJ≤ε≤εKwhere εH, εI, εJ, and εK are the strains at pointsH,I,J, and K, respectively, ε′=ε-εH, εJ′=εJ-εH, and b1, b2, b3, b4, b5, and b6 are fitting parameters.The constitutive equation of the fatigue test at the temperature of 25°C, the stress-strength ratio of 0.3, and the loading rate of 10 mm/min was obtained by fitting the data. The fitting results are listed in Table8, and the squared correlation coefficient (R2) is larger than 0.98. The fitting and the test curve are shown in Figure 18. As shown in the figure, the fatigue performances of the asphalt mixture are well represented by the proposed constitutive equation.Table 8
Fitting results for fatigue destruction.
Fitting parameters b 1 b 2 b 3 b 4 b 5 b 6 Value 13.50 3.09×10−4 0.48 -0.13 -0.45 -0.023Figure 18
Comparison ofσ-ε curve for fatigue destruction based on the test and the fitting curve.


## 4.1. Construction of Constitutive Equations of the Damage Accumulation Stage

The elasticity-power hardening model can be used to describe the stress-strain relationship of the material [21] and is expressed as(8)σ=Eε,0≤σ≤σ0kεn,σ>σ0whereσis the stress,ε is the strain,Eis the elastic modulus, and k and n are material constants obtained by fitting the test data.The elasticity-power hardening model, a widely used mechanical model, can not only well describe the plastic deformation of the material, but also can well describe the elastic deformation of the material [21].Figure15 shows that constitutive relation of the loading stage consisted of two parts. The material in partAB is elastic, and the material in partBC is elastic-plastic. The asphalt mixture in the partAB has the characteristics of linear deformation; the asphalt mixture in partBC has the characteristic of nonlinear deformation. Therefore, the elasticity-power hardening model was used to describe the constitutive relation of damage accumulation stage in the loading stage. Constitutive relation of the unloading stage consisted of two parts. The asphalt mixture in the partCD has the characteristics of linear deformation. The asphalt mixture in the partDE has the characteristics of nonlinear deformation. The elasticity-power hardening model indicates that linear function can well describe the elastic deformation of the material, and the power function can well describe the elastic-plastic deformation of the material. Therefore, the elasticity-power hardening model was used to describe the constitutive relation of damage accumulation stage in the unloading stage. In this paper, the constitutive equations for the elastic and the elastic-plastic state are expressed by the linear function and the power function, respectively [9, 21, 23].The constitutive equations for the damage accumulation state are expressed as follows.Loading stage is(9)σ=E01-Dε∗=E01-∑i=12Ai1-e-N/Nf/Kiε∗,εA≤ε<εB1-Da1ε∗-a2a3=1-∑i=12Ai1-e-N/Nf/Kia1ε∗-a2a3,εB≤ε<εCUnloading stage is(10)σ=E01-Dε∗∗=E01-∑i=12Ai1-e-N/Nf/Kiε∗∗,εC≤ε<εD1-Da4ε∗∗∗+1a5=1-∑i=12Ai1-e-N/Nf/Kia4ε∗∗∗+1a5,εD≤ε≤εEwhere εA, εB, εC, εD, and εE are the stains at pointsA,B,C,D, and E, respectively, ε∗= ε-εA, ε∗∗=ε-εB, ε∗∗∗=ε-εC, a1, a2, a3, a4, and a5 are fitting parameters.The properties of asphalt mixture are stable after 30 cycles of loading. The constitutive equation of the 30th loading cycle of the fatigue test at the temperature of 25°C, the stress-strength ratio of 0.4, and the loading rate of 10 mm/min was obtained by fitting the data. The fitting results are listed in Table7, and the squared correlation coefficient (R2) is larger than 0.98. The fitting and the test curve are shown in Figure 17. As shown in the figure, the fatigue behavior of the asphalt mixture is well represented by the proposed constitutive equation. Because the characteristics of the constitutive relations before the last cyclic loading are the same, the proposed constitutive equations of the damage accumulation stage are typical.Table 7
Fitting results for damage accumulation.
Fitting parameters a 1 a 2 a 3 a 4 a 5 Value 146.20 -3.01×10−4 0.79 0.12 1658.19Figure 17
Comparison ofσ-εcurve for damage accumulation based on the test and fitting curve.


## 4.2. Construction of Constitutive Equations of the Damage Failure Stage

TheSidoroff damage model can be used to describe the stress-strain relationship of the material [5], and it is expressed as(11)σ=Eε,0≤ε≤εfEεfεfε3,ε>εfwhere εf is the strain at the peak stress.Figure16 shows that the constitutive relation of damage failure stage consists of three parts. The material in partHI is elastic, and the material in partIJ is elastic-plastic. The constitutive relation of the partHJ, similar to the constitutive relation of the partAC, is described with the elasticity-power hardening model. The material in partJK is elastic-plastic, and the asphalt mixture in the partJK has the characteristics of nonlinear deformation. The Sidoroff damage model assumes that damage occurs and cracks are generated when the material reaches its peak stress. The Sidoroff damage model can well describe the nonlinear deformation of the material, so it was used to describe the constitutive relation of the unloading part in the damage failure stage [7, 21, 23].The constitutive equations of the damage failure stage are expressed as follows:(12)σ=E01-Dε′=E01-∑i=12Ai1-e-N/Nf/Kiε′,εH≤ε<εI1-Db1ε′-b2b3=1-∑i=12Ai1-e-N/Nf/Kib1ε′-b2b3,εI≤ε<εJb4b5ε′-b63,εJ≤ε≤εKwhere εH, εI, εJ, and εK are the strains at pointsH,I,J, and K, respectively, ε′=ε-εH, εJ′=εJ-εH, and b1, b2, b3, b4, b5, and b6 are fitting parameters.The constitutive equation of the fatigue test at the temperature of 25°C, the stress-strength ratio of 0.3, and the loading rate of 10 mm/min was obtained by fitting the data. The fitting results are listed in Table8, and the squared correlation coefficient (R2) is larger than 0.98. The fitting and the test curve are shown in Figure 18. As shown in the figure, the fatigue performances of the asphalt mixture are well represented by the proposed constitutive equation.Table 8
Fitting results for fatigue destruction.
Fitting parameters b 1 b 2 b 3 b 4 b 5 b 6 Value 13.50 3.09×10−4 0.48 -0.13 -0.45 -0.023Figure 18
Comparison ofσ-ε curve for fatigue destruction based on the test and the fitting curve.


## 5. Conclusions

Laboratory investigations of the fatigue properties of asphalt mixture were conducted by three-point bending fatigue tests. Test results were analyzed to build fatigue damage constitutive equations. With the study above, the following conclusions can be drawn.(1) Fatigue life decreases with the increase of stress-strength ratio. The linear function can well describe the relationship between logarithmic fatigue life and stress-strength ratio. The healing of the asphalt mixtures improves as temperature increases, and the corresponding fatigue life increases. The rate of the damage accumulation decreases with the increase of loading rate, and the corresponding fatigue life decreases. The above conclusions may provide a theoretical base for the establishing of fatigue life equation for asphalt mixture.(2) The coupled multifactor (stress-strength ratio, temperature, and loading rate) fatigue life equation has been established, which can well describe the relationship of the logarithmic fatigue life to the stress-strength ratio, the temperature, and the loading rate and predict the fatigue life of the asphalt mixture and provide a theoretical base for the establishing of fatigue damage model.(3) Both the damage model and the damage evolution equation have been established based on theE-N curve, which indicate that fatigue damage evolution is nonlinear and consists of three stages. The sensitivity analysis of damage model parameters indicates that each parameter has different effects on the three stages of damage evolution. The damage model can be used to build the constitutive equations coupling fatigue damage.(4) The fatigue damage constitutive equations were built based on theσ-εcurves, which consist of two parts: the damage accumulation stage and the fatigue failure stage. The constitutive equation can well characterize the fatigue damage performances of the asphalt mixtures under cyclic loading.



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*Source: 3489082-2018-07-25.xml*